Proof. The additive idempotents of the trace cone are exactly the ideal weights \(\tau_I\). Indeed \(\tau+\tau=\tau\) forces every value to be zero or infinity, and the zero set is a closed ideal. Moreover \[I\subseteq J\quad\Longleftrightarrow\quad \tau_I+\tau_J=\tau_I.\] Thus \(F\) determines an order isomorphism of ideal lattices. If \(F(\tau_I)=\tau_J\), then \[[a]\in\mathop{\mathrm{Cu}}(J) \quad\Longleftrightarrow\quad d_{F\tau_I}(a)=0 \quad\Longleftrightarrow\quad d_{\tau_I}(G[a])=0 \quad\Longleftrightarrow\quad G[a]\in\mathop{\mathrm{Cu}}(I).\] This proves compatibility with \(G\).

The topology recovers the two other ideals through \[ \lim_{r\downarrow0}r\tau=\tau_{\mathop{\mathrm{Fin}}(\tau)}, \qquad \lim_{r\uparrow\infty}r\tau=\tau_{\mathop{\mathrm{Ker}}(\tau)}. \tag{10}\] These are (Elliott et al. 2011, Equation (3.4)). They can also be checked directly in (2): all positive cuts of an element of \(\mathop{\mathrm{Fin}}(\tau)\) have finite trace by (9), whereas an element outside that ideal has infinite trace; for the second limit use whether its trace is zero or positive. Continuity and positive homogeneity of \(F\) therefore preserve both limits.

An ideal-lattice isomorphism preserves prime ideals and their hull–kernel topology. In a separable \(C^*\)-algebra prime ideals are primitive (Dixmier 1960, Corollary 1), which gives the asserted homeomorphism. Finally, the zero algebra is distinguished by the cone: every nonzero algebra has the nonzero weight \(\tau_{\{0\}}\). We may henceforth assume that \(P,Q\ne0\). ◻

Proof. Choose a norm-dense sequence \((a_j)\) of positive contractions in \(Q\). The supports of \((a_j-r)_+\), for positive rational \(r\), form a countable open basis, after taking finite unions. For example, to find such a support inside \(U\) and containing \(x\in U\), choose \(a\in Q(U)_+\) nonzero in the quotient at \(x\), and approximate it sufficiently closely by some \(a_j\). A rational cut larger than the approximation error belongs to \(Q(U)\), and a cut smaller than the quotient norm remains nonzero at \(x\). The support of \((a_j-r)_+\) is compactly contained, in the ideal lattice, in the support of \((a_j-r/2)_+\). Indeed, if a directed union of ideals contains the latter element, norm approximation puts the more deeply cut element in one member of that union. Choosing two rational cut levels inside the preceding construction gives a basis open compactly contained in the prescribed \(U\).

Let \(h\) be strictly positive in \(Q(U)\). The ideals generated by \((h-1/k)_+\) increase with dense union \(Q(U)\). Compact containment gives \(k\) such that \(Q(V)\subseteq\mathop{\mathrm{Ideal}}_Q((h-1/k)_+)\); take this cut as \(b_{V,U}\). If \(c\) satisfies (11) inside \(Q(V)\), then \([c]\ll[z]\) for some \(z\in Q(V)_+\). The Cuntz-semigroup ideal generated by \([b_{V,U}]\) consists of the elements below \(\sup_N N[b_{V,U}]\). Thus \([z]\leq\sup_N N[b_{V,U}]\), and compact containment proves (12). ◻

Proof. Use (8) once to exhibit \(Q=C\otimes\mathcal Q\otimes\mathcal K\). For a matrix amplification choose a Hilbert-space unitary \(U_k:\mathbb C^k\otimes\ell^2\to\ell^2\) and fold by \(\mathop{\mathrm{id}}_{C\otimes\mathcal Q}\otimes\mathop{\mathrm{Ad}}(U_k)\). This preserves the unnormalized matrix extension of every trace. For a finite set \(E\subset Q\), choose a finite-rank projection \(e\) in the last factor such that \(\|a-eae\|\) is small for \(a\in E\). A unitary of \(B(\ell^2)\) can send the folded first-corner copy of \(e\ell^2\) back to \(e\ell^2\), agreeing with the specified identification there. Conjugating the folding by this unitary fixes the compressed finite tests. The two compression errors give the desired approximation on \(E\).

Write \(\lambda_i=l_i/L\). In an additional UHF factor choose orthogonal projections \(e_i\) with normalized traces \(l_i/L\) and sum one. After folding the internal matrices, form \[h(s)=\sum_i h_i(s)\otimes e_i.\] Orthogonality makes its norm the maximum of the block norms, and functional calculus is blockwise for functions vanishing at zero. This proves complete positivity, contractivity, and (14)–(15) before the final UHF identification.

Choose that identification by interleaving tensor factors of a fixed infinite tensor-product realization of \(\mathcal Q\). Finite tensor permutation unitaries identify its first-factor embedding with the identity to any prescribed accuracy on finite tests. For a finite matrix direction the trace identity on matrix units gives \(\widetilde\tau(a\otimes e)=\operatorname{tr}(e)\tau(a)\), where \(\tau(a)=\widetilde\tau(a\otimes1)\) and \(\operatorname{tr}\) is normalized. Increasing finite matrix algebras and positive cuts give this factorization for the UHF direction, including extended weights. Thus the identification preserves the displayed formulas even when \(Q\) is nonunital. Multiplier conjugation also preserves every extended trace, by the trace identity applied to \(a^{1/2}u\). The choices are made separately at each coordinate; finite sets, matrix sizes, and accuracies may therefore vary with that coordinate. ◻

Proof. Pull \(\tau_n\) back to \(\ell^\infty(Q)\) by its \(n\)-th coordinate homomorphism. The compact Hausdorff trace cone has a unique limit along \(\omega\), say \(\widetilde\rho\). For every positive \(x\) in the product, the defining topology gives \[\lim_\omega\tau_n((x_n-\varepsilon)_+) \leq\widetilde\rho(x),\qquad \widetilde\rho((x-\varepsilon)_+) \leq\lim_\omega\tau_n((x_n-\varepsilon)_+).\] Taking suprema over \(\varepsilon\) proves (16) on the product. Every fixed cut of an element of \(c_0(Q)_+\) is eventually zero, so \(\widetilde\rho\) vanishes on that ideal and descends to a lower-semicontinuous trace \(\rho\) on the quotient. This also proves independence of the representative. The same two inequalities for constant sequences give \(\rho|_Q=\sigma\).

For completeness, rescale the representatives so that \(0\leq x_n\leq1\), and let the right side of (17) be \(L\). For every \(k\) and \(t>0\), \[\tau_n((x_n^{1/k}-t)_+) \leq d_{\tau_n}((x_n-t^k)_+).\] Regularization and then the supremum over \(k\) show \(d_\rho(x)\leq L\). Conversely, choose a continuous function \(g_\varepsilon\) with values in \([0,1]\), zero on \([0,\varepsilon/2]\), and one on \([\varepsilon,1]\). For \(0<t<1\), \[\tau_n((g_\varepsilon(x_n)-t)_+) \geq(1-t)d_{\tau_n}((x_n-\varepsilon)_+).\] The support of \(g_\varepsilon(x)\) is below the support of \(x\). Thus \(d_\rho(x)\geq\rho(g_\varepsilon(x))\), and regularization followed by \(t\downarrow0\) gives \(d_\rho(x)\geq\lim_\omega d_{\tau_n}((x_n-\varepsilon)_+)\). Taking the supremum over \(\varepsilon\) proves the reverse inequality. ◻

Proof. We first extend the rank identities to all elementary tensors, then identify the ideal where the weights are densely finite, and finally compare their bounded restrictions inside that ideal.

Extension of the tests. Suppose that positive contractions \(f_j,a_i\) approximate \(f,a\) within \(\eta\). The perturbation cut inequalities and their tensor products give \[(f-3\eta)_+\otimes(a-3\eta)_+ \precsim(f_j-2\eta)_+\otimes(a_i-2\eta)_+ \precsim f\otimes a.\] The middle tensors belong to the test collection when \(\eta\) is rational; their enlarged versions give the same comparison with any fixed positive cuts of \(f\) and \(a\) on the left. Ranks preserve increasing suprema of such cuts. Applying these comparisons to both \(\theta\) and \(m\otimes\nu\), and letting \(\eta\downarrow0\), proves (18) for every elementary tensor with compact function support. Exhaustion gives the remaining elementary tensors.

Identification of the finite ideal. Set \(I=\mathop{\mathrm{Fin}}(\nu)\). The finite ideal of \(\theta\) is exactly \(C_0(Y)\otimes I\). For one inclusion, a compactly supported function and a compactly Cuntz supported positive element of \(I\) give a tensor of finite \(\theta\)-rank. Their ideals generate \(C_0(Y)\otimes I\). For the other inclusion, let an open rectangle \(V\times U\) be contained in the support of \(\mathop{\mathrm{Fin}}(\theta)\). Take nonzero \(f\in C_c(V)_+\) and cut-supported \(a\in P(U)_+\). Choose \(f_0\in C_c(V)_+\) equal to one on \(\mathop{\mathrm{supp}}(f)\), and \(b\in P(U)_+\), \(\delta>0\), with \(a\precsim(b-\delta)_+\). Then \[f\otimes a\precsim(f_0\otimes b-\delta/2)_+.\] The larger tensor belongs to \(\mathop{\mathrm{Fin}}(\theta)\), so \(f\otimes a\) is compactly Cuntz supported there and has finite \(\theta\)-rank by (9). Formula (18) and \(0<m(f>0)<\infty\) imply \(d_\nu(a)<\infty\). Such elements generate \(P(U)\), so \(P(U)\subseteq I\). Rectangles exhaust the open support of any ideal, proving the claim.

Equality on the finite ideal. It remains to identify the densely finite traces on this common ideal; there is nothing to prove if \(I=0\). Choose a strictly positive \(h\in I\), put \(e_n=(h-1/n)_+\), and choose increasing positive compactly supported functions \(f_n\) whose open supports exhaust \(Y\). The hereditary algebras \[H_n=\mathop{\mathrm{Her}}(f_n\otimes e_n) =C_0(\{f_n>0\})\otimes\mathop{\mathrm{Her}}(e_n)\] increase with dense union in \(C_0(Y)\otimes I\). Both traces have finite support rank on \(H_n\), because \(m(f_n>0)d_\nu(e_n)<\infty\). Their restrictions are bounded and extend normally to \(H_n^{**}\). For a positive elementary tensor in \(H_n\), two-variable spectral integration and the already established elementary rank identities give \[\begin{align*} \theta(f\otimes a) &=\int_0^{\|f\|}\int_0^{\|a\|} d_\theta((f-t)_+\otimes(a-u)_+)\,du\,dt\\ &=\int_0^{\|f\|}\int_0^{\|a\|} m(f>t)d_\nu((a-u)_+)\,du\,dt =\Bigl(\int_Y f\,dm\Bigr)\nu(a). \end{align*}\] The first integrand is the value of the normal extension on the product of the two spectral projections. Tonelli’s theorem applies to these nonnegative functions. Boundedness and algebraic tensor density now give equality of the two traces on \(H_n\). Choose a positive contractive approximate unit \((v_n)\) of the common ideal with each \(v_n\) in the union of the \(H_j\). For every \(x\geq0\) in that ideal, \(v_nxv_n\) belongs to one of these hereditary algebras, and \[\theta(v_nxv_n) =\theta(x^{1/2}v_n^2x^{1/2})\leq\theta(x), \qquad v_nxv_n\longrightarrow x.\] Lower semicontinuity gives \(\theta(x)=\lim_n\theta(v_nxv_n)\), and the same argument holds for \(m\otimes\nu\). Equality on the localizations therefore gives equality on the common finite ideal. Both weights are infinity outside it. Thus they agree everywhere. ◻

Proof. Choose \(v\) with \(\|v^*(b-\delta)_+v-a\|<\eta\), and put \[g_\delta(t)= \begin{cases} \sqrt{(t-\delta)_+/t},&t>0,\\ 0,&t=0, \end{cases} \qquad w=g_\delta(b)v.\] Then \(w^*bw=v^*(b-\delta)_+v\), while \(g_\delta(b)^2\leq\delta^{-1}(b-\delta)_+\). This proves (20). ◻

Proof. Choose bounded positive representatives \(x_n,y_n\). If comparison fails, there is \(\varepsilon>0\) with \((x-2\varepsilon)_+\not\precsim y\). For every fixed \(\delta>0\), there must be arbitrarily late coordinates with \[ (x_n-\varepsilon)_+\not\precsim(y_n-\delta)_+. \tag{22}\] Otherwise 4, applied for each approximation accuracy, supplies bounded sequences of witnesses showing \((x-\varepsilon)_+\precsim y\), a contradiction.

Choose increasing indices \(n_k\) where (22) holds for \(\delta=1/k\). Pointwise order in \(\mathop{\mathrm{Cu}}(Q)\), from (7), gives a trace whose rank on the source is strictly larger than its rank on the target. The smaller rank is finite. If the larger rank is finite, divide the trace by that rank; if it is infinite, divide by one plus the smaller rank. Thus we may choose \(\tau_k\) with \[d_{\tau_k}((x_{n_k}-\varepsilon)_+)\geq1, \qquad d_{\tau_k}((y_{n_k}-1/k)_+)\leq1.\] Take a free ultrafilter on these indices and the corresponding regularized limit. Formula (17) gives \(d_\rho(x)\geq1\). For each fixed \(t>0\), the cut \((y_{n_k}-t)_+\) is eventually below \((y_{n_k}-1/k)_+\), so the same formula gives \(d_\rho(y)\leq1\). This contradicts (21). All comparisons used to select \(\tau_k\) took place in \(Q\). Matrix folding preserves those comparisons and ranks by 2. ◻

Proof. For a regularized limit \(\rho\), put \(\sigma=\rho|_Q\) and \(t=d_{F\sigma}(a)\). The ranks of \(c_r\) and \(p(s)\) are \(rt\) and \(m_0t\). Their ratio has the required strict scalar slack in the appropriate direction, so 5 proves (23). This reasoning includes \(t=0\) and \(t=\infty\); the separating normalization in that lemma forces a finite nonzero value whenever it is needed for a contradiction.

Apply the lower comparison to \(s'=(f-\eta)_+\otimes(a-\delta)_+\). Its support lies in the spectral region \(f\otimes a>\eta\delta\), so elementary functional calculus gives \(s'\precsim(f\otimes a-\kappa)_+\) whenever \(0<\kappa<\eta\delta\). Applying \(p\) proves (24).

For the upper comparison, choose \(m_0<r<r'\) and \(\delta>0\) with \(\|f\|\delta<\varepsilon/2\). The norm perturbation from \(f\otimes a\) to \(f\otimes(a-\delta)_+\) gives \[(p(f\otimes a)-\varepsilon)_+ \precsim p(f\otimes(a-\delta)_+) \precsim c, \qquad [c]=rG[(a-\delta)_+].\] Since positive scalar multiplication preserves compact containment, \[[c]\ll rG[a]\leq r'G[a]=[b].\] There is therefore \(\kappa>0\) with \([c]\leq[(b-\kappa)_+]\). This choice uses only the indicated constant classes, not \(p\).

Conversely, apply any regularized rank to the fixed-cut comparisons. Let \(\delta,\eta\downarrow0\) and let the rational lower and upper scalars approach \(m_0\). The resulting inequalities give (18) for the pullback trace on every compact test. Supremum of cuts handles zero or infinite ranks, and compact exhaustion handles infinite-measure supports. 3 now gives (19). ◻

Proof. For a nonzero elementary tensor of finite function support, the upper sandwiches put its image in the ideal generated by the corresponding constants. The lower trimmed sandwiches recover the ideals of all cuts of \(a\), hence the whole constant ideal \(Q(U_a)\). For general \(s\), the ideal generated by \(s\) in \(S\) is generated by positive elementary tensors whose rectangle supports lie in its open support. Their projected supports exhaust \(U_s\). Applying the elementary lower bounds gives \(\mathop{\mathrm{Ideal}}_D(Q(U_s))\subseteq\mathop{\mathrm{Ideal}}_D(p(s))\). Approximating \(s\) by finite sums of elementary tensors from \(S(U_s)\) and applying the upper bounds gives the reverse inclusion.

For the second assertion, start with the separable algebra generated by the stated ranges, constants, and extra elements. For every countable test, every rational cut and scalar in 3, and every model, adjoin the matrix entries of comparison witnesses at accuracies tending to zero. The resulting algebra \(D_0\) is separable, and all these comparisons hold in \(\mathop{\mathrm{Cu}}(D_0)\), not just in \(\mathop{\mathrm{Cu}}(D)\). For \(\gamma\in T(D_0)\), write \(\sigma=\gamma|_Q\). The rank of a constant representing \(rG[a]\) is \(r d_{F\sigma}(a)\). Applying \(d_\gamma\) to the retained sandwiches therefore gives the elementary rank formula for each \(\gamma\circ p_i\). The proof of the converse in 3 and 3 applies without requiring \(\gamma\) to be a regularized limit. This proves (27). ◻

Proof. We recall the cut estimates that will be used. For bounded operators in a \(C^*\)-algebra and every tracial rank, the following weaker forms of the spectral cut inequalities suffice: \[\begin{align*} d_\tau((|x+y|-\varepsilon)_+) &\leq d_\tau((|x|-\varepsilon/4)_+) +d_\tau((|y|-\varepsilon/4)_+), \tag{31}\\ d_\tau((|bx|-\varepsilon)_+),\ d_\tau((|xb|-\varepsilon)_+) &\leq d_\tau((|x|-\varepsilon/(2M))_+) \quad(\|b\|\leq M, M>0). \tag{32}\end{align*}\] Also \(d_\tau((|x^*|-t)_+)=d_\tau((|x|-t)_+)\), and \[ \|x-y\|<\varepsilon/4 \quad\Longrightarrow\quad d_\tau((|x|-\varepsilon)_+) \leq d_\tau((|y|-\varepsilon/2)_+). \tag{33}\] Here is an explicit cut reduction. Set \[x_t=x f_t(|x|),\qquad f_t(s)=\begin{cases}(s-t)_+/s,&s>0,\\0,&s=0.\end{cases}\] Then \(\|x-x_t\|\leq t\) and \(|x_t|=(|x|-t)_+\). Use the perturbation comparison \(\|z-w\|<\varepsilon\Rightarrow(|z|-\varepsilon)_+\precsim|w|\). For the sum take \(w=x_{\varepsilon/4}+y_{\varepsilon/4}\), and use \(|w|\precsim|x_{\varepsilon/4}|\oplus|y_{\varepsilon/4}|\). For a product take \(w=bx_{\varepsilon/(2M)}\), whose absolute value is Cuntz below \(|x_{\varepsilon/(2M)}|\). For (33) take \(w=y_{\varepsilon/2}\), so that \(\|x-w\|<3\varepsilon/4\). The identities for adjoints give the other product order. These comparisons follow from the positive cut perturbation lemma and \([v^*v]=[vv^*]\); applying the additive rank functional proves the displays for extended weights as well.

If \(x,y\in J(U)\), choose their controllers at tolerance \(\varepsilon/4\). A folded finite block sum of the controllers is compactly Cuntz supported in \(Q(U)\). Taking the maximum of the two decay sequences proves the bound for \(x+y\) by (31). Equation (32) proves closure under multiplication on either side by an arbitrary bounded sequence. The equality for adjoints proves self-adjointness. Equation (33) proves independence of the representative. For norm closure, first fix \(\varepsilon\), then choose one element of the approximating sequence within \(\varepsilon/8\) in quotient norm. Its representatives are eventually within \(\varepsilon/4\), so its controller at tolerance \(\varepsilon/2\) works for the desired output cut. Monotonicity in \(U\) follows directly from the definition.

To prove (29), fix a cut of \(|x|\), choose its controller \(c\precsim(b-\delta)_+\) in \(Q(U)\), and wait until \(r_n\leq1\). Pointwise order in \(\mathop{\mathrm{Cu}}(Q)\) gives \((|x_n|-\varepsilon)_+\precsim c\precsim(b-\delta)_+\). The bounded witnesses of 4 give comparison with the constant \(b\) in the quotient. Every positive cut of \(|x|\) is consequently in \(\mathop{\mathrm{Ideal}}_D(Q(U))\), and so is \(x\).

For vanishing, fix \(\varepsilon\) and the same type of controller. For each coordinate weight, \[ d_{\tau_n}(c) \leq\frac4\delta\, \tau_n((b-3\delta/4)_+). \tag{34}\] Indeed, on the spectral region \(b>\delta\), the function \((b-3\delta/4)_+\) is at least \(\delta/4\). Put \(h=(b-\delta/2)_+\). Dense finiteness on \(Q(U)\) and (9) give \(\sigma(h)<\infty\), while the trace-cone convergence inequalities give \[\lim_\omega\tau_n((b-3\delta/4)_+) =\lim_\omega\tau_n((h-\delta/4)_+) \leq\sigma(h).\] The ranks in (34) are therefore bounded on an \(\omega\)-large set. Multiplying by \(r_n\to0\) in (28) and using (17) yields \(d_\rho(|x|)=0\). Hence \(\rho\) vanishes on \(J(U)_+\). The proof uses only finite matrix sums and unnormalized ranks, so it also proves the matrix assertions, with foldings as in 2. ◻

Proof. The common controller and the coordinate choice. Enumerate all algebraic relations and positive rational cut tests. Reindex the rows so that row \(j\) includes the first \(j\) requirements and their quotient norm errors tend to zero. Include the bounds on each prescribed error and comparison element among these requirements. Set any as yet unspecified early-row errors to zero. In the following construction let \(l\) enumerate pairs consisting of a prescribed local error and a positive rational tolerance \(\varepsilon_l\); repeat that error’s opens and representatives with each tolerance. Choose once and for all the controller \(b_l=b_{V_l,U_l}\) from 1.

In row \(j\), the \(l\)-th error has, after folding, a bound \[d_\tau((|e^{(j)}_{l,n}|-\varepsilon_l)_+) \leq r^{(j)}_{l,n}d_\tau(c^{(j)}_l), \qquad r^{(j)}_{l,n}\longrightarrow0,\] uniformly for \(\tau\in T(Q)\) eventually in \(n\). The controller \(c^{(j)}_l\) is compactly Cuntz supported in \(Q(V_l)\). There is therefore a finite integer \(N_{j,l}\) with \[[c^{(j)}_l]\leq N_{j,l}[b_l].\] This integer absorbs internal matrix sizes, repeated blocks, and any other finite multiplicity at that stage. No bound on it as \(j\to\infty\) is required.

Fix a sequence \(\eta_j\downarrow0\). After the controllers and the integers \(N_{j,l}\) have been chosen for \(l\leq j\), choose strictly increasing coordinate thresholds \(n_j\geq j\) so large that \[ N_{j,l}r^{(j)}_{l,n}\leq\eta_j \qquad(n\geq n_j,\ l\leq j). \tag{35}\] At the same threshold impose the first \(j\) norm and algebraic requirements, with an additional coordinate error at most \(\eta_j\). This is possible because a quotient norm is the limsup of the coordinate norms. Choose representatives for the countable values; linearity of those representatives is unnecessary, since the linearity relations themselves are among the tests.

For \(n_j\leq n<n_{j+1}\), put \(j(n)=j\) and use row \(j\). The diagonal norm bounds make each prescribed value a bounded sequence. For any fixed local test \(l\), (35) then gives, eventually, \[d_\tau((|e_{l,n}|-\varepsilon_l)_+) \leq\eta_{j(n)}d_\tau(b_l), \qquad j(n)\longrightarrow\infty.\] The controller on the right is fixed, so this is precisely the required \(J(U_l)\) bound. Enumerating all rational cut tolerances proves membership in that ideal. The diagonal norm requirements give \(\|L(a)\|\leq\|a\|\), and all algebraic errors vanish. Thus the quotient values extend uniquely to a contractive homomorphism on \(S\).

Extension of local errors. For an arbitrary open \(U\), the ideals \(S(V)\), with basis opens \(V\Subset U'\subseteq U\), have dense union in \(S(U)\). Indeed they generate \(S(U)\) as a closed sum, and finite sums already lie in one member: replace finitely many pairs \(V_i\Subset U'_i\) by their finite unions, using the basis from 1. Thus the union of the scheduled localized dense sets is dense in \(S(U)\). Each retained error on those sets lies in \(J(U')\subseteq J(U)\). Closedness of \(J(U)\) and the assumed continuity extend this containment to every localized input. For errors with several variables, approximate each variable on its scheduled dense set, using the global lists for unrestricted variables. The continuity used here is either uniform equicontinuity of the coordinate errors or continuity of their expression in the completed quotient maps. Continuity of each row separately would not suffice.

Comparison witnesses with a fixed target cut. Suppose a row comparison has uniformly bounded source \(a^{(j)}\) and uniformly bounded \(b^{(j)}\), with target \((b^{(j)}-\delta)_+\) and \(\delta>0\) independent of \(j\). Include both source and target norms in the coordinate requirements, so their diagonals belong to the sequence algebra. For a chosen positive source cut and comparison accuracy, use 4 with uncut target \((b^{(j)}-\delta')_+\) and cut size \(\delta-\delta'\). Its squared norm is bounded by the source norm plus the accuracy, divided by \(\delta-\delta'\). Lift the witness to a bounded coordinate sequence with arbitrarily small additional norm and equation errors, and put its finite-test equation among the requirements imposed at \(n_j\). This retained target slack gives bounded witnesses on the diagonal. Enumerating source cuts and accuracies proves the desired Cuntz comparisons there. Thus the order of choices is: fixed tests and target cuts, row controllers and witnesses, comparison multiplicities, and finally the coordinate thresholds.

For rows of models, the lower bounds (24) use target thresholds \(\kappa<\eta\delta\) determined by fixed trimmed input factors. The upper bounds (25) use cuts of fixed constant representatives, determined by the scalar slack and the fixed source tolerance. Both kinds of target threshold are independent of the row. They can therefore be retained by the preceding procedure. The converse part of 3 identifies the diagonal model data. On increasing intervals the same argument uses the common compact tests and their unchanged measure; no condition is imposed on the moving endpoints. Restriction to any subsequence retains all fixed-cut comparisons, which proves the final assertion. ◻

Proof of 3(i). Take c.p.c. lifts \(\chi_n:S\to Q\) of \(\chi\), using nuclearity and the Choi–Effros lifting theorem (Choi and Effros 1976, Theorem 3.10). The constant-ideal assertion of 1 gives \[\mathop{\mathrm{Ideal}}_D(\chi(x_l))=\mathop{\mathrm{Ideal}}_D(Q(U_{x_l})).\] Compact support of the controller in (41), with an extra cut before passing to coordinates, consequently gives an integer \(M_q\) and \(d_q>0\) such that \[ d_\tau(c_q) \leq M_q d_\tau((\chi_n(x_l)-d_q)_+) \quad\text{eventually, for every }\tau. \tag{43}\] Here is the cut detail. Write \([c_q]\leq[(z-\gamma)_+]\), with \(z\in Q(U_{x_l})_+\). The constant \((z-\gamma/2)_+\) lies in the ideal generated by \(\chi(x_l)\). Its suitable smaller cut is therefore below finitely many copies of a positive cut of \(\chi(x_l)\). Leave a further cut between \((z-\gamma)_+\) and that source before lifting the comparison. The source perturbation lemma then gives (43) at every late coordinate. Both its multiplicity and target tolerance depend only on the request, not on the coordinate.

For an upper test, choose a strictly positive \(h_i\in Q(V_i)\). The constant-ideal assertion gives \(\chi(a_i)\in\mathop{\mathrm{Ideal}}_D(Q(V_i))=\mathop{\mathrm{Ideal}}_D(h_i)\). Compact containment between two positive cuts of \(\chi(a_i)\) therefore gives a finite integer \(N_i\) and \(\kappa_i>0\) with \[[(\chi(a_i)-\varepsilon_i/2)_+]\leq N_i[(h_i-\kappa_i)_+].\] Let \(c_i\in Q(V_i)_+\) be the folded sum of these \(N_i\) cuts. It is compactly Cuntz supported in \(Q(V_i)\). The common-controller lemma then gives \([c_i]\leq L_i[b_i]\) for a finite integer \(L_i\). Lift a comparison for the less deeply cut source with error smaller than the remaining source slack. The positive perturbation lemma gives \[ d_\tau((\chi_n(a_i)-\varepsilon_i)_+) \leq L_i d_\tau(b_i) \quad\text{eventually, for every }\tau. \tag{44}\] Thus both the controller \(b_i\) and its multiplicity are fixed before the UHF fractions are selected.

At stage \(j\), choose a positive rational \(t_j\) with \[t_j\leq j^{-1}, \qquad t_jL_i\leq j^{-1}\quad(i\leq j).\] Only after choosing \(t_j\), choose an increasing sequence of coordinate thresholds \(N_j\) so that all the first \(j\) coordinate comparisons hold and \[ M_qr_q(n)\leq t_j \qquad(q\leq j, n\geq N_j). \tag{45}\] For \(N_j\leq n<N_{j+1}\), let \(p_n\) be a projection of normalized trace \(t_j\) in an extra UHF factor and use \[\theta_n(a)=\chi_n(a)\otimes p_n.\] After the fixed trace-preserving identifications these are c.p.c. maps into \(Q\). Their multiplicative defects tend to zero because those of \(\chi_n\) do, so they define a homomorphism \(\theta:S\to D\). Equation (44) gives (40) with \(s_{i,n}\leq j^{-1}\) on the \(j\)-th stage. Moreover, \[\begin{align*} d_\tau(A_{q,n}) &\leq r_q(n)d_\tau(c_q)\\ &\leq M_qr_q(n)d_\tau((\chi_n(x_l)-d_q)_+) \leq d_\tau((\theta_n(x_l)-d_q)_+). \tag{46}\end{align*}\] All inequalities use extended nonnegative arithmetic; the scalar factors are strictly positive. Pointwise order in \(\mathop{\mathrm{Cu}}(Q)\) gives the coordinate comparison. Apply (42) to the target \((\theta_n(x_l)-d_q/2)_+\), cut further at \(d_q/2\). Bounded witnesses now pass to \(D\), proving (38) for the enumerated tolerances, and hence for every tolerance.

It remains to check (37). If \(y\in S(U_x)_+\), then \(\chi(y)\in\mathop{\mathrm{Ideal}}_D(\chi(x))\). For every source tolerance, a slightly larger source cut is below finitely many copies of a fixed cut of \(\chi(x)\). Passing to coordinates with slack gives this rank inequality eventually. Tensoring both sides with the same \(p_n\) multiplies both ranks by its positive trace, so the inequality remains true. Pointwise coordinate order and (42) give \((\theta(y)-\varepsilon)_+\in\mathop{\mathrm{Ideal}}_D(\theta(x))\). Letting \(\varepsilon\downarrow0\), then using linearity and closure, proves (37). Finally \(Nt_j\leq1\) eventually for every fixed \(N\), so \(N\) equal rational subprojections fit in the original UHF unit. They give the stated copies inside the spare model. ◻

Proof. We first choose the lower threshold, once for each request. Write \[[c_q]\leq[(z_q-\gamma_q)_+], \qquad z_q\in Q(U_{x_l})_+,\quad\gamma_q>0.\] For \(z\in X\), denote by \((x_l)_z\) the image in \(C_0((0,1])\otimes(P/P_z)\), where \(P_z\) is the primitive ideal labeled by \(z\). The sets \[O_q(h)=\{z:\|(x_l)_z\|>8h\}\] are open and increase to \(U_{x_l}\) as \(h\downarrow0\). Approximation in this increasing union of ideals allows us to choose \(h_q>0\) such that \[ z'_q=(z_q-\gamma_q/2)_+\in Q(O_q(h_q)), \qquad [c_q]\ll[z'_q]. \tag{48}\] Keep \(h_q\) fixed from now on.

For a point \(z\in O_q(h_q)\), choose an open neighborhood \[N_z\subseteq O_q(h_q)\cap \bigcap_{\{i:(a_i)_z\ne0\}}\mathop{\mathrm{supp}}_X(b_i).\] It exists because \((a_i)_z\ne0\) implies \(z\in V_i\), and \(\mathop{\mathrm{supp}}_X(b_i)\) contains \(V_i\). Functional calculus on a positive element of \(Q(N_z)\) nonzero at \(z\) gives positive contractions \(h_z,k_z\) with \[ h_zk_z=k_z, \qquad (k_z)_z\ne0, \qquad h_z\text{ compactly Cuntz-supported in }Q(N_z). \tag{49}\] For example, first take a nonzero positive cut, make \(h_z\) equal to one on a smaller spectral region, and support \(k_z\) inside that region. The ideal supports of all these \(k_z\)’s cover \(O_q(h_q)\). By (48), finitely many of them suffice for \[ [c_q]\leq M_q\sum_{t=1}^{\ell_q}[k_{qt}] \tag{50}\] for some integer \(M_q\). This uses compact containment in the Cuntz semigroup: the ideal generated by all the \(k_z\)’s contains \(z'_q\), and \([c_q]\ll[z'_q]\) is captured by a finite sum with finite multiplicity. No compactness of the whole open set \(O_q(h_q)\) is assumed.

For each selected center \(z\), put \(S_z=C_0((0,1])\otimes(P/P_z)\), and let \(\alpha_t\), \(0\leq t\leq1\), be cone scaling: \[(\alpha_t f)(r)=f(tr), \qquad \alpha_0=0.\] The scaled images of a finite test set form norm-compact families. Choose finite nets in those families and their products. Cone quasidiagonality, followed by contractivity to pass from the nets to the families, gives a c.p.c. map \(\phi_z:S_z\to M_{d_z}\) with the required uniform multiplicativity error and \[ \|\phi_z((x_l)_z)\|>7h_q. \tag{51}\] Define a matrix-valued cone function by \[t\longmapsto\phi_z(\alpha_t(s_z)).\] It is norm-continuous and zero at \(t=0\). Evaluate its entries at \(h_z\). The resulting map \(\psi_z:S\to M_{d_z}(Q)\) is c.p.c.; it has the same multiplicativity bound because the evaluation map is a homomorphism. Its range lies in \(M_{d_z}(\mathop{\mathrm{Her}}(h_z))\).

For the lower estimate, choose a unit vector \(v\in\mathbb C^{d_z}\) which is an eigenvector of the positive matrix \(\phi_z((x_l)_z)\) for an eigenvalue \(\lambda>7h_q\). Equation (49) implies that evaluation on \(k_z^{1/2}\) is evaluation at parameter one. Consequently the rectangular column \[w=(\lambda-4h_q)^{-1/2}\,v\otimes k_z^{1/2}\] satisfies \[ w^*(\psi_z(x_l)-4h_q)_+w=k_z, \qquad \|w\|\leq(3h_q)^{-1/2}. \tag{52}\] Combining these comparisons with (50) gives the second inequality of (47) for the direct sum of all the selected blocks.

For an upper test, a block is zero if \((a_i)_z=0\). Otherwise \(h_z\) is compactly Cuntz-supported in an ideal contained in \(\mathop{\mathrm{Ideal}}_Q(b_i)\). It is therefore below finitely many copies of \(b_i\). Since the block range is supported by \(1_{d_z}\otimes h_z\), its rank is at most \(d_zd_\tau(h_z)\). Summing these finite bounds gives the first inequality of (47). The direct sum is contractive, since its norm is the maximum of the block norms. Its internal matrix trace is the sum of the block traces. This proves the lemma. ◻

Proof of 3(ii). At stage \(j\), apply 7 to the first \(j\) upper tests, lower requests, and an increasing dense list of algebra tests, with multiplicativity error at most \(j^{-1}\). Write the resulting map as \(\Phi_j\), and its constants as \(L_{ij}\) and \(M_{qj}\). Choose a positive rational \(t_j\) satisfying \[ t_j\leq j^{-1}, \qquad t_jL_{ij}\leq j^{-1}\quad(i\leq j), \qquad \beta_{qj}:=t_j/M_{qj}>0. \tag{53}\] After all the stage maps and fractions have been chosen, choose increasing coordinate thresholds \(N_j\) with \[ r_q(n)\leq\beta_{qj} \qquad(q\leq j, n\geq N_j). \tag{54}\] On \(N_j\leq n<N_{j+1}\), tensor \(\Phi_j\) with a UHF projection of trace \(t_j\), and fold its finite matrices to obtain \(\theta_n:S\to Q\). These maps are c.p.c. and asymptotically multiplicative. Their quotient is therefore a homomorphism \(\theta\).

For a fixed upper test, (47) and (53) give (40) eventually with bound \(j^{-1}\). Thus \(\theta(S(U))\subseteq J(U)\). For a lower request \(q=(l,\eta)\), \[ d_\tau(A_{q,n}) \leq r_q(n)d_\tau(c_q) \leq d_\tau((\theta_n(x_l)-4h_q)_+) \tag{55}\] eventually at every coordinate. Pointwise trace order in \(\mathop{\mathrm{Cu}}(Q)\) converts this into Cuntz comparison. Put \(B_{q,n}=(\theta_n(x_l)-2h_q)_+\), choose comparison witnesses with error at most \(n^{-1}\), and apply (42) with tolerance \(2h_q\). The resulting witnesses have squared norms at most \[\frac{\sup_n\|A_{q,n}\|+1}{2h_q},\] independently of the stage, its centers, and its matrix size. They prove \[[(|e_l|-\eta)_+] \leq[(\theta(x_l)-2h_q)_+] \quad\text{in }\mathop{\mathrm{Cu}}(D).\] This establishes (38) and then (36). A very small \(\beta_{qj}\) changes only the delay \(N_j\); it never changes the fixed cut \(h_q\). All comparisons may be made in the common block sequence algebra before folding. Include the finitely many prescribed error representatives in each stage’s folding tests in 2. The coordinate conjugacies then preserve their original quotient classes, and the norm bounds above are unchanged. This proves the separate-corner assertion. ◻

Proof. Choose a countable family \(\mathcal B\subseteq S_+\) of positive cuts which is an ideal basis: for every \(x\in S_+\), the elements of \(\mathcal B\cap\mathop{\mathrm{Ideal}}_S(x)\) generate \(\mathop{\mathrm{Ideal}}_S(x)\). Such a family is obtained from a countable norm-dense positive set and rational cuts, using the positive-element perturbation lemma. In particular, \[ U_x=\bigcup_{y\in\mathcal B\cap\mathop{\mathrm{Ideal}}_S(x)}U_y. \tag{57}\] For each \(y\in\mathcal B\), prescribe the errors \(E_j(a,z_1,\ldots,z_{r_j})\), where \(a\) runs through a countable dense set in \(S(U_y)\), and the other variables run through countable dense sets in their parameter spaces. These are countably many requests with controlling element \(y\), to which 3 applies. Continuity gives the desired containment for every input in \(S(U_y)\) and every parameter value.

Now fix \(x\). If \(y\in\mathcal B\cap\mathop{\mathrm{Ideal}}_S(x)\), the homomorphism property gives \(\theta(y)\in\mathop{\mathrm{Ideal}}_D(\theta(x))\). By (57), \(S(U_x)\) is the closed sum of the ideals \(S(U_y)\) in that equation. Linearity in the localized input and closedness of the target ideal give (56). Dense parameter sets and local norm continuity handle the asserted continuous families. Notice that the argument uses \(y\in\mathop{\mathrm{Ideal}}_S(x)\), not merely \(U_y\subseteq U_x\). Thus it does not presume coarse fullness of the cone stabilizer. ◻

Proof. Rank evaluation is lower semicontinuous on the compact trace cone, so \(K_b\) is closed. Its convexity follows from additivity and positive homogeneity of rank. For \(\eta>0\) the assumed rank bound gives \[ 0\leq\gamma(x)-\gamma((x-\eta)_+)\leq\eta L, \qquad \gamma\in K_b, \tag{58}\] and \(\gamma(x)\leq\|x\|L\). If \(\gamma_i\to\gamma\) in \(K_b\), the defining cut inequalities for the trace topology imply \[\gamma(x)\leq\liminf_i\gamma_i(x) \leq\limsup_i\gamma_i(x) \leq\gamma(x)+\eta L.\] Letting \(\eta\downarrow0\) proves continuity. Affineness follows from the cone operations; it does not require cancellation in that cone. Squaring a positive element does not change its support rank. Finally, support subadditivity and unitary invariance give \[d_\gamma\bigl(\lvert u x_1u^*-x_2\rvert^2\bigr) \leq d_\gamma(x_1)+d_\gamma(x_2),\] which proves the last assertion. ◻

Proof. Approximate \(\mu\) weakly by finitely supported probability measures \(\mu_j=\sum_k t_{jk}\delta_{\gamma_{jk}}\), omitting terms of weight zero. Their finite barycenters \(\beta_j=\sum_k t_{jk}\gamma_{jk}\) belong to \(K_b\). Take a subnet converging in the compact set \(K_b\) to \(\bar\gamma\). For every \(v\) in the statement, \[v(\bar\gamma)=\lim_jv(\beta_j) =\lim_j\int v\,d\mu_j=\int v\,d\mu.\] The same subnet works for all these functions. Neither uniqueness of \(\bar\gamma\) on discontinuous evaluations nor subtraction of extended weights is asserted or used. ◻

Proof. The equality of pullback weights implies equality of all their ranks. By (Ciuperca et al. 2013, Lemma 5.1), the support projections of \(p(s)\) and \(q(s)\) are equivalent in \(C^{**}\) for every \(s\in S_+\). Their images remain equivalent under the normal extension of every representation of \(C\).

The finite-rank assumption puts \(p(s_i)\) and \(q(s_i)\) in \(\mathop{\mathrm{Fin}}(\gamma_i)\). After zero padding, \(M_2(\mathop{\mathrm{Fin}}(\gamma_i))\) is invariant under \(M_2(C)^{\sim}\)-unitaries, so the conjugates and squared differences below remain in that finite ideal. For each selected weight, take its trace representation \(\pi_i\) on its densely finite ideal after quotienting by its zero ideal, and extend it nondegenerately to \(C\). Let \(\mathop{\mathrm{Tr}}_i\) be the resulting normal semifinite trace on \(M_i=\pi_i(C)''\). Form \(\pi=\pi_0\oplus\pi_1\oplus\cdots\oplus\pi_r\), with \(\pi_0\) a faithful representation on a separable Hilbert space, and put \(M=\pi(C)''\). The trace representations can also be chosen separably acting: their densely finite ideals are separable and have countable approximate units of finite-trace cuts. Thus \(M\) has separable predual. Restriction to the \(i\)th reducing Hilbert summand gives a normal surjective homomorphism \(q_i:M\to M_i\). Its kernel is \(M(1-z_i)\) for a central projection \(z_i\), and \(q_i\) identifies \(Mz_i\) with \(M_i\). Define \[\widehat\gamma_i=\mathop{\mathrm{Tr}}_i\circ q_i.\] This is the transported normal semifinite trace on \(Mz_i\), extended by zero on \(M(1-z_i)\). The projections \(z_i\) may overlap; the construction is made separately for each weight. On the densely finite ideal of \(\gamma_i\) one has \(\widehat\gamma_i\pi=\gamma_i\), so it computes all designated finite-rank tests and their squares.

We apply von Neumann uniqueness only after a common zero padding and unitization. Let \(S^\dagger\) be the forced unitization, and define unital homomorphisms into \(M_2(M)\) by \[\widehat p(a+\lambda1)=(\pi p(a)\oplus0)+\lambda1_{M_2(M)},\qquad \widehat q(a+\lambda1)=(\pi q(a)\oplus0)+\lambda1_{M_2(M)}.\] These maps have equal support ranks on every positive element of \(S^\dagger\). If its scalar part is zero, this follows from the support equivalences already proved. If the scalar part is positive, then on the properly infinite central part of \(M\) both support projections contain the second matrix corner, which is equivalent to the unit of \(M_2(M)\) there. On the finite central part, every normal finite trace has equal pullbacks along \(\pi p\) and \(\pi q\); this follows by restricting it to \(C\) and using the hypothesis. The unital extensions also have equal pullbacks, hence equal ranks by monotone spectral calculus. Normal traces determine equivalence of projections in a finite von Neumann algebra. This proves the claim on both parts.

The source \(S^\dagger\) is nuclear. We use the unital form of von Neumann uniqueness in (Ciuperca et al. 2013, Proposition 2.1, Corollary 3.10, and Section 4). For completeness, the properly infinite factor argument needs only the partial-isometry approximation of (Ciuperca et al. 2013, Lemma 3.6) and the following unital completion. Including \(1\) among the inputs gives partial isometries \(v\) whose initial projections \(e=v^*v\) tend strongly to \(1\). For finitely many normal states choose \(e_0\leq e\) arbitrarily close to \(e\) on those states, with \(1-e_0\sim1-ve_0v^*\sim1\): take \(e_0=e\) if \(e\) is finite, and otherwise remove an infinite tail decreasing strongly to zero. Complete \(ve_0\) to a unitary \(u\). Since \((u-v)e_0=0\), Cauchy–Schwarz gives \[|\rho(u^*xu-v^*xv)|\leq4\|x\|\rho(1-e_0)^{1/2}.\] Thus the partial-isometry conjugacies can be replaced by unitary conjugacies in the weak star topology; (Ciuperca et al. 2013, Lemma 3.9) makes them strong star. Section 4 then assembles these unital factor conclusions by central decomposition; separability lets one impose the support comparisons on one countable dense set of positive cuts before selecting the factorwise unitaries. We obtain \(w_j\in M_2(M)\) with \[w_j(\pi p(a)\oplus0)w_j^*\longrightarrow\pi q(a)\oplus0 \quad\text{strong star},\qquad a\in S.\] The unital extensions are needed here: support ranks of nonunital maps alone do not compare the complements of their support projections.

For clarity, the passage to \(L^2\) uses more than strong convergence. Put \(x_j=w_j(\pi p(s_i)\oplus0)w_j^*\) and \(x=\pi q(s_i)\oplus0\). Write \(T_i=\widehat\gamma_i\otimes\mathop{\mathrm{Tr}}_2\). Their \(T_i\)-norms are finite and satisfy \[\|x_j\|_{2,T_i}^2 =\gamma_i(p(s_i^2)) =\gamma_i(q(s_i^2)) =\|x\|_{2,T_i}^2.\] For a bounded test \(y\in L^1(T_i)\cap L^2(T_i)\) supported on trace-finite projections, its pairing with \(x_j-x\) is the normal bounded functional \(t\mapsto T_i(y^*t)\) and tends to zero. Such tests are dense in \(L^2(T_i)\); the common \(L^2\) bound therefore gives weak \(L^2\) convergence. Equality of norms now gives \(\|x_j-x\|_{2,T_i}\to0\).

Choose one \(w_j\) meeting the finite list to sufficiently small tolerance. A bounded self-adjoint logarithm of this unitary can be approximated strong star, within its norm bound, by self-adjoint elements of \(M_2(\pi(C))^{\sim}\), by Kaplansky density (Kaplansky 1951, Theorem 1). Exponentiation gives unitaries from \(M_2(C)^{\sim}\) converging strong star to \(w_j\). Their conjugates again have the same designated \(L^2\) norms, so the preceding finite-trace-test argument gives \(L^2\) convergence for this approximation too. On these conjugates the normal traces compute the original weights, since their finite ideals are invariant under \(M_2(C)^{\sim}\)-unitaries. This proves (60). ◻

Proof. By 1, choose a separable subalgebra \(D_0\) containing the two ranges, the required constant representatives, and the comparison witnesses, so that \(p\) and \(q\) agree on all traces of \(D_0\). Work in a common stabilized corner when needed.

Choose a countable local positive test family \(s_i\), dense in linear span in each of the basis ideals used in 6. Each test is cut-supported in both factors and has compact support in \(Y\). Choose an enlarged elementary test \(\bar s_i\) and a fixed \(\kappa_i>0\) with \(s_i\precsim(\bar s_i-\kappa_i)_+\), using the positive cut in each factor. Hence \(p(s_i)\precsim(p(\bar s_i)-\kappa_i)_+\), and the same holds for \(q\). The model-independent upper sandwich, applied to this fixed cut of \(\bar s_i\), supplies a common compact-Cuntz-supported constant; dominating it in the chosen containing open gives \(b_i\in Q(U_i)_+\) and a finite constant \(L_i\) such that \[ d_\gamma(p(s_i))+d_\gamma(q(s_i)) \leq L_i d_\gamma(b_i),\qquad\gamma\in T(D_0), \tag{62}\] for a finite \(L_i\). The constants and the comparisons are included in \(D_0\). One can retain \(b_i\) whenever later finite lists grow.

Fix a finite initial list and \(\delta>0\). For \(u\in\mathcal U(M_2(D_0)^{\sim})\), define the nonnegative continuous functions \[f_{u,i}(\gamma) =(\gamma\otimes\mathop{\mathrm{Tr}}_2) \bigl(\lvert u(p(s_i)\oplus0)u^*-(q(s_i)\oplus0)\rvert^2\bigr), \qquad \gamma\in K_{b_i}.\] Here \(\mathop{\mathrm{Tr}}_2\) is unnormalized: the common zero padding leaves (62) unchanged. Continuity follows from 8. We claim that zero lies in the closed convex hull of the tuples \((f_{u,i})_i\) in \(\bigoplus_i C(K_{b_i},\mathbb R)\). Otherwise a separating real functional would be bounded below by a positive constant on all these tuples. Represent its components by signed measures. Because \(f_{u,i}\geq0\), replacing each measure by its positive part retains that lower bound. By 9, the resulting functional on these functions is a finite sum of positive multiples of evaluations at individual weights in the corresponding caps. 10 makes all these evaluations simultaneously arbitrarily small, a contradiction.

It follows that a finite convex combination satisfies \[ \sup_{\gamma\in K_{b_i}} \sum_{k=1}^r t_k f_{u_k,i}(\gamma)<\delta \qquad\text{for every tested }i. \tag{63}\] The coefficients may be chosen positive and rational: the finite functions being averaged are bounded, so sufficiently small changes of the coefficients preserve the strict inequality. Write \(t_k=m_k/l\). Before folding, work in \(B_l=M_l(M_2(Q))\) with trace \(\tau\otimes\mathop{\mathrm{Tr}}_2\otimes\operatorname{tr}_l\): only this last, additional averaging direction is normalized. The two averaged maps are diagonal copies of \(\mathop{\mathrm{Ad}}(u_k)(p\oplus0)\) and of \(q\oplus0\), with \(m_k\) occurrences of the \(k\)th block. The original algebra \(D_0\) has the fixed corner embedding \[\iota_l:D_0\longrightarrow(B_l)_\infty, \qquad x\longmapsto(x\oplus0)\otimes1_l.\]

We record explicitly the resulting coordinate estimate. Lift the finitely many unitaries to unitary multiplier sequences, and use c.p.c. representatives of \(p,q\). Write \(z_{i,n}\) for the difference of their averaged values, after folding into \(Q\). For every fixed \(t>0\), we first prove the absolute estimate \[ d_\tau((|z_{i,n}|-t)_+) \leq \frac{4\delta}{t^2} \qquad\bigl(d_\tau(b_i)\leq1\bigr) \tag{64}\] for every sufficiently large \(n\), uniformly over this cap. Suppose that this absolute bound fails arbitrarily late, and choose violating coordinates and weights \(\tau_n\). The trace identities of 2 pull these weights back, before folding, to \(\tau_n\otimes\mathop{\mathrm{Tr}}_2\otimes\operatorname{tr}_l\). The subsequent multiplier conjugations preserve these identities. Positive cuts commute with the coordinate folding maps, so regularization commutes with them as well. Let \(\rho_l\) be the regularized trace before folding and put \(\gamma=\rho_l\circ\iota_l\). It is a trace on the same original \(D_0\), and 2 gives \[d_\gamma(b_i) =\sup_{\eta>0}\lim_\omega d_{\tau_n}((b_i-\eta)_+)\leq1.\] Thus \(\gamma\in K_{b_i}\). Put \(e_{k,i}=u_k(p(s_i)\oplus0)u_k^*-(q(s_i)\oplus0)\in M_2(D_0)\). If \(z_i\) denotes the unfolded diagonal difference, matrix-unit traciality and finite additivity give \[ \rho_l(z_i^*z_i) =\sum_k\frac{m_k}{l}(\gamma\otimes\mathop{\mathrm{Tr}}_2)(e_{k,i}^*e_{k,i})<\delta. \tag{65}\] Indeed, restriction to the first internal corner gives \(\gamma\), so the matrix extension on \(M_2(D_0)\) is \(\gamma\otimes\mathop{\mathrm{Tr}}_2\). For every positive \(a\in M_2(D_0)\), \(\rho_l(a\otimes e_{jj})=(\gamma\otimes\mathop{\mathrm{Tr}}_2)(a)/l\): the \(l\) averaging corners are equivalent and sum to \(a\otimes1_l\). This uses only finite positive sums, so also holds when a value is infinite. It follows that (65) is valid without a bounded-weight assumption. On the other hand, functional calculus gives \[\tau\bigl((|z_{i,n}|^2-t^2/2)_+\bigr) \geq \frac{t^2}{2}d_\tau((|z_{i,n}|-t)_+).\] Regularization and the violations of the absolute estimate therefore give a square trace at least \(2\delta\), contrary to (65). This proves (64). If \(0<d_\tau(b_i)<\infty\), apply that estimate to \(\tau/d_\tau(b_i)\). If \(d_\tau(b_i)=0\), apply it to every positive multiple of \(\tau\), forcing the left side to be zero. For \(d_\tau(b_i)=\infty\) the desired proportional bound is automatic. Thus, at the same sufficiently late coordinates, \[ d_\tau((|z_{i,n}|-t)_+) \leq \frac{4\delta}{t^2}d_\tau(b_i) \qquad\text{for all }\tau\in T(Q). \tag{66}\]

At stage \(j\) use the first \(j\) local tests, the cut tolerances \(1,1/2,\ldots,1/j\), and choose \(\delta_j<2^{-j}/(4j^2)\). Then advance far enough in that row for (66), the chosen algebraic tests, and their norm bounds all to hold. Pass through the rows slowly as in 6. The averaging matrix size is fixed while each row’s uniform coordinate estimate is proved. For a fixed local test and any fixed cut tolerance, this gives its \(J(U_i)\) estimate against the fixed \(b_i\). Closure and the local density of the tests give (61) for all open ideals.

For completeness, these operations preserve the asserted relation to the original maps. At each coordinate, a diagonal sum of conjugates of a map is conjugate, by the diagonal of the same unitaries, to its tensor-first copy in the averaging matrix factor. The coordinate folding can be chosen, using 2, to carry that copy within the stage tolerance of the original on every current input. The first-corner clause of 2 removes the common internal zero padding at the same time. Make that tolerance tend to zero during the same diagonal choice. Inner conjugacy preserves every coordinate weight, and vanishing norm changes preserve regularized values by cuts, so the models keep their data. The argument applies to a designated block of a direct sum, amplifying the spare by the same averaging factor. Its unused part can always be placed in a separate stable corner before folding. ◻

Proof. The multiplier-pair theorem (Gabe 2024, Theorem 13.9) supplies a weakly \(X\)-nuclear representation \(\Theta\) and an identity-plus-compact unitary \(u\) comparing \(\phi\oplus\Theta\) with \(\psi\oplus\Theta\) to any prescribed finite-test tolerance \(\delta\). Here the pair class may be read in the nuclear theory, since \(S\) is nuclear. Put \(\sigma=\Theta\oplus\theta^\infty\). By 5, there is a module unitary \(V\) from the repeat module to the module of \(\sigma\) such that \[\|V^*\sigma(a)V-\theta^\infty(a)\|<\eta, \qquad a\in\mathcal F.\] With \(W_0=1\oplus V\), set \[v=W_0^*(u\oplus1)W_0.\] It is a unitary, and \(v-1\) is compact because conjugating a compact module operator by a module unitary preserves compactness. The error in (69) is at most \(\delta+2\eta\). Choose these parameters with \(\delta+2\eta<\varepsilon\). This is the finite-test common-stabilizer transfer underlying (Dadarlat and Eilers 2001, Lemmas 2.4 and 3.4); no initial value of a continuous unitary path is needed here. ◻

Proof. Since \(\mathcal W\) is simple and nuclear, every ideal of its spatial tensor product with another \(C^*\)-algebra is a tensor-product ideal (Gupta and Jain 2018, Theorem 3.1). Together with the transported labels of 1, this gives, after reordering the tensor factors, an \(X\)-algebra \(C=C_0(Y)\otimes A\otimes\mathcal K\) such that \[S=\mathcal W\otimes C,\qquad S(U)=\mathcal W\otimes C(U).\] The identity of \(\mathcal W\) has zero ordinary Kasparov class. Exterior tensoring its stabilized cycle relation and its homotopy with \(C\) retains weak equivariance. Explicitly, a typical coefficient of the tensor representation, for \(c\in C(U)\), is a finite sum \[\sum_{i,j}\langle\xi_i,\pi(w)\xi_j\rangle \otimes c_i^*c c_j\ \in\ \mathcal W\otimes C(U).\] The same formula applies throughout the homotopy. It follows that the identity cycle of \(S\) is zero for these labels. Taking its cycle product with an \(S\)–\(H\) cycle preserves this stabilized null relation. The identity-product homotopy identifies the same product with the original cycle, so that cycle is zero as well.

The cycle relation and mixed product statements used here are (Gabe 2024, Propositions 12.17 and 12.20 and Lemmas 12.18, 12.19, and 12.23). They require the first algebra to be separable and the remaining coefficient algebras to be sigma-unital. Thus they apply to this \(H\) even when it is nonseparable. Nuclearity of \(S\) also makes the cycles and homotopies weakly \(X\)-nuclear, as needed above. ◻

Proof. The defining positive element is strictly positive in its hereditary algebra, so \(H\) is sigma-unital. Since \(r e^{1/2}\) and \(e^{1/2}r\) belong to \(I\), multiplication of \((e^{1/2}\oplus k^{1/2})M_2(C)(e^{1/2}\oplus k^{1/2})\) on either side by \(r\oplus0\) remains in \(H\). These elements are dense in \(H\). Under the additional hypothesis, the same products also lie in \(L\), proving (70). The last identity is the ideal correspondence for a hereditary subalgebra: an approximate unit of \(H\) compresses the ambient ideal generated by \(x\) to the ideal it generates in \(H\). ◻

Proof. Choose \(u=1+k\) with error less than \(\varepsilon/2\), and put \(R=\max(1,\max_{a\in\mathcal F}\|a\|)\). If \(E_N\) denotes the specified finite-summand projection, choose \[\|k-E_NkE_N\|<\eta, \qquad 0<\eta<\min\{1/4,\varepsilon/(8R)\}.\] The operator \(z_N=1+E_NkE_N\) is within \(\eta\) of a unitary, so it is invertible and the spectrum of \(|z_N|\) is contained in \([1-\eta,1+\eta]\). Its polar unitary satisfies \[v_N=z_N(z_N^*z_N)^{-1/2},\qquad \|v_N-z_N\|\leq\eta, \qquad\|v_N-u\|\leq2\eta.\] It is the identity off \(E_N\) and differs from the identity by a finite matrix over \(H\). Include successively the identical zero slots as well, if present, so that \(E_N\to1\) strongly. The change in the conjugacy error is at most \(4R\eta\). Since the two representations are diagonal and agree on the omitted tail, the remaining estimate is exactly the displayed finite-copy estimate.

For localization, a compact operator which is weakly equivariant on an input in \(S(U)\) has all matrix coefficients in \(H(U)\), hence belongs to \(H(U)\otimes\mathcal K\). All module unitaries preserve the submodules \(E_HH(U)\), so conjugation preserves this statement. It applies to the compact differences here. ◻

Proof. Put \(C_1=C^*(p(S),q(S))\), and let \(I\) be the ideal of \(C_1\) generated by the differences \(p(a)-q(a)\). This is a separable ideal. For each open \(U\), \[ p(S(U))I+q(S(U))I+Ip(S(U))+Iq(S(U))\subseteq J(U). \tag{71}\] Indeed, modulo \(J(U)\), the two maps agree on \(S(U)\). Their common image is an ideal in the algebra generated by both images, because multiplication from either side by \(p(a)\) or \(q(a)\) corresponds to multiplication inside the source ideal. It annihilates each difference: for \(s\in S(U)\), \[p(s)(p(a)-q(a))=p(sa)-q(sa)=0\pmod{J(U)},\] and the other side is identical. It therefore annihilates the ideal generated by the differences, proving (71).

Use the existing-model version of 3, with the spare \(\chi\), to obtain a locally small homomorphism \(\theta\). Impose the coefficient dominations \[ p(S(U_x))I+q(S(U_x))I+Ip(S(U_x))+Iq(S(U_x)) \subseteq\mathop{\mathrm{Ideal}}(\theta(x)),\qquad x\in S_+. \tag{72}\] Apply 2 to the four bounded bilinear maps \((a,b)\mapsto p(a)b\), \(q(a)b\), \(bp(a)\), and \(bq(a)\), with \(b\in I\). Equation (71) gives their local hypotheses, and that corollary supplies the displayed inclusion for every positive \(x\). The first version also ensures \[ \theta(S(U_x))\subseteq\mathop{\mathrm{Ideal}}(\theta(x)). \tag{73}\] Moreover \(\theta\) is represented by the same coordinate maps as \(\chi\), tensored with rational UHF projections of trace tending to zero.

Choose strictly positive elements \(e\in I\) and \(k\in\theta(S)\), and form \[H=\mathop{\mathrm{Her}}_{M_2(D)}(e\oplus k).\] The maps \(p,q\), placed in the first corner and zero in the second, idealize \(H\), by 13. View them as representations on the module \(H\), and then stabilize that module. Equip \(H\) with the order-preserving ideals \[ H(U)=\mathop{\mathrm{Ideal}}_H(0\oplus\theta(S(U))). \tag{74}\] No intersection property of this assignment is needed. From (72), 13, and its ideal-intersection identity, values of the first-corner maps on \(S(U_x)\) multiply \(H\) into \(\mathop{\mathrm{Ideal}}_H(0\oplus\theta(x))\). Thus \(p,q\) are weakly equivariant for (74). By (73), \(0\oplus\theta(x)\) generates \(H(U_x)\), so this compact-valued map is \(X\)-full. The algebra \(H\) is sigma-unital; it is not required to be separable.

The multiplier pair agrees modulo \(H\), since its differences belong to \(I\oplus0\subseteq H\). Pad the module \(H\) by zero copies to the standard module; the resulting pair class (68) is zero by 12. The coarse source action is lower semicontinuous, since the projection \(Y\times X\to X\) is open. For 5, place \(0\oplus\theta\) in the first compact corner of the standard module. Its infinite repeat is, after reindexing, the diagonal of copies of \(\theta\) on \(H\) with identical zero slots. Thus 14 applies on the original \(H\) summands, yielding finite matrices over \(H\), rather than retaining an additional untruncated compact direction inside each slot. For any finite \(\mathcal F\) and \(\varepsilon>0\) this gives a finite \(N\) and an identity-plus-finite-matrix unitary comparing \[ p\oplus\theta^{\oplus N} \quad\text{and}\quad q\oplus\theta^{\oplus N} \tag{75}\] within \(\varepsilon\), with zero slots permitted.

These estimates hold for the actual ambient elements, as follows. If \(v=1+c\) is the obtained finite unitary and \(A,B\) denote the two finite diagonal maps, then \[ vA(a)v^*-B(a) =(A(a)-B(a))+cA(a)+A(a)c^*+cA(a)c^*. \tag{76}\] Every term belongs to a finite matrix over \(H\), because the maps idealize \(H\) and their difference belongs to it. The multiplier norm of this element is its intrinsic \(H\)-norm and hence its ambient norm in the matrix algebra over \(D\). This argument does not assume that restriction of the entire ambient idealizer to \(\mathcal M(H)\) is faithful. Only its isometric restriction to \(H\) is used.

There is also no lifting obstruction for the finite unitary. Write it as \(1+c\) in the unitization of a finite matrix algebra over \(D\), and choose bounded representatives \(c_n\) over \(Q\). The two unitary defects of \(1+c_n\) tend to zero. For all sufficiently large \(n\) its polar part \[v_n=(1+c_n)\bigl((1+c_n)^*(1+c_n)\bigr)^{-1/2}\] is a unitary in the corresponding matrix multiplier algebra and differs from \(1+c_n\) by a sequence tending to zero. Thus a strict quotient estimate, first obtained with a smaller tolerance, holds at every sufficiently late coordinate.

Finally fix this \(N\) before advancing the coordinates. Since the UHF trace of the projection defining \(\theta_n\) tends to zero, \(N\) disjoint equivalent copies fit inside the positive spare fraction eventually. Using exactly the same \(\chi_n\) in those copies identifies them with (75). Move the unused spare to a separate stable corner and leave it unchanged. Consequently the finite conjugacy need not preserve an internal complement of its original matrix slots. Apply the coordinate folding and the multiplier permutations from 2, on all current inputs and all these blocks. This proves eventual finite-test conjugacy of \(p\oplus\chi\) and \(q\oplus\chi\). On increasing finite sets with tolerances tending to zero, choose each coordinate beyond the corresponding threshold. The resulting single sequence of multiplier unitaries proves the proposition. ◻

Proof of 4. Split the tensor-first copy of each map along two equivalent UHF projections of trace \(1/2\). Denote its two halves by \(p_0,p_1\) and \(q_0,q_1\). Both halves have the original model data multiplied by \(1/2\), and their sums are coordinate approximately unitarily equivalent to the original maps by 2.

First compare \(p_0\oplus p_1\) with \(q_0\oplus p_1\), retaining \(p_1\) as a common spare. Apply 11 in the changing block to put its difference in every appropriate \(J(U)\), carrying the spare through the same diagonal amplifications. Then 6 gives the required conjugacy. The coordinate conjugacies relating the aligned blocks to the originals may be undone, with their vanishing errors, so the conclusion applies to the displayed original half-sums. Next compare \(q_0\oplus p_1\) with \(q_0\oplus q_1\), using \(q_0\) as the common spare and interchanging the two matrix positions. The same argument applies. Composing the two coordinate conjugacies and the folding adjustments proves the assertion on every finite test set and tolerance. A diagonal choice on a dense increasing family gives the stated sequence \(u_n\).

Finally, any two c.p.c. lifts of a fixed model differ in norm by a sequence tending to zero on each input. The conclusion therefore holds for the lifts specified in the statement. ◻

Proof. The central decomposition of a semifinite hyperfinite algebra gives a countable direct sum of algebras \[L^\infty(X_j,\mu_j^0)\,\overline\otimes\,N_j, \qquad N_j\in\{M_n,\ B(\ell^2),\ R,\ R_\infty\}.\] Here the measure spaces are standard and the factor traces are the usual matrix trace or the standard finite or semifinite trace. This is the semifinite form of the hyperfinite central decomposition; see (Haagerup et al. 2003, sec. 6, p. 49, after (6.48) and (6.50)). Relative to these traces and a finite measure \(\mu_j^0\) in the measure class, \(T\) has a central Radon–Nikodym density \(h_j\). Faithfulness and semifiniteness give \(0<h_j<\infty\) almost everywhere. Replacing \(\mu_j^0\) by \(\mu_j=h_j\mu_j^0\) absorbs the density and leaves a standard \(\sigma\)-finite measure space.

There is a normal, trace-preserving injection \[L^\infty(X_j,\mu_j)\longrightarrow L^\infty([0,\infty),dt).\] Indeed, partition \(X_j\) into countably many pieces of finite measure. On each piece assign an interval of the appropriate length to each atom; the remaining diffuse part is isomorphic, modulo null sets, to the remaining interval. Pullback by the resulting measure-preserving map, extended by zero outside the union of the intervals, gives the injection. Moreover, \(L^\infty([0,\infty))\) embeds trace-preservingly as \(L^\infty([0,1])\overline\otimes\ell^\infty\) in \(R\overline\otimes B(\ell^2)=R_\infty\).

Each standard factor \(N_j\) embeds trace-preservingly in \(R_\infty\): use \(1_R\otimes M_n\), \(1_R\otimes B(\ell^2)\), \(R\otimes e_{11}\), or the identity map, respectively. Tensor these embeddings with the preceding abelian embeddings and use the trace-preserving isomorphism \(R_\infty\overline\otimes R_\infty\cong R_\infty\). The countably many summands can be placed in the diagonal corners of \(R_\infty\overline\otimes B(\ell^2)\). The orthogonal strong sum is normal, injective, and trace preserving.

Finally, in an infinite corner of an arbitrary factor of type \(\mathrm{II}_\infty\), take a projection of trace one, nested unital matrix algebras \(M_{2^n}\) in its finite corner, and their weak closure. They give a trace-preserving copy of \(R\). Matrix amplification in the infinite corner gives \(R_\infty\). Composition completes the proof. ◻

Proof. Stability of \(Q\) supplies multiplier isometries whose normal images show that \(M\) is properly infinite. Since \(M\) is separably acting, the usual scalar trace-dimension comparison applies in its semifinite factor cases; in its type III case all nonzero projections are equivalent (Blackadar 2006, III.1.7.9–10 and III.1.3.7). Suppose first that \(M\) is of type I. The rank function \(d_{T\circ\pi}\) on \(\mathop{\mathrm{Cu}}(Q)\) takes only the values \(0\) and \(\infty\). Otherwise real scalar divisibility of \(\mathop{\mathrm{Cu}}(Q)\), applied to a class of finite nonzero integer rank, would give a class of noninteger rank in a type I factor. Thus the right side of (82) also takes only these two values. Its zero ideal on \(P\) is the ideal transported from \(\ker\pi\). A faithful separable representation of the corresponding quotient of \(S\), repeated with infinite multiplicity and embedded in \(M\), realizes the required ranks. For type III the same construction realizes the zero ideal; all nonzero support projections are equivalent. The assertion is also immediate when the prescribed quotient is zero.

Suppose now that \(M\) has type \(\mathrm{II}_\infty\), and put \[\nu=m\otimes F(T\circ\pi),\qquad I=\mathop{\mathrm{Fin}}(\nu), \qquad K=\mathop{\mathrm{Ker}}(\nu).\] Here \(\mathop{\mathrm{Fin}}(\nu)\) is the closed ideal on which \(\nu\) is densely finite, and \(K\subseteq I\) is its zero ideal. The trace representation of \(I/K\) has weak closure \(N\) carrying the faithful normal semifinite trace induced by \(\nu\). Nuclearity of \(I/K\) makes \(N\) hyperfinite (Connes 1976a); the nuclear-representation consequence is also stated in (Connes 1976b, Theorem 4.1 and the following paragraph). The algebra \(N\) has separable predual. Its nondegenerate representation extends from \(I\) to \(S\) by the action of \(S\) on the ideal \(I\). 15 embeds this representation trace-preservingly in one infinite corner of \(M\).

In an orthogonal infinite corner, choose a copy of \(B(\ell^2)\) whose minimal projections have a fixed positive finite trace. In this copy place an infinite multiple of a faithful separable representation of \(S/I\). On \(I\) this second summand is zero. If \(x\in S_+\) does not belong to \(I\), its second image has infinite trace and infinite support rank. This is exactly what is required, since \(\nu(x)=\infty\) off \(I\). On \(I\), the first summand gives \(\nu\) and all its ranks, including infinite values. The combined homomorphism therefore satisfies \(T\circ l=\nu\), has zero ideal \(K\), and proves (82). If \(I=0\) or \(I=S\), simply omit the empty summand. ◻

Proof. Use the standard central decomposition of \(M\), modulo null sets, into separably acting factors \(M_x\) (Blackadar 2006, III.5.1.14–15). We explain the measurable selection in a form that does not choose a normalization of the factor traces. For each elementary test \(s_j=f_j\otimes a_j\) fix \(c_j\in Q_+\) with \[ [c_j]=m(\{f_j>0\})G[a_j]. \tag{83}\] Realification and stability provide these representatives. Code a contractive homomorphism on a countable dense rational \(*\)-algebra in \(S\) by its values in the strong-\(*\) bounded balls of the factor field. Include norm-convergent approximations to the chosen positive tests. Multiplication on bounded balls, adjoints, norm bounds, and membership in the measurable algebra field give countably many Borel conditions. Membership may be checked against the strong-\(*\) dense measurable sections of the field. Adjoin contraction coordinates \(w_j\in M_x\) subject to \[ w_j^*w_j=\mathop{\mathrm{supp}}(l_x(s_j)),\qquad w_jw_j^*=\mathop{\mathrm{supp}}(\pi_x(c_j)). \tag{84}\] The support map is the strong limit of \(t\mapsto t(t+1/n)^{-1}\), so these are again Borel conditions.

Thus the augmented admissible-code relation is Borel in a countable product of standard measurable bounded operator fields. Its sections are nonempty by 16: equal factor dimensions give the partial isometries in (84). Apply Jankov–von Neumann measurable uniformization to this augmented analytic relation, retaining both the homomorphism coordinates and all the contraction witnesses \(w_j\). The selector is measurable for the completed central measure. Replace its countably many coordinates by measurable representatives and discard one common null set on which any selected algebraic equation or support identity fails. This is precisely the measured-field setting, as distinguished from an everywhere Borel field in (Vaes and Wouters 2025, sec. 8, especially Lemma 8.6(1), and the proof of Proposition 3.2). The uniform operator bounds make the selected values, including the \(w_j\), decomposable operators. Their countably many algebraic relations define a homomorphism \(l:S\to M\), and the assembled witnesses establish (84) in \(M\). Every elementary cut test needed below is included in the code, so its prescribed rank data hold in this field. ◻

Proof. Fix one selected \(\tau\) and put \(I=\mathop{\mathrm{Fin}}(\tau)\). Its trace representation \(\pi_\tau\) on \(I/\mathop{\mathrm{Ker}}(\tau)\) has weak closure \(N_\tau\) with faithful normal semifinite trace \(T_\tau\). The nondegenerate extension of \(\pi_\tau\) to \(Q\) still has image in \(N_\tau\): for an approximate unit \((u_\lambda)\) of \(I\), the operators \(\pi_\tau(q)\pi_\tau(u_\lambda)\in\pi_\tau(I)\) converge strongly to \(\pi_\tau(q)\). Hence \(\pi_\tau(Q)''=N_\tau\).

The projection onto this representation summand commutes with \(M=\pi(Q)''\). Restriction to that summand is therefore a normal surjective homomorphism \(\theta_\tau:M\to N_\tau\). Its kernel has the form \((1-z_\tau)M\) for a central projection \(z_\tau\), and its restriction to \(z_\tau M\) is a normal isomorphism. Define \[\widetilde\tau=T_\tau\circ\theta_\tau.\] This is a normal semifinite trace, zero on \((1-z_\tau)M\), and it agrees with \(\tau\) on \(\pi(I)_+\). Normality also gives \(d_{\widetilde\tau}(\pi(c))=d_\tau(c)\) for \(c\in I_+\). The projections \(z_\tau\) for different selected weights may overlap: each trace is defined through its own normal quotient, with no disjointness or additional contribution from other representation summands. The construction includes the zero trace representation. Agreement with \(\tau\) outside \(I\) is neither asserted nor needed.

Include the rectangle tests \((f_i-r)_+\otimes(a_i-t)_+\) for rational \(r,t\geq0\), and choose \(c_{r,t}\in Q_+\) with \[[c_{r,t}]=m(\{f_i>r\})G[(a_i-t)_+].\] Their ranks are at most \(m(Y)d_\tau(b_i)<\infty\), so \(c_{r,t}\in I\). Retaining the selected witness coordinates in (84) gives, in \(M\), equivalence of the support of \(l((f_i-r)_+\otimes(a_i-t)_+)\) with \(\mathop{\mathrm{supp}}(\pi(c_{r,t}))\). Applying \(\widetilde\tau\) directly to these equivalent projections gives \[\widetilde\tau\bigl(\mathop{\mathrm{supp}}l((f_i-r)_+\otimes(a_i-t)_+)\bigr) =m(\{f_i>r\})d_{F(\tau)}((a_i-t)_+).\] In particular, \(\mathop{\mathrm{supp}}(l(s_i))\) has finite \(\widetilde\tau\)-trace.

In the support corner of \(l\), the multiplier images of \(f_i\otimes1\) and \(1\otimes a_i\) commute, and their product is \(l(s_i)\). The last display specifies their finite joint spectral measure on every upper rectangle in the positive quadrant. It is the product of the spectral measure of \(f_i\) under \(m\) and that of \(a_i\) under \(F(\tau)\). Increasing positive step approximations to the product function \((u,v)\mapsto u^kv^k\), using rational spectral partitions away from zero, therefore give (85) by normal monotone convergence, for \(k=1,2\). All rank identities used here were evaluated on constants in \(\mathop{\mathrm{Fin}}(\tau)\). ◻

Proof. Reduce a separator to finitely many weights. Suppose a continuous real linear functional strictly separates the target from this convex hull. On each \(C(K_i,\mathbb R)\) coordinate it is integration against a finite signed measure. Split each measure into its positive and negative parts. By 9, each nonzero part has a barycenter in \(K_i\) reproducing every finite continuous affine evaluation used here; its mass is an ordinary scalar coefficient. This holds simultaneously for the candidate evaluations and the target evaluations, including candidates with different support constants. Thus the separating functional involves only finitely many selected traces. On the algebra coordinates it involves finitely many bounded selfadjoint functionals.

Realize the selected values exactly. Choose a faithful separably acting representation \(\pi\) of \(Q\) containing representations for these bounded functionals and the induced representations of the densely finite ideals of the selected traces. Write \(M=\pi(Q)''\) and identify \(Q\) with its image. [tc:measurable-realization,tc:finite-trace-realization] give a homomorphism \(l:S\to M\) attaining the selected trace values. We next attain the lower norm coordinates in \(M\).

Let \(q_j\) be the support projection of \(l((f_j-\delta_j)_+\otimes(a_j-\delta_j)_+)\). Functional calculus in the two commuting factors gives \[q_j\leq1_{(\delta_j^2,\infty)}(l(s_j)).\] The rank data and (87) give, in every factor, \(\mathop{\mathrm{supp}}(y_j)\precsim q_j\). This comparison also covers zero and infinite dimensions and the type III parts. Comparison of projections in a direct integral therefore gives a partial isometry \(v_j\in M\) with initial projection \(\mathop{\mathrm{supp}}(y_j)\) and final projection at most \(q_j\). Hence \[ y_jv_j^*g_j(l(s_j))v_jy_j=y_j^2. \tag{92}\]

Approximate inside fixed supports. It remains to approximate these finite data by candidates satisfying (89) on the entire caps. A tested value must vanish exactly in a block where its controller vanishes; strong approximation alone would not retain this condition. We therefore split into central sectors and factor each sector map through the required source quotient before approximating it. Let \(z_i\) be the central support of \(b_i\) in \(M\), and form the finitely many central sectors \[z_E=\prod_{i\in E}z_i\prod_{i\notin E}(1-z_i), \qquad E\subseteq\{1,\ldots,n\}.\] If \(i\notin E\), the rank data and \(G[a_i]\leq[b_i]\) imply \(z_El(s_i)=0\). Let \(J_E^S\) be the ideal of \(S\) generated by these \(s_i\), and let \[I_E=\bigcap_{i\in E}\mathop{\mathrm{Ideal}}_Q(b_i), \qquad I_{\varnothing}=Q.\] The weak closure of \(I_E\) has central support dominating \(z_E\). Choose positive contractions \(e_E\in I_E\) with compact Cuntz support, tending strongly to the identity on this sector. The maps \[ \psi_E(s)=e_Ez_El(s)e_E \tag{93}\] take values in \(\mathop{\mathrm{Her}}(e_E)''\), are completely positive contractions, and factor through \(S/J_E^S\). This quotient is taken before the approximations below. The element \(e_E\) itself need not vanish outside the sector; (93) does vanish there. Write \[D_0(s)=\operatorname{diag}_E(z_El(s)),\qquad D_e(s)=\operatorname{diag}_E(e_Ez_El(s)e_E).\] Thus \(D_0\) is the exact sector representation and \(D_e\) is its completely positive compression. As the \(e_E\) increase along approximate units, \(D_e(s)\to D_0(s)\) strongly-\(*\) for each \(s\).

For the selected trace belonging to test \(i\), the sector contributes zero if \(i\notin E\). If \(i\in E\), the positive operators \(z_El(s_i)\) and \(z_El(s_i^2)\) are integrable, and \(z_El(s_i)\) is square integrable, by (85). Strongly converging bounded approximate units therefore make the compressions converge in the required \(L^1\) and \(L^2\) seminorms. For the \(L^1\) assertion apply \(L^2\) convergence to the square root of the integrable positive operator and use Cauchy–Schwarz; for the square evaluation apply it directly to \(z_El(s_i)\).

For the lower signal set \(V_j=(z_Ev_j)_E\). This column is a contraction, and, because \(g_j(0)=0\), the exact sector representation satisfies \[y_jV_j^*g_j(D_0(s_j))V_jy_j=y_j^2.\] Bounded strong-\(*\) functional calculus gives \(g_j(D_e(s_j))\to g_j(D_0(s_j))\). Fix the \(e_E\) sufficiently far out that replacing \(D_0\) by \(D_e\) makes the errors small against the selected normal functionals, as well as meeting the preceding finite trace requirements. The compressed expression need not equal \(y_j^2\) exactly.

We give the remaining approximation explicitly. Nuclearity of the quotient \(R_0=S/J_E^S\) provides completely positive contractive factorizations through finite matrix algebras approximating its identity on the finite tests (Choi and Effros 1978); see also (Han and Paulsen 2011, Corollary 3.3). For nonunital \(R_0\), apply the latter unital statement to its nuclear unitization. Restrict the incoming map to \(R_0\) and compress the outgoing map \(\beta\) to \(x\mapsto e\beta(x)e\), where \(e\in R_0\) is a positive contractive approximate-unit element. Choose \(e\) first so that \(\|eae-a\|<\delta/2\) on the finite tests, and then choose the factorization with error less than \(\delta/2\) there. Contractivity bounds the total error by \(\delta\), and the compressed map takes values in \(R_0\). Compose this outgoing map with \(\psi_E\). A completely positive map from \(M_d\) into \(\mathop{\mathrm{Her}}(e_E)''\) is specified by its positive Choi matrix. Approximate the square root of that matrix strongly-\(*\), with bounded norm, by matrices over \(\mathop{\mathrm{Her}}(e_E)\); the products give positive Choi matrices over the hereditary algebra. If the resulting map \(\beta\) has \(t=\beta(1)\), replace it by \[x\longmapsto \max(1,t)^{-1/2}\beta(x)\max(1,t)^{-1/2}.\] The functional calculus is in the unitization. This replacement is completely positive and contractive, stays in \(\mathop{\mathrm{Her}}(e_E)\), and retains the strong-\(*\) limit, because the limiting unit value is at most one. We have obtained completely positive contractions \[H_E:S/J_E^S\longrightarrow\mathop{\mathrm{Her}}(e_E)\subseteq Q\] approximating (93) on the required tests.

The support element \(e_E\) was fixed before this Choi-matrix approximation, and every approximating value remains in \(\mathop{\mathrm{Her}}(e_E)\). For each \(i\in E\), compact Cuntz support inside \(I_E\) gives a finite integer \(N_{E,i}\) with \([e_E]\leq N_{E,i}[b_i]\). Thus the support of every output in this block has rank at most \(N_{E,i}d_\tau(b_i)\), for every extended trace \(\tau\). On the selected traces this is one fixed finite normal trace support. Bounded strong-\(*\) convergence on that support gives convergence of the trace and square-trace evaluations, with no loss of mass. For \(i\notin E\), the quotient-first factorization kills \(s_i\) and \(s_i^2\) exactly, so no trace-convergence argument is required in that block. This proves the trace approximation, as well as the required uniformity on the whole caps.

Put the sector maps in separate matrix blocks: \(H=\bigoplus_E H_E\). This is a completely positive contraction since the norm of a direct sum is the maximum of the norms. Its unnormalized matrix trace is the sum of the sector traces. The blocks with \(i\notin E\) kill \(s_i\) and \(s_i^2\) exactly, while \[d_\tau(H(s_i)),\ d_\tau(H(s_i^2)) \leq\left(\sum_{E\ni i}N_{E,i}\right)d_\tau(b_i).\] The maps \(H_E\) approximate the fixed compressed map \(D_e\), not the exact map \(D_0\). Rectangular Kaplansky density approximates the columns \(V_j\) strongly-\(*\) by contraction columns over \(Q\). Bounded strong-\(*\) functional calculus and multiplication therefore make the resulting algebra coordinates arbitrarily close, against the selected normal functionals, to those for \(D_e\). Combined with the preceding choice of the \(e_E\), this approximates \(y_j^2\) to the required functional accuracy. Those functionals are normal in \(\pi\) by construction.

Thus candidate vectors approximate the value of every finite functional obtained from a supposed separator, arbitrarily closely to the target value. This contradicts strict separation and proves the proposition. ◻

Proof. Write the displayed average as \(T\) and let \(p=1_{(\varepsilon,\infty)}(y)\) in the bidual. The norm estimate gives \(pTp\geq(\varepsilon^2/2)p\). Since the \(v_\alpha\) are contractions and \(0\leq g(t)\leq1_{(\eta,\infty)}(t)\), positivity and traciality give \[\frac{\varepsilon^2}{2}\tau(p) \leq \|y\|^2\sum_\alpha\lambda_\alpha d_\tau((H_\alpha(s)-\eta)_+).\] The sum is exactly the rank on the right of (95), by the normalized averaging convention.

Here is a justification when \(\tau\) is not densely finite. If the right side is infinite, there is nothing to prove. If it is finite, discard zero coefficients. Every support of \(g(H_\alpha(s))\) then has finite rank. The norm estimate gives \((y^2-\varepsilon^2/2)_+\precsim T\), and \(T\) is Cuntz below the finite direct sum of the \(g(H_\alpha(s))\), by its finite row factorization. Since \(p\) is the support of \((y^2-\varepsilon^2)_+\), it has finite rank and belongs to the normal trace’s finite part. One may initially use a slightly smaller spectral projection inside \(p\). The displayed computation is now a computation for the induced normal trace on this finite part. Let the additional cut slack decrease to zero. This gives the inequality for \(p\) by normal monotone convergence and proves the assertion for every extended weight. ◻

Proof of 6. Fix the comparisons before forming averages. Fix the test family, its larger pre-cuts, its local controllers, and all lower data before choosing the sequence. For each nonzero \(f\otimes a\in\mathcal E\), take countably many pairs \((\delta,r)\) with \[\delta\downarrow0,\qquad 0<r<m(\{f>\delta\}),\] and positives \(y\) representing \(rG[(a-\delta)_+]\). Include all positive rational cuts of these \(y\) in the lower requirements. Their ideals exhaust \(Q(U_a)\): multiplication by a positive real scalar does not change a Cuntz ideal, the cuts of \(a\) exhaust its ideal, and \(G\) identifies ideals. For each requirement fix its function \(g\), its lower cutoff \(\eta\), and its source cutoff \(\varepsilon\). In particular, none of these cutoffs depends on the eventual averaging denominator.

At stage \(n\), apply 7 to the first \(n\) trace tests and first \(n\) lower requirements. Choose rational averages and fold them into \(Q\), obtaining a completely positive contraction \(h_n:S\to Q\). Given any prescribed sequence \(\gamma_n\downarrow0\), the trace errors can be made to satisfy, for \(i\leq n\), \[ \begin{split} |\tau(h_n(s_i))-t_i^{(1)}(\tau)| &\leq\gamma_n d_\tau(b_i),\\ |\tau(h_n(s_i^2))-t_i^{(2)}(\tau)| &\leq\gamma_n d_\tau(b_i),\\ |\tau(h_n(s_i)^2)-t_i^{(2)}(\tau)| &\leq\gamma_n d_\tau(b_i) \end{split} \tag{96}\] whenever \(d_\tau(b_i)<\infty\). Indeed, the uniform estimate on \(K_i\) rescales when this rank is nonzero; when it is zero, apply the cap estimate to every positive scalar multiple of \(\tau\) and let the scalar increase. All evaluations in that argument are finite, by the support bounds. Simultaneously choose the algebra norm errors below \(\varepsilon^2/2\) for each of the first \(n\) fixed lower requirements. 19 then supplies the corresponding lower rank inequalities with their fixed constants.

Localize the multiplicative defects. The coordinate maps define a completely positive contraction \(h:S\to D\). Put \(\Delta_{n,i}=h_n(s_i^2)-h_n(s_i)^2\geq0\) by Schwarz’s inequality. For \(t>0\), (96) gives \[ d_\tau((\Delta_{n,i}-t)_+) \leq\frac{2\gamma_n}{t}d_\tau(b_i). \tag{97}\] For zero controller rank this follows by the same rescaling argument; for infinite controller rank the upper bound is automatic. Thus \(h(s_i^2)-h(s_i)^2\in J(U_i)\). In \(D/J(U_i)\) the positive element \(s_i\) belongs to the multiplicative domain of the induced completely positive contraction. Hence both mixed defects with \(s_i\) belong to \(J(U_i)\), for every \(z\in S\).

For an arbitrary open \(U\), use the tests localized in basis ideals inside \(U\). Their linear spans are dense in \(S(U)\). Inclusion and closedness of the ideals \(J(V)\subseteq J(U)\) therefore give (79) for all \(s\in S(U)\). The local scheduling in 6 ensures the same conclusion when a controller is first selected in a smaller open: dominate it by a finite multiple of a fixed controller in the chosen larger open, and reduce the stage error before passing to the next row. These multiplicities never change an already fixed cut threshold or lower comparison.

Recover the square moments. Let \(\rho\) now be a regularized limit of coordinate weights \(\tau_n\), with \(\sigma=\rho|_Q\) densely finite on the chosen controlling ideal \(Q(U_i)\). Compact Cuntz support, and the larger controller fixed at the start, imply that \(d_{\tau_n}(b_i)\) is bounded along the defining ultrafilter. The constant restrictions converge to \(\sigma\) in the trace topology of 2. Bounded-rank continuity from 8, applied also after \(F\) using (78), gives \[\lim_\omega t_i^{(k)}(\tau_n)=t_i^{(k)}(\sigma),\qquad k=1,2.\] Consequently (96) gives convergence of the uncut square traces to \(t_i^{(2)}(\sigma)\), and a bounded first moment \(\lim_\omega\tau_n(h_n(s_i))<\infty\).

There is no loss of square mass on regularization. For \(x\geq0\) and \(t>0\), scalar functional calculus gives \[ 0\leq\tau(x^2)-\tau((x^2-t)_+) \leq\sqrt t\,\tau(x) \tag{98}\] whenever the right side is finite. Apply this with \(x=h_n(s_i)\), take the ultralimit, and then let \(t\downarrow0\). The definition of \(\rho\) and (98) yield exactly (80).

Retain the lower ideal support. Fix one lower datum \((y,\varepsilon,\eta)\) for the test \(s=f\otimes a\). For all sufficiently large \(n\), 19 gives \[d_\tau((y-\varepsilon)_+) \leq C\,d_\tau((h_n(s)-\eta)_+) \quad\text{for every }\tau\in T(Q), \qquad C=2\|y\|^2/\varepsilon^2.\] The order determination in 2 implies \[[(y-\varepsilon)_+]\leq \lceil C\rceil[(h_n(s)-\eta)_+].\] If \((y-\varepsilon)_+=0\), this comparison has no content and needs no witness. Otherwise \(C>0\); set \(N=\lceil C\rceil\geq1\). The uniform comparison implementation in 4 gives rectangular columns \(w_n\) such that \[\left\|w_n^*\operatorname{diag}_N(h_n(s))w_n-(y-\varepsilon)_+\right\| <1/n, \qquad \|w_n\|^2\leq\frac{\|y\|+1/n}{\eta}.\] This bound is independent even of \(N\). The columns pass to the sequence algebra and put \((y-\varepsilon)_+\) in \(\mathop{\mathrm{Ideal}}_D(h(s))\). Letting the source cut decrease to zero puts \(y\) itself in this ideal. The previously chosen \(y\)’s and their cuts generate \(Q(U_a)\), proving (81). ◻

Proof. Fix \(U\) and pass to \(D/J(U)\), writing bars for the quotient maps. For \(s\in S(U)\) and \(a,b\in S\), the mixed multiplication identities give \[\bar h(s)\bigl(\bar h(ab)-\bar h(a)\bar h(b)\bigr) =\bar h(sab)-\bar h(sa)\bar h(b)=0.\] The analogous right product is zero. More generally a word in \(\bar h(S)\) between \(\bar h(s)\) and a defect can be absorbed into the localized input one factor at a time. The resulting input is still in \(S(U)\). Polynomial approximation in \(E\) therefore proves (100) for the whole generated ideal \(I\).

The algebra \(E\), and hence \(I\), is separable. Apply 2 to the bounded bilinear map \((s,b)\mapsto h(s)b\), with \(b\in I\). Include its adjoints in the same countable family. This gives a locally small cone homomorphism \(\theta\) satisfying, simultaneously for every \(x\in S_+\), \[ h(S(U_x))I+Ih(S(U_x)) \subset\mathop{\mathrm{Ideal}}_{M_2(D)}(\theta(x)). \tag{102}\] The first-corner and second-corner placements are understood in this formula; the ideal on the right is the ambient matrix ideal.

Choose strictly positive elements \(e\in I_+\) and \(k\in\theta(S)_+\), omitting a zero summand when necessary, and set \[ H=\mathop{\mathrm{Her}}_{M_2(D)}(e\oplus k). \tag{103}\] It is \(\sigma\)-unital, since \(e\oplus k\) is strictly positive for \(H\). No separability assertion about \(H\) is needed. Since \(I\lhd E\), \(h(a)I+Ih(a)\subset I\) for \(a\in S\). Functional-calculus approximate units of \(I\) show that the first-corner map idealizes the first support. They also show that it preserves all the off-diagonal corners in (103): on the dense linear span of products \[f(e\oplus k)\,z\,g(e\oplus k),\qquad z\in M_2(D),\] left multiplication by \(h(a)\) replaces the first support factor by an element of \(I\), and right multiplication has the corresponding property. Approximation by the support approximate unit proves idealization of \(H\). The defects themselves lie in \(I\subset H\).

For \(a\in S(U_x)\), the same calculation, now using (102), puts \(h(a)H\) and \(Hh(a)\) in \(H\cap\mathop{\mathrm{Ideal}}_{M_2(D)}(\theta(x))\). The hereditary ideal correspondence gives \[H\cap\mathop{\mathrm{Ideal}}_{M_2(D)}(\theta(x)) =\mathop{\mathrm{Ideal}}_H(\theta(x)).\] This proves (101); equivalently, one may apply 13. All matrix placements preserve the local smallness of \(\theta\). ◻

Proof. Use the Hilbert-module Stinespring construction in (Kasparov 1980, proof of Theorem 3, pp. 140–141), applied to the unital extension \(d^+\) of \(d\) on the forced unitization of \(S\). The source is separable and \(H\) has a countable approximate unit, as required there. The resulting representation acts on the completion of the algebraic Stinespring module. Its adjointable isometry \(V:H\to\mathcal E\) is given by \(V\xi=1\otimes\xi\) and \(V^*(b\otimes\eta)=d^+(b)\eta\), so \(V^*\Pi(a)V=d(a)\). The module is countably generated: use a countable dense subset of the unitized source and a countable approximate unit of \(H\). Decompose \(\mathcal E=VH\oplus\mathcal E'\) and identify \(VH\) with \(H\).

For \(B(a)=(1-VV^*)\Pi(a)V\), \[ B(a)^*B(a)=d(a^*a)-d(a^*)d(a)\in H =\mathcal K_H(H). \tag{106}\] An adjointable operator whose square modulus is compact is compact, by functional calculus in the compact-operator ideal. Thus \(B(a)\) is compact, and so is \(C(a)=B(a^*)^*\). Kasparov stabilization (Kasparov 1980, Theorem 2) of \(\mathcal E'\) after adding a zero representation identifies its enlargement with the standard module. This leaves \(VH\) fixed and preserves compactness and the indicated diagonal compressions.

We check the ideal assertion before using the homotopy. For \(a\in S(U_x)\), coefficients on the dense Stinespring generators have the form \[ \langle b\otimes\xi,\Pi(a)(c\otimes\eta)\rangle =\xi^*d(b^*ac)\eta\in H_x, \tag{107}\] where \(b,c\) belong to the forced unitization. The containment uses \(b^*ac\in S(U_x)\) and (101). It extends to all vectors. Passing to the quotient module by \(\mathcal E H_x\) shows that \(\Pi(a)\mathcal E\subset\mathcal E H_x\); the same holds for the adjoint. Stabilization and module unitaries preserve these ideal submodules.

Now take \(a\in S(O)\) and put \(x=|a|\). Scaling does not enlarge its projected support: \[ \alpha_t(a)\in S(U_x)\quad(0\le t\le1). \tag{108}\] Indeed a point quotient in the \(P\) coordinate which kills \(a\) also kills every rescaling of \(a\). Since \(x\in S(O)\), \(H_x\subset H(O)\), and (107) applied to \(\alpha_t(a)\) proves weak equivariance of \(\Pi_t\). It proves the same assertion for the compressions. Source ideals and quotients are nuclear, so these coefficient conditions also give weak nuclearity, as in 5.

This argument does not assert that \(\alpha_t\) preserves \(S(O)\). The stronger estimate for the entire projected ideal \(S(U_x)\) is the reason that the scaled representation has the required fine ideal bounds. ◻

Proof. Absorb a finite chain of dilations. Include \(\mathcal F^*\) and all products from \(\mathcal F\) in the finite list on which estimates will be imposed. By norm continuity of \(t\mapsto\alpha_t(a)\), choose \[0=t_0<t_1<\cdots<t_k=1\] so that adjacent values differ in norm by less than \(\eta\) on this enlarged list. Write \(d_i=d_{t_i}\), \(d'_i=d'_{t_i}\) and \(\Pi_i=\Pi_{t_i}\), using the same first module \(H\) and the same complementary module in all these dilations.

For the absorption application put \(\widehat\theta(a)=\theta(a)\otimes e_{11}\in H\otimes\mathcal K\), with target labels \(H(O)\otimes\mathcal K\). If \(O_a\) is the fine support of \(a\in S_+\), then \(S(O_a)=\mathop{\mathrm{Ideal}}_S(a)\) and \[\mathop{\mathrm{Ideal}}_H(\theta(S(O_a)))=\mathop{\mathrm{Ideal}}_H(\theta(a)).\] Thus \(\widehat\theta\) is full for these fine labels, independently of any coarse fullness assertion. Apply 5 to a literal stable infinite repeat \(\Theta\) of \(\widehat\theta\) and the finite sum of the \(\Pi_i\). On the standard Hilbert \(H\)-module, this repeat can be reindexed as \[ \Theta(a)=\operatorname{diag} (\theta(a),0,\theta(a),0,\ldots). \tag{113}\] Indeed each repeated copy of \(\widehat\theta\) acts by \(\theta\) on one \(H\) slot and by zero on the other slots of its standard module. Both the active slots and the remaining zero slots are countably infinite, giving (113) by a fixed permutation. We retain every zero slot throughout absorption and finite compression. The bare diagonal repeat with all these zero slots removed need not absorb a zero representation. The hypotheses hold: the tightly labeled source is separable, exact and lower semicontinuous; \(H\otimes\mathcal K\) is \(\sigma\)-unital and stable; and 21 supplies weak nuclearity of the representations being absorbed. Absorption identifies the repeat module with a module carrying \[\Theta\oplus\Pi_1\oplus\cdots\oplus\Pi_k\] so that its difference from the original repeat is compact on every input. More precisely, absorption in (Gabe 2024, Definition 5.4) gives a repeat-side module unitary \(U\) with \[U^*(\Theta\oplus\textstyle\bigoplus_i\Pi_i)(a)U-\Theta(a) \in\mathcal K_H(H^{\oplus\infty})\qquad(a\in S);\] the compact membership holds for every input for each chosen unitary, in addition to the finite-test norm approximation. Only this compact-membership conclusion from absorption is needed below. Its finite-test approximation error does not enter our bound: after the unitary identification we keep the absorbed representation itself. All these identifications act on the repeat side; the initial module carrying \(d=d_k\) is fixed.

Shift the first summands along the chain. Denote the initial copy of \(H\) by \(H_0\), the first copy in \(\Pi_i\) by \(H_i\), and its complement by \(\mathcal E'_i\). Delete the compact off-diagonals of the absorbed \(\Pi_i\). Apart from the unused repeat, the diagonal summands are \[ d_k\big|_{H_0}\ \oplus \bigoplus_{i=1}^k \left(d_i\big|_{H_i}\oplus d'_i\big|_{\mathcal E'_i}\right). \tag{114}\] Pair \(H_0\) with \(\mathcal E'_k\), and, for \(i<k\), pair \(H_{i+1}\) with \(\mathcal E'_i\). The unused first module is \(H_1\). Restore the compact off-diagonals of \(\Pi_i\) in the pair whose second component is \(\mathcal E'_i\). In this ordered decomposition the resulting map is \[ T'=\Theta\oplus d_1\oplus \bigoplus_{i=1}^{k-1} \left[\Pi_i+ \begin{pmatrix}d_{i+1}-d_i&0\\0&0\end{pmatrix}\right] \oplus\Pi_k. \tag{115}\] It is a compact perturbation of the map in (114), including its repeat: the diagonal entries were only permuted, and the restored entries are compact. The representation \[ R'=\Theta\oplus0\oplus \bigoplus_{i=1}^{k-1}\Pi_i\oplus\Pi_k \tag{116}\] is within \(\eta\) of \(T'\) on the enlarged finite list. Indeed \(\|d_1(a)\|\le\|\alpha_{t_1}(a)\|<\eta\) there, since \(\alpha_0=0\), and every other difference in (115) has norm less than \(\eta\). No compactness of \(d_{i+1}(a)-d_i(a)\) has been used.

Pull both maps back through the block permutation and the absorption identification to the original module \[\mathcal L=H\oplus H^{\oplus\infty}.\] Writing \(T_0=d\oplus\Theta\), we obtain \[ T=T_0+K,\qquad K(a)\in\mathcal K_H(\mathcal L),\qquad \|T(a)-R(a)\|<\eta \tag{117}\] on the enlarged list, where \(R\) is a representation. The maps in this construction are bounded linear and \(*\)-preserving. They are weakly equivariant: the block compressions and off-diagonal entries have the ideal bounds of 21, and any Hilbert-module unitary preserves the submodule associated to an ideal. Thus (109) applies to \(K\). For \(a,b\in\mathcal F\), comparison with the contractive representation \(R\) gives \[ \|T(a)\|\le\|a\|+\eta, \qquad \|T(ab)-T(a)T(b)\|\le3\eta+\eta^2. \tag{118}\]

Compress the compact correction. Let \(E_N\) be the projection onto the initial copy of \(H\) and the first \(N\) slots in the original diagonal repeat (113), including both active and zero slots. It commutes with \(T_0(a)\) for all \(a\). The projections \(E_N\) tend strictly to the identity on the compact operators. Choose \(N\) so that both tails of every \(K(a)\) on the enlarged list have norm less than \(\eta\). Then \[\begin{align*} &E_NT(ab)E_N-E_NT(a)E_NT(b)E_N\\ &\quad=E_N\bigl(T(ab)-T(a)T(b)\bigr)E_N +E_NK(a)(1-E_N)K(b)E_N. \tag{119}\end{align*}\] The second term has norm at most \(\eta^2\). Compression preserves the norm estimate in (118).

The entries of \(E_NK(a)E_N\) lie in \(H\), so the compressed map has an actual ambient realization with diagonal baseline \(h\) followed by the retained \(\theta\) and zero slots. To obtain exactly (110), let \(Z_N(a)\) be the finite diagonal matrix which places \(\theta(a)\) in each retained zero repeat slot and zero in every other slot, including the initial one. Define \[ K_N(a)=E_NK(a)E_N-Z_N(a). \tag{120}\] Replacing each zero baseline slot by \(\theta(a)\) and subtracting it in (120) leaves the compressed map \(L\) unchanged. The map \(Z_N\) is bounded linear and \(*\)-preserving, has entries in \(H\), and is locally in \(J(U)\) because \(\theta\) is locally small. Thus \(K_N\) has the required compact and local properties. This rebasing is performed only after finite compression; the tail estimate used the original \(T_0\) with all its zero slots.

Return to the ambient sequence algebra. To transfer the multiplier estimates, let \(p_H\) be the support projection of \(H\) in \(M_2(D)^{**}\). Idealization implies that \(h(a)\) commutes with \(p_H\). The correction entries, and the values of \(\theta\), are supported on \(p_H\). On its matrix amplification the ambient map is exactly the compressed multiplier map just estimated. On the orthogonal complement all corrections vanish; the remaining first-corner map \(h\) is contractive and multiplicative there, since its defects lie in \(I\subset H\). Taking the maximum of these two reducing parts proves (111) in the ambient norm. Finally (112) follows from the local compact ideal statement for \(K\). ◻

Proof. If the defect ideal is zero, \(h\) is already a homomorphism. Otherwise use the preceding construction. Nuclearity and separability of \(S\) provide a c.p.c. coordinate lift of \(h\) by (Choi and Effros 1976, Theorem 3.10); fix one such lift. Fix a countable rational \(*\)-subalgebra dense in \(S\), together with countable dense local tests in \(S(V)\) for a compact-containment basis and all finite unions of its members. Include rational linear combinations, products, adjoints, and positive cuts needed for the algebraic and ideal tests. Fix \(\eta_j\downarrow0\). At stage \(j\) rescale the first \(j\) tests and their relevant products into the unit ball. Apply 22 with a tolerance small enough that, after rescaling back, its norm and multiplicativity errors on these tests are at most \(\eta_j\). Denote the resulting map by \(L_j\) and its finite repeat size by \(N_j\). In particular, \(\|L_j(a)\|\leq\|a\|+\eta_j\) for every current input \(a\).

Write \(h^{\mathrm{pad},j}=h\oplus0^{\oplus N_j}\) in the current row, and subtract it from \(L_j\). The remaining terms are its compact correction and the \(N_j\) small copies of \(\theta\). They belong to the appropriate matrix ideal \(J(V)\) on every tested input in \(S(V)\). On each fixed input \(a\), these row differences also satisfy \[\|L_j(a)-h^{\mathrm{pad},j}(a)\| \leq2\|a\|+\eta_j\] for all sufficiently large \(j\). Thus they meet the boundedness hypothesis for prescribed errors in 6; no uniform bound on the global norms of the maps \(L_j\) is needed. For each fixed positive cut tolerance their row estimates therefore have the form \[ d_\tau\bigl((|z_{j,n}|-\varepsilon)_+\bigr) \le r_{j,n}\,d_\tau(c_j),\qquad r_{j,n}\longrightarrow0\quad(n\to\infty), \tag{122}\] where \(c_j\) is compactly Cuntz supported in \(Q(V)\) and the bound is uniform in \(\tau\).

For each scheduled pair \(V\Subset U\), first fix a cut-positive \(b_{V,U}\in Q(U)\) whose ideal contains \(Q(V)\). Compact support gives \([c_j]\le M_j[b_{V,U}]\) for a finite integer \(M_j\). At stage \(j\), wait in its coordinate row until \(M_jr_{j,n}\le 1/j\) for all the finitely many current tests and cut tolerances. Wait also until their coordinate norm, involution and product estimates agree with the rescaled finite-test bounds, with an additional error tending to zero. The integers \(M_j\) and \(N_j\) are chosen before this waiting time; neither needs a bound independent of \(j\). This is the diagonal procedure of 6.

Choose coordinate representatives linearly on each finite rational span, preserving the representatives already needed for its displayed algebraic tests. The preceding eventual estimates allow successive rows to be joined. Matrix folding in 2 is applied only after these choices. Its corner identification is adjusted to be within \(1/j\) of the identity on the first \(j\) constants of a fixed dense subset of \(Q\). The resulting corner map on the sequence algebra fixes \(Q\) pointwise. Apply the corner embeddings to the fixed coordinate lift of \(h\) to define \(h^{\mathrm{pad}}\).

The joined corrected representatives define a contractive, linear, \(*\)-preserving, multiplicative map on the dense rational \(*\)-subalgebra: each asserted identity and norm bound is one of the eventually imposed tests. It extends uniquely to a homomorphism \(\lambda:S\to D\). The joined estimates (122), now against the fixed \(b_{V,U}\), put its difference from \(h^{\mathrm{pad}}\) in \(J(U)\) on the tests from \(S(V)\). Both resulting maps are continuous. Since the ideals \(S(V)\) with \(V\Subset U\) span densely in \(S(U)\) and \(J(U)\) is closed, this proves (121) for every open \(U\).

For completeness, if \(\iota_n:Q\to Q\) are the coordinate corner embeddings, they induce a homomorphism \(\iota:D\to D\) with \(\iota|_Q=\mathop{\mathrm{id}}_Q\) and \(h^{\mathrm{pad}}=\iota h\). The pullback of a regularized trace along \(\iota\) is the regularized trace of the coordinate weights \(\tau_n\iota_n\), with the same restriction \(\sigma\) to \(Q\). Thus every prescribed formula for \(h\) of this kind is preserved. Applying \(\iota\) to an ideal comparison involving constants also preserves that comparison and those constants. ◻

Proof. Write \(\omega_1,\omega_2\) for the traces, and \(\omega\) for either one when making an estimate. Finite second moments give finite ranks on all positive cuts of a test \(s\): for \(\varepsilon>0\), \[d_\omega((s-\varepsilon)_+)\le\varepsilon^{-2}\omega(s^2).\] The preceding observation proves dense finiteness for both traces. Fix \(s\) and a continuous function \(g\) on its squared spectrum which vanishes near zero. Approximate \(g(t)/t\) uniformly by polynomials. Equality of the moments \(\omega(s^{2k})\) therefore gives equality on \(g(s^2)\): the error is bounded by the uniform polynomial error times \(\omega(s^2)\). Increasing such functions to spectral support indicators gives equality of the ranks of every positive cut of \(s^2\).

The test supply contains cuts from dense sets of the two factors, so these squares approximate every positive elementary tensor in \(S(V)\). The comparison \[\|a-b\|<\delta \quad\Longrightarrow\quad (a-t-\delta)_+\precsim(b-t)_+ \qquad(t\ge0)\] transfers equality of all cut ranks to each such elementary limit, by allowing the cut slack to decrease to zero. Spectral integration now gives equality of trace values on all positive elementary tensors.

To pass from elementary tensors to the whole algebra, let \(L\) be a relatively compact open subset of \((0,1]\) and let \(H_c=\mathop{\mathrm{Her}}(c)\subset P(V)\), where \(c\) is compactly Cuntz supported in \(P(V)\). Choose \(f_0\in C_c((0,1])_+\) equal to one on \(\overline L\), and write \(c\precsim(b-\delta)_+\) with \(b\in P(V)_+\) and \(\delta>0\). Then \[f_0\otimes c\precsim(f_1\otimes b-\delta/2)_+\] for a compactly supported \(f_1\) equal to one on \(\mathop{\mathrm{supp}}(f_0)\). The larger tensor belongs to the common finite ideal \(S(V)\) already established above. Thus \(f_0\otimes c\) has finite rank for both traces. The support of \(C_0(L)\otimes H_c\) is below its support, so both traces are bounded on this localized algebra, with norms bounded by that rank. A positive continuous \(P\)-valued function there is uniformly approximated by finite positive elementary sums, using a scalar partition of unity. Additivity and boundedness yield equality on this localized algebra. Finally take increasing scalar cutoffs and an increasing cut-supported approximate unit of \(P(V)\). Traciality writes the corresponding compressions as increasing positive approximations under the square root of the element being tested. Lower semicontinuity extends the equality to every positive element of \(S(V)\). ◻

Proof. We first preserve the lower ideal signal through correction. For a fixed weight we will then identify its trace on the prescribed finite ideal, and use that signal to rule out any larger finite ideal.

Preserve the lower signal. Take \(h\) from 6 and correct it by 8. For each controlled positive test \(s=f\otimes a\), put \(\delta(s)=\lambda(s)-h^{\mathrm{pad}}(s)\). It belongs to \(J(U_s)\). Apply the cone version of 3 once more to obtain a locally small homomorphism \(\theta_1:S\to D\) with \[ \delta(s)\in\mathop{\mathrm{Ideal}}_D(\theta_1(s)) \tag{124}\] for every such test. The countable coefficient form permits all the required cuts and localized families simultaneously. Set \(p=\lambda\oplus\theta_1\) and fold the two corners, again fixing the constants. In the following comparisons we retain matrix notation for this folding.

The ideal of \(p(s)\) contains both \(\lambda(s)\) and \(\theta_1(s)\), hence contains \(h^{\mathrm{pad}}(s)\) by (124). The lower signal of 6 and the preservation of constants give \[ Q(U_a)\subset\mathop{\mathrm{Ideal}}_D(p(f\otimes a)) \qquad(f\ne0) \tag{125}\] for each controlled test. Also \[ (p-h^{\mathrm{pad}})(S(U))\subset J(U), \tag{126}\] where zero padding of \(h^{\mathrm{pad}}\) is understood.

Identify the trace on its prescribed finite ideal. Fix a regularized trace \(\rho\) and write \(Q(V)=\mathop{\mathrm{Fin}}(\sigma)\). By 1, \(\mathop{\mathrm{Fin}}(F\sigma)=P(V)\). Put \[\omega=\rho\circ p,\qquad \nu=m\otimes F\sigma.\] The trace \(\rho\) vanishes on \(J(V)\) by 4. It therefore factors through the quotient by this zero ideal. For each controlled \(s=f\otimes a\) localized with its controller in \(V\), 6 and (126) give \[ \omega(s^2) =\rho\bigl(h^{\mathrm{pad}}(s)^2\bigr) =\int f^2\,dm\, F\sigma(a^2) =\nu(s^2)<\infty. \tag{127}\] Apply the same formula to the positive powers in the supply. The cuts of these localized elementary tests generate \(S(V)\), including when \(V\) is obtained as an increasing union of the basis opens. Thus 23 proves \[ S(V)\subset\mathop{\mathrm{Fin}}(\omega),\qquad \omega|_{S(V)}=\nu|_{S(V)}. \tag{128}\]

Exclude an additional finite region. We show that the first inclusion is equality. Suppose the open set corresponding to \(\mathop{\mathrm{Fin}}(\omega)\) contains a rectangle \(L\times U\) not contained in \((0,1]\times V\). Use a scalar test with compact support in \(L\) and a cut positive \(a\) from the basis supply in \(P(U)\). The tests may be chosen with slightly enlarged supports still in this rectangle. Their tensor products are then compactly Cuntz supported in \(\mathop{\mathrm{Fin}}(\omega)\), and consequently have finite \(\omega\)-rank. For any such nonzero scalar test, write \(s=f\otimes a\). If \(c\in Q(U_a)_+\) and \(\varepsilon>0\), (125) gives, for suitable finite \(M\) and \(\eta>0\), \[ [(c-\varepsilon)_+]\le M[(p(s)-\eta)_+] \quad\hbox{in }\mathop{\mathrm{Cu}}(D). \tag{129}\] It follows that \[d_\sigma((c-\varepsilon)_+) =d_\rho((c-\varepsilon)_+) \le M d_\omega(s)<\infty.\] Hence \(\sigma\) is densely finite on \(Q(U_a)\), so \(U_a\subset V\). The supports \(U_a\) of these cut tests exhaust \(U\). This implies \(U\subset V\), contradicting the choice of the rectangle. Therefore \[ \mathop{\mathrm{Fin}}(\omega)=S(V)=\mathop{\mathrm{Fin}}(\nu). \tag{130}\] Both traces take value infinity on positive elements outside this ideal. Together with (128), this proves (123). The argument also applies when \(V\) is empty: there is no finite region to identify, and the retained signal excludes every nonzero additional finite ideal. ◻

Proof. For \(j\geq2\), put \(I_j=(-j,j]\) and give it Lebesgue measure. This is a finite full-support cone measure, so 7, transported along a homeomorphism \((0,1]\cong I_j\), gives a model \(\lambda^{(j)}:C_0(I_j)\otimes P\to D\). Choose \(0\leq\eta_j\leq1\) in \(C_c((-j,j))\), equal to one on \([-j+1,j-1]\). The maps \[R_j:C_0(\mathbb R)\otimes P\longrightarrow C_0(I_j)\otimes P, \qquad R_j(s)=\eta_j\,s|_{I_j},\] are c.p.c.: restriction followed by multiplication by the scalar positive contraction \(\eta_j\) is completely positive and contractive, and its values vanish at the omitted left endpoint. The domain \(C_0(I_j)\otimes P\) of each \(\lambda^{(j)}\) is separable and nuclear. The maps \(R_j\) are exactly multiplicative on every fixed finite set of compactly supported inputs once \(j\) is large enough to contain their supports in the region where \(\eta_j=1\).

Take c.p.c. coordinate lifts of each \(\lambda^{(j)}\), by (Choi and Effros 1976, Theorem 3.10), and compose with \(R_j\). Apply 6 to an increasing dense list of compactly supported algebra tests and the elementary rank tests of 3. Here the tests are fixed on the line before increasing \(j\): each function, its positive trims, and its enlarged compact support all lie inside \((-j+1,j-1)\) in every sufficiently late row. Their Lebesgue masses are therefore unchanged by \(R_j\). For every sandwich test choose the strict scalar slack, the enlarged source, and the trimmed target before selecting coordinates in that row. The constant Cuntz representatives in \(Q\) are the same for all these rows. The fixed target tolerances give the bounded comparison witnesses of 4; thus the row diagonal retains the comparisons in \(D\), even when the row thresholds vary. Choose the coordinates for the first \(j\) tests only after these choices, with algebra errors at most \(j^{-1}\). The resulting c.p.c. sequence \(\lambda_n\) is asymptotically multiplicative. Its quotient \(\lambda\) satisfies the elementary rank identities for every regularized limit, and 3 proves the asserted model identity.

Lebesgue measure is translation invariant, including the extended product convention. Hence \(\lambda\) and \(\lambda\circ\alpha\) are models for the same data. Apply 4 to their specified coordinate lifts to obtain (131). ◻

Proof. Choose \(\eta\in C_c(\mathbb R)_+\) strictly positive on \([0,1]\), and set \[H(t)=\sum_{n\in\mathbb Z}\eta(t-n)^2,\qquad \zeta(t)=H(t)^{-1/2}\eta(t).\] The sum defining \(H\) is locally finite, positive, continuous, and one-periodic. The partition identity for \(\zeta\) follows immediately. Integrating that identity over \([0,1]\) gives the asserted normalization. Put \[ p=\sum_{n\in\mathbb Z}p_n v^n, \qquad p_n(t)=\zeta(t)\zeta(t-n). \tag{134}\] Only finitely many coefficients are nonzero. The involution formula gives \(\alpha_n(p_{-n})=p_n\), so \(p^*=p\). Convolution gives, for every \(k\in\mathbb Z\), \[(p^2)_k(t) =\zeta(t)\zeta(t-k)\sum_{n\in\mathbb Z}\zeta(t-n)^2 =p_k(t).\] Thus \(p\) is a projection.

Choose finitely many real functions \(q_i\in C_c(\mathbb R)\), each supported in an interval of diameter less than one, such that \[q:=\sum_iq_i^2=1\quad\hbox{on }\mathop{\mathrm{supp}}(\zeta).\] For completeness, choose bump functions \(r_i\) in such intervals whose squared sum is positive near \(\mathop{\mathrm{supp}}(\zeta)\), and a cutoff \(\kappa\) equal to one on that support and supported where this sum is positive. Then \(q_i=\kappa r_i/(\sum_jr_j^2)^{1/2}\), extended by zero, have the required properties. The coefficient formula for \(p\) yields \[ qp=p=pq,\qquad \sum_iq_i p q_i=\zeta^2. \tag{135}\] To verify the second identity, its \(n\)-th coefficient is \[\zeta(t)\zeta(t-n)\sum_iq_i(t)q_i(t-n).\] This vanishes when \(n\ne0\), because each support has diameter less than one, and equals \(\zeta(t)^2\) when \(n=0\).

For \(a\in(R_0)_+\), set \(y_i=q_i p\otimes a^{1/2}\). These are elements of \(E\otimes R_0\), and (135) gives the exact positive identities \[\sum_i y_i^*y_i=p\otimes a,\qquad \sum_i y_i y_i^*=\zeta^2\otimes a.\] Additivity and \(\omega(y_i^*y_i)=\omega(y_i y_i^*)\) prove (133). There are only finitely many positive summands. In particular, this calculation uses no subtraction of infinite values and requires no finiteness assumption on \(\omega\). For the Lebesgue trace assertion, the extended product formula on the right has value \((\int\zeta^2)\nu(a)=\nu(a)\).

We also verify fullness and the corner description. Equation (135) puts \(\zeta^2\) in \(\mathop{\mathrm{Ideal}}_E(p)\). Conjugation by \(v^n\) preserves this ideal, so it contains every \(\alpha_n(\zeta^2)\). For \(f\in C_c(\mathbb R)\), the partition identity expresses \[f=\sum_{n\in\mathbb Z}f\,\alpha_n(\zeta^2)\] as a finite sum of elements of that ideal. It therefore contains \(C_0(\mathbb R)\), and then the dense finite Fourier sums in \(E\). Hence \(p\) is full.

For \(f\in C_c(\mathbb R)\) and \(n\in\mathbb Z\), direct convolution gives \[ p(fv^n)p=g_{f,n}p,\qquad g_{f,n}(t)=\sum_{k\in\mathbb Z} \zeta(t-k)f(t-k)\zeta(t-k-n). \tag{136}\] The function \(g_{f,n}\) is continuous and one-periodic. Conversely, for a continuous one-periodic \(g\), the element \(gp\) belongs to \(E\), since all its Fourier coefficients are compactly supported. Periodicity makes \(g\) a central multiplier of \(E\), so \(gp\in pEp\), and the map \(g\mapsto gp\) is a unital homomorphism \(C(\mathbb R/\mathbb Z)\to pEp\). It is injective: its zeroth coefficient is \(g\zeta^2\), and every orbit has a point where \(\zeta\ne0\). Equation (136) and density of compactly supported Fourier sums prove surjectivity. ◻

Proof. Use (132) and 24 to define \(\varphi(a)=\Psi(p\otimes a)\). For a regularized limit \(\rho\), the extended trace \(\omega=\rho\circ\Psi\) restricts on \(C_0(\mathbb R)\otimes P\) to Lebesgue measure tensored with \(F(\sigma)\), where \(\sigma=\rho|_Q\). Thus, for every \(a\in P_+\), \[\rho(\varphi(a)) =\omega(p\otimes a) =\omega(\zeta^2\otimes a) =F(\sigma)(a).\] The positive partition calculation applies equally when this common value is infinite. ◻

Proof. Take c.p.c. lifts \(\varphi_n:P\to Q\) of the point model in 3, using nuclearity and (Choi and Effros 1976, Theorem 3.10). For every finite \(\mathcal E\subseteq P\) and every \(\varepsilon>0\), there is \(N\) such that \[ \inf_{w\in\mathcal U(M(Q))} \max_{a\in\mathcal E} \|\mathop{\mathrm{Ad}}(w)(\varphi_m(a))-\varphi_n(a)\|<\varepsilon \qquad(m,n\geq N). \tag{137}\] If this failed, choose pairs \(m_k,n_k\to\infty\), with both index sequences strictly increasing, for which the displayed infimum is at least \(\varepsilon\). Every subsequence of a model is a model for the same data: given coordinate traces \(\tau_k\) on indices \(n_k\) and a free ultrafilter \(\omega\) on \(k\), set \(\widetilde\tau_{n_k}=\tau_k\) and use the zero trace at the remaining coordinates. Define \(A\in\widetilde\omega\) if and only if \(\{k:n_k\in A\}\in\omega\). Then \(\widetilde\omega\) is a free ultrafilter concentrated on the chosen indices, and every cut ultralimit agrees with the corresponding subsequence ultralimit. Thus \((\varphi_{m_k})\) and \((\varphi_{n_k})\) define two point models. 4 contradicts the choice of the pairs.

Let \(\mathcal E_k\) be increasing finite sets with dense union in the unit ball of \(P\). Use (137) to choose an increasing subsequence \(n_k\) and \(w_k\in\mathcal U(M(Q))\) with \[\max_{a\in\mathcal E_k} \|\mathop{\mathrm{Ad}}(w_k)(\varphi_{n_{k+1}}(a)) -\varphi_{n_k}(a)\|<2^{-k}.\] Choose \(n_k\) beyond the threshold for the \(k\)-th test, so that both indices in this inequality are eligible. Starting with \(W_1=1\), set \(W_{k+1}=W_k w_k\) and \(\psi_k=\mathop{\mathrm{Ad}}(W_k)\circ\varphi_{n_k}\). The sequence \(\psi_k(a)\) is Cauchy for every \(a\) in the dense union, with summable successive errors, and then for every \(a\in P\), by contractivity. Its limit \(\phi:P\to Q\) is c.p.c. and multiplicative, since the multiplicative defects of the original coordinate lifts tend to zero. It is therefore a homomorphism.

We check the invariant without assuming continuity of unbounded trace evaluation under norm convergence. Fix \(\tau\in T(Q)\), and let \(\rho_\tau\) be its regularized constant-coordinate limit. The maps \(\psi_k\) and the constant map \(\phi\) give the same homomorphism into \(Q_\infty\). For every \(a\in P_+\) and every cut tolerance, coordinate unitary invariance gives \[\tau((\psi_k(a)-\varepsilon)_+) =\tau((\varphi_{n_k}(a)-\varepsilon)_+).\] Regularization and the subsequence model identity therefore imply \[\tau(\phi(a)) =\rho_\tau((\psi_k(a))_k) =F(\tau)(a).\] Here \(\rho_\tau|_Q=\tau\) follows from 2. Applying the equality to positive powers gives \[d_\tau(\phi(a))=d_{F(\tau)}(a) =d_\tau(G[a]).\] This calculation also applies to finite matrices. If \(x=(x_{ij})\in M_k(P)_+\), the unnormalized matrix traces give \[\begin{align*} (\tau\otimes\mathop{\mathrm{Tr}}_k)(\phi^{(k)}(x)) &=\sum_i\tau(\phi(x_{ii})) =\sum_i F(\tau)(x_{ii})\\ &=(F(\tau)\otimes\mathop{\mathrm{Tr}}_k)(x), \end{align*}\] including infinite values. Apply this identity to positive powers and then use positive cuts of finite-matrix approximations in \(P\otimes\mathcal K\). Pointwise trace determination in \(\mathop{\mathrm{Cu}}(Q)\) gives \(\mathop{\mathrm{Cu}}(\phi)=G\). Since \(G\) is injective, \(\phi(a)=0\) for positive \(a\) implies \([a]=0\), hence \(a=0\); thus \(\phi\) is injective. Finally, for any regularized limit \(\rho\), put \(\sigma=\rho|_Q\). Its value on the constant map is \[\rho\circ\phi=\sigma\circ\phi=F(\sigma).\] So the constant map is itself a point model. ◻

Proof. Apply 10 to \(F\) and to \(F^{-1}\). This gives homomorphisms \(\phi:P\to Q\) and \(\psi:Q\to P\) with inverse trace and Cuntz data. Their compositions have identity trace data. The constant maps \(\psi\phi\) and \(\mathop{\mathrm{id}}_P\) are therefore point models for the same data, and similarly for \(\phi\psi\) and \(\mathop{\mathrm{id}}_Q\). By 4, the compositions can be conjugated arbitrarily close to the corresponding identity on each finite set. We give the intertwining construction to ensure that the conclusion is an isomorphism in \(P,Q\).

Fix increasing finite sets \(\mathcal P_n,\mathcal Q_n\) with dense unions in the unit balls, and put \(\varepsilon_n=2^{-n}\). Start with \(f_1=\phi\), \(g_0=\psi\), and empty \(\mathcal E_0,\mathcal F_0\). Suppose \(f_n\) and \(g_{n-1}\) have been chosen. Choose a finite set \[\mathcal E_n\supseteq \mathcal E_{n-1}\cup\mathcal P_n \cup g_{n-1}(\mathcal F_{n-1}).\] The composition \(g_{n-1}f_n\) still has identity trace data, because every adjustment made below is unitary conjugacy. Choose \(v_n\in\mathcal U(M(P))\), and set \(g_n=\mathop{\mathrm{Ad}}(v_n)\circ g_{n-1}\), so that \[ \|g_nf_n(a)-a\|<\varepsilon_n \qquad(a\in\mathcal E_n). \tag{138}\] Next choose a finite set \[\mathcal F_n\supseteq \mathcal F_{n-1}\cup\mathcal Q_n\cup f_n(\mathcal E_n).\] Choose \(z_n\in\mathcal U(M(Q))\), and set \(f_{n+1}=\mathop{\mathrm{Ad}}(z_n)\circ f_n\), with \[ \|f_{n+1}g_n(b)-b\|<\varepsilon_n \qquad(b\in\mathcal F_n). \tag{139}\] This second adjustment is possible by the same identity-data argument for \(f_ng_n\).

The inclusions of the image tests make these sequences Cauchy. If \(a\in\mathcal E_n\), contractivity and (138)–(139) give \[\begin{align*} \|f_{n+1}(a)-f_n(a)\| &\leq\|f_{n+1}(a-g_nf_n(a))\|\\ &\quad+\|f_{n+1}g_n(f_n(a))-f_n(a)\| <2\varepsilon_n. \end{align*}\] If \(n\geq2\) and \(b\in\mathcal F_{n-1}\), then \(g_{n-1}(b)\in\mathcal E_n\), and \[\begin{align*} \|g_n(b)-g_{n-1}(b)\| &\leq\|g_n(b-f_ng_{n-1}(b))\|\\ &\quad+\|g_nf_n(g_{n-1}(b))-g_{n-1}(b)\| <\varepsilon_{n-1}+\varepsilon_n. \end{align*}\] These summable bounds and density give point-norm limits \(f_\infty:P\to Q\) and \(g_\infty:Q\to P\), both homomorphisms. Equations (138) and (139) imply \[g_\infty f_\infty=\mathop{\mathrm{id}}_P,\qquad f_\infty g_\infty=\mathop{\mathrm{id}}_Q.\] For example, contractivity controls the moving input \(f_n(a)\) by its limit \(f_\infty(a)\), after which point-norm convergence of \(g_n\) applies to the fixed limit input. Density gives these equalities on the whole algebras. Thus \(f_\infty\) is the required isomorphism.

The isomorphism realizes the prescribed trace data as well. Each \(f_n\) has Cuntz data \(G\), since the changes to \(\phi\) are unitary conjugacies. For \(a\in P_+\) and \(\varepsilon>0\), choose \(n\) with \(\|f_n(a)-f_\infty(a)\|<\varepsilon\). Positive cut comparison gives, for every \(\tau\in T(Q)\), \[d_{F\tau}((a-\varepsilon)_+)\leq d_\tau(f_\infty(a)),\qquad d_\tau((f_\infty(a)-\varepsilon)_+)\leq d_{F\tau}(a).\] Taking the supremum over cuts proves equality of ranks; spectral integration gives \(\tau\circ f_\infty=F(\tau)\), including infinite values. This argument uses no continuity of an unbounded weight on an uncut norm-convergent sequence. The matrix argument in 10 also gives \(\mathop{\mathrm{Cu}}(f_\infty)=G\). Set \(\Phi=f_\infty\). Substituting \(P=A\otimes\mathcal W\otimes\mathcal K\) and \(Q=B\otimes\mathcal W\otimes\mathcal K\) proves 1. ◻