Proof. Choose an approximate unit \((e_l)\) for \(D\) consisting of positive contractive continuous functions of \(P\) which vanish at zero. Such elements satisfy \(d_\tau(e_l)\leq d_\tau(P)\). For any positive contraction \(z\in D\), the elements \(e_lze_l\) converge to \(z\) in norm, and \[\tau(e_lze_l)\leq\tau(e_l^2)\leq d_\tau(P).\] Lower semicontinuity therefore gives \(\tau(z)\leq d_\tau(P)\). Monotonicity and additivity of \(d_\tau\) on Cuntz classes give the second inequality in (7). Thus \(\tau\) is bounded on the positive unit ball of \(D\) and extends to the asserted bounded positive functional. Its norm bound proves (8). This argument also applies when \(d_\tau(b)=0\), in which case the restriction is identically zero. ◻

Proof. Write \(A_\infty=\ell^\infty(A)/c_0(A)\), identifying \(A\) with its constant sequences. There are commuting positive contractions \(u_1,\ldots,u_m\in A_\infty\cap A'\) such that \(u_j\) is represented by positive contractions in \(I_j\) and \(u_jx_j=x_j\). Indeed, at coordinate \(i\) choose successively an element of a quasicentral approximate unit of each \(I_j\). Require its commutators with the first \(i\) elements of a fixed dense subset of \(A\) to have norm less than \(1/i\), require that it act on \(x_j\) within \(1/i\), and require that it commute within \(1/i\) with the elements already chosen at that coordinate. These are finitely many approximate-unit and quasicentrality requirements. They give exactly the stated relations on passing to \(A_\infty\).

Choose \(N\) pairwise disjoint closed intervals contained in \((0,1)\). For the \(r\)th interval choose continuous functions \(g_r^-,g_r^+\colon[0,1]\to[0,1]\) satisfying \[g_r^-g_r^+=0,\qquad 0\leq g_r^-+g_r^+\leq1,\] whose sum is one outside that interval. We choose \(g_r^-\) equal to one below the interval and zero above it, and \(g_r^+\) equal to zero below the interval and one above it. Within the interval the two functions can decrease and increase, respectively, meeting at zero. Set \(h_r=g_r^-+g_r^+\). At every point of \([0,1]\) at most one interval can contribute a loss, so \[ 1-\frac1N\leq\frac1N\sum_{r=1}^N h_r\leq1. \tag{11}\] For \(t=(t_1,\ldots,t_m)\in\mathcal T\) define \[P_{S,t}= \prod_{j\in S}g_{t_j}^+(u_j) \prod_{j\in\{1,\ldots,m\}\setminus S}g_{t_j}^-(u_j).\] These are central positive contractions, and for fixed \(t\) they are pairwise orthogonal as \(S\) varies. Families for different values of \(t\) may overlap and are combined by averaging. Joint functional calculus and independent averaging of the coordinates of \(t\) give \[\begin{align*} Q&=\frac1{N^m}\sum_{t\in\mathcal T}\sum_S P_{S,t} =\prod_{j=1}^m\left(\frac1N\sum_{r=1}^N h_r(u_j)\right), \tag{12}\\ (1-1/N)^m1&\leq Q\leq1, \qquad \|1-Q\|\leq m/N. \tag{13}\end{align*}\] If \(j\in S\), then \(P_{S,t}\) belongs to the image of \(\ell^\infty(I_j)\) in \(A_\infty\), since \(g_{t_j}^+(0)=0\). If \(1\leq j\leq m\) and \(j\notin S\), then \(g_{t_j}^-(u_j)x_j=g_{t_j}^-(1)x_j=0\), whence \(P_{S,t}x_j=0\).

We construct the required orthogonal representatives directly, before imposing exact ideal membership. Start with positive contractive representatives of a finite orthogonal family, whose pairwise products tend to zero. To separate a first representative \(a_i\) from a later representative \(b_i\), choose \(r_i>0\) tending to zero slowly enough that \(\|a_ib_i\|/r_i\to0\). Let \(f_i\colon[0,1]\to[0,1]\) be one on \([0,r_i/2]\) and zero on \([r_i,1]\), with \(1-f_i(t)\leq2t/r_i\). Replace the first element by \((a_i-r_i)_+\) and every later one by \(f_i(a_i)b_if_i(a_i)\). These replacements are orthogonal to the first cut, and \[\|b_i-f_i(a_i)b_if_i(a_i)\| \leq4\|a_ib_i\|/r_i\longrightarrow0.\] Choose \(r_i\) using the maximum of the finitely many relevant products. Repeating with the later elements preserves orthogonality already obtained and changes no quotient element. This gives, for every \(t\), orthogonal positive contractive lifts \(R_{S,t,i}\) of \(P_{S,t}\).

For \(j\in S\), the distance from \(R_{S,t,i}\) to \(I_j\) tends to zero. Let \(\varepsilon_i>0\) tend to zero and dominate all these distances, over the finite collection of triples \((S,t,j)\) under consideration. The first cuts \[T_{S,t,i}=(R_{S,t,i}-2\varepsilon_i)_+\] belong exactly to every \(I_j\) with \(j\in S\): their images in \(A/I_j\) vanish by functional calculus. Choose another sequence \(\sigma_i>0\) tending to zero, and put \[P_{S,t,i}=(T_{S,t,i}-\sigma_i)_+.\] Because \(T_{S,t,i}\in\operatorname{Ideal}(b_j)\), the usual ideal criterion for Cuntz comparison gives a finite integer \(K_{S,t,i,j}\) for which (9) holds. Here the strict positivity of the second cut is essential: every positive cut of an element in \(\operatorname{Ideal}(b_j)\) is Cuntz below a finite direct sum of copies of \(b_j\). Both cuts preserve orthogonality and change the representatives by at most \(2\varepsilon_i+\sigma_i\to0\). Consequently they preserve all the quotient relations established above, while imposing all the ideal and finite-multiple conditions simultaneously.

Finally, centrality of \(P_{S,t}\) gives \(P_{S,t}^{1/2}aP_{S,t}^{1/2}=P_{S,t}a\) in \(A_\infty\). Thus (13) implies (10), and \(P_{S,t}x_j=0\) proves the required vanishing of the omitted compressions. No bound on \(K_{S,t,i,j}\) uniform in \(i\) has been asserted or used. ◻

Proof. Fixing the corners. Choose \(N\) so large that \(m/N<\varepsilon/4\), and use Lemma 17. Write \[l_{S,t,i}(a)=P_{S,t,i}^{1/2}aP_{S,t,i}^{1/2}.\] Asymptotic centrality implies, for \((c,d)\in\mathcal O\), \[\|l_{S,t,i}(c)l_{S,t,i}(d)\| =\|P_{S,t,i}^{1/2}cP_{S,t,i}dP_{S,t,i}^{1/2}\| \longrightarrow0.\] Together with (10) and the omitted compression estimates, this permits us to fix one coordinate \(i\) such that \[\begin{align*} \left\|\frac1{N^m}\sum_{t,S}l_{S,t,i}(a)-a\right\| &<\varepsilon/3 &&(a\in\mathcal F), \tag{18}\\ \|l_{S,t,i}(x_j)\|&<\varepsilon &&(1\leq j\leq m,\ j\notin S), \tag{19}\\ \|l_{S,t,i}(c)l_{S,t,i}(d)\|&<\varepsilon/2 &&((c,d)\in\mathcal O). \tag{20}\end{align*}\] All these requirements range over finite sets. From now on this coordinate is fixed. For each occurrence indexed by \((S,t)\), suppress \(i\) and write \[P=P_{S,t,i},\quad D_P=\operatorname{her}(P),\quad l_P(a)=P^{1/2}aP^{1/2},\quad K_{P,j}=K_{S,t,i,j}.\] Every subscript \(P\) below retains this \((S,t)\) occurrence index. Thus coincident positive elements, including zero pieces, still receive separate corner approximations and outcome labels. In particular, the finitely many constants \(K_{P,j}\) have already been fixed before any approximation accuracy within \(D_P\) is chosen.

Approximating within the corners. Each \(D_P\) is separable and nuclear. The Brown–Carrión–White approximation theorem, Theorem 6, supplies finite-dimensional approximations with asymptotically order-zero first maps and return maps which are finite convex combinations of completely positive contractive order-zero maps. For a fixed corner, take a sequence of these first maps \(\zeta_r\colon D_P\to E_r\). The asymptotic product estimate (2) gives \[ \limsup_{r\to\infty}\|\zeta_r(y)\zeta_r(z)\| \leq\|yz\|\qquad(y,z\in D_P). \tag{21}\] Thus sufficiently late first maps have product norm less than \(\varepsilon\) on the finitely many pairs \((l_P(c),l_P(d))\) in (20). This uses the approximation sequence for the fixed corner; it requires no uniformity in a varying hereditary subalgebra.

Choose positive numbers \(\xi_P\) such that \[ \max_P\xi_P<\varepsilon/3, \qquad \frac1{N^m}\sum_{t\in\mathcal T}\sum_{S\ni j} K_{P_{S,t,i},j}\xi_{P_{S,t,i}}<\varepsilon/2 \quad(1\leq j\leq m). \tag{22}\] There are only finitely many constants in these requirements, so such choices are possible. Choose a corresponding approximation in each \(D_P\), denoted \[D_P\xrightarrow{\zeta_P}E_P \xrightarrow{\varphi_P}D_P, \qquad \varphi_P=\sum_{r=1}^{h_P}\lambda_{P,r}\beta_{P,r},\] where the \(\lambda_{P,r}\) are nonnegative and sum to one, and each \(\beta_{P,r}\) is completely positive contractive and order zero. All component return maps for this corner have the same domain \(E_P\) and use the same first map \(\zeta_P\). Arrange that \[\begin{align*} \|\varphi_P\zeta_Pl_P(a)-l_P(a)\|&<\xi_P &&\bigl(a\in\mathcal F\cup\{x_1,\ldots,x_m\}\bigr), \tag{23}\\ \|\zeta_Pl_P(c)\zeta_Pl_P(d)\|&<\varepsilon &&((c,d)\in\mathcal O). \tag{24}\end{align*}\] For a zero corner take \(E_P=0\) and the zero maps, with a single return component of weight one.

Individual outcomes and their retained parts. Consider the finite probability space whose outcomes \(\nu\) consist of a tuple \(t\in\mathcal T\) and, for each subset \(S\), a choice \(r_S\in\{1,\ldots,h_{P_{S,t,i}}\}\). Give \(t\) uniform probability and make the return-map choices independently with the probabilities \(\lambda_{P,r}\). Write \(\lambda_\nu\) for the resulting probability, and let \[E_t=\bigoplus_S E_{P_{S,t,i}},\qquad \alpha_t(a)=\bigoplus_S\zeta_{P_{S,t,i}}l_{P_{S,t,i}}(a),\] \[\beta_\nu((z_S)_S)= \sum_S\beta_{P_{S,t,i},r_S}(z_S), \qquad T_\nu=\beta_\nu\alpha_t.\] The map \(\alpha_t\) is completely positive contractive because a finite direct sum has the maximum norm. The hereditary subalgebras \(D_{P_{S,t,i}}\) are mutually orthogonal for the single tuple \(t\) used by the outcome. Consequently \(\beta_\nu\) is completely positive contractive and order zero: each summand has these properties, and products between distinct summand ranges vanish. Moreover, (24) bounds the product defect of every \(\alpha_t\) by \(\varepsilon\), uniformly over the outcomes.

Independence of the return-map choices gives \[ \sum_\nu\lambda_\nu T_\nu(a) =\frac1{N^m}\sum_{t\in\mathcal T}\sum_S \varphi_{P_{S,t,i}}\zeta_{P_{S,t,i}}l_{P_{S,t,i}}(a). \tag{25}\] For each fixed \(t\), the differences in (23) lie in orthogonal hereditary subalgebras, so their sum has norm at most \(\max_P\xi_P\). Equations (18), (22), and (25) therefore imply \[ \left\|\sum_\nu\lambda_\nu T_\nu(a)-a\right\| <2\varepsilon/3\qquad(a\in\mathcal F). \tag{26}\]

For \(1\leq j\leq m\), retain only the corners indexed by \(S\ni j\): \[ W_{\nu,j}= \sum_{S\ni j}\beta_{P_{S,t,i},r_S} \zeta_{P_{S,t,i}}l_{P_{S,t,i}}(x_j). \tag{27}\] This is a positive contraction. Contractivity of each component and orthogonality of the omitted corners give, for every individual outcome, \[ \|T_\nu(x_j)-W_{\nu,j}\| \leq\max_{S\not\ni j}\|l_{P_{S,t,i}}(x_j)\| <\varepsilon. \tag{28}\] The maximum over an empty family is taken to be zero. In particular, the estimate holds before taking any average.

The uniform relative trace estimate. Fix \(j\) and a lower semicontinuous extended trace \(\tau\) with \(d_\tau(b_j)<\infty\). In every retained corner, Lemma 16 applies with the already fixed constant \(K_{P,j}\). Thus (23) implies \[\tau(\varphi_P\zeta_Pl_P(x_j)) \leq\tau(l_P(x_j))+\xi_PK_{P,j}d_\tau(b_j) \qquad(j\in S).\] All quantities in this inequality are finite. For each fixed \(t\), traciality and \(\sum_{S\ni j}P_{S,t,i}\leq1\) give \[\begin{align*} \sum_{S\ni j}\tau(l_{P_{S,t,i}}(x_j)) &=\tau\left(x_j^{1/2} \Bigl(\sum_{S\ni j}P_{S,t,i}\Bigr)x_j^{1/2}\right)\\ &\leq\tau(x_j). \end{align*}\] Taking expectations in (27) and using (22) proves \[ \sum_\nu\lambda_\nu\tau(W_{\nu,j}) \leq\tau(x_j)+(\varepsilon/2)d_\tau(b_j). \tag{29}\] For later use there is also an individual bound \[ \tau(W_{\nu,j})\leq M_{\nu,j}d_\tau(b_j), \qquad M_{\nu,j}=\sum_{S\ni j}K_{P_{S,t,i},j}<\infty. \tag{30}\] Every coefficient here was chosen independently of \(\tau\). If \(d_\tau(b_j)=0\), then each retained output has trace exactly zero, as does \(x_j\). Hence the same inequalities hold in this case without an absolute additive error. No condition is being placed on traces for which \(d_\tau(b_j)=\infty\).

Rounding the probabilities and forming the matrix model. After all corner maps and the constants \(M_{\nu,j}\) have been fixed, choose \(k=2^s\) and nonnegative integers \(n_\nu\) with \(\sum_\nu n_\nu=k\) so that \(\mu_\nu=n_\nu/k\) satisfies \[ \sum_\nu|\mu_\nu-\lambda_\nu|<\varepsilon/3, \qquad \sum_\nu|\mu_\nu-\lambda_\nu|M_{\nu,j} <\varepsilon/2\quad(1\leq j\leq m). \tag{31}\] Such choices exist for arbitrarily large \(s\): round the finitely many numbers \(k\lambda_\nu\) down to integers and distribute the remaining units among the outcomes. The resulting probabilities stay in the probability simplex, and their total absolute error tends to zero as \(s\to\infty\). Contractivity of the \(T_\nu\) shows that replacing \(\lambda_\nu\) by \(\mu_\nu\) changes the expected map on the unit ball by less than \(\varepsilon/3\). Equations (30) and (31) show that the expected retained trace changes in absolute value by at most \((\varepsilon/2)d_\tau(b_j)\).

List the outcomes with their multiplicities as \(\nu_1,\ldots,\nu_k\), and write \(t_l\) for the threshold tuple in \(\nu_l\). Define \[F=\bigoplus_{l=1}^k E_{t_l},\qquad \alpha(a)=\bigoplus_{l=1}^k\alpha_{t_l}(a),\qquad \beta((z_l)_{l=1}^k) =\operatorname{diag}\bigl(\beta_{\nu_l}(z_l)\bigr)_{l=1}^k.\] The map \(\alpha\) is completely positive contractive, and \(\beta\) is completely positive contractive and order zero, by the orthogonality of the matrix diagonal positions. We have \[\theta(a)=\operatorname{diag}(T_{\nu_l}(a))_{l=1}^k, \qquad L_k(\theta(a))=\sum_\nu\mu_\nu T_\nu(a).\] Consequently (26) and (31) give (14), and the maximum norm in the direct sum gives (15). Finally, put \[w_j=\operatorname{diag}(W_{\nu_l,j})_{l=1}^k.\] These are positive contractions, and the estimate for individual outcomes in (28) proves (16). Since \(L_k(w_j)=\sum_\nu\mu_\nu W_{\nu,j}\), (29) and the trace part of the rounding estimate prove (17). ◻

Proof of Lemma 15. Fix a countable dense subset \((c_l)_{l\geq1}\) of the unit ball of \(A\). The space \[\{(c,d)\in A_+\times A_+: \|c\|,\|d\|\leq1,\ cd=0\}\] is a separable metric space, so choose a countable dense family \((c'_l,d'_l)_{l\geq1}\) in this space. At stage \(n\), apply Lemma 18 to the first \(n\) tests, \(\mathcal F_n=\{c_1,\ldots,c_n\}\), \(\mathcal O_n=\{(c'_1,d'_1),\ldots,(c'_n,d'_n)\}\), and \(\varepsilon=2^{-n}\). Denote the resulting maps by \(\alpha_n\) and \(\beta_n\), and the retained elements by \(w_{n,j}\) for \(j\leq n\).

Contractivity extends (14) from the dense family to point-norm convergence on \(A\). Likewise, if \(c,d\) are orthogonal positive contractions and \((c',d')\) is one of the chosen orthogonal pairs, then \[\|\alpha_n(c)\alpha_n(d)\| \leq\|c-c'\|+\|d-d'\| +\|\alpha_n(c')\alpha_n(d')\|.\] First taking the limit superior in \(n\), then approximating \((c,d)\) by such pairs, proves asymptotic preservation of orthogonality. Scaling proves the assertion for arbitrary orthogonal positive elements.

For \(j>n\) set \(w_{n,j}=0\). Take \(\delta_{n,j}=2^{-n}\) for all \(n,j\), and set \[\eta_{n,j}= \begin{cases} 2^{-n},&j\leq n,\\ 1,&j>n. \end{cases}\] Equations (5) and (6) now hold for every \(n,j\), and both error sequences tend to zero for each fixed \(j\). At each stage the trace inequality holds simultaneously for all the indicated extended traces. Thus the diagonalization involves only the countable lists of elements, tests, and orthogonal pairs; it requires neither a countable family of traces nor a restriction on the ideal lattice of \(A\). ◻

Proof. Use the notation and the two unitaries of Proposition 32. Write \(k^{[0]}=k\) and \(k^{[1]}=1-k\). Each unitary \(U_i\in M_2(R_\omega)\) has unitary representatives \(U_{i,n}\in M_2(R)\). Indeed, a bounded representative is asymptotically unitary on both sides. On an \(\omega\)-large set its coordinates are invertible and polar correction gives unitaries with the same quotient class; put the identity on the remaining coordinates.

For the matrix-model maps \(\alpha_n:A\to F_n\) and \(\beta'_n:F_n\to A\otimes\mathcal Z_1\subset R\), define \[\varphi_{i,n}(b) =U_{i,n}\begin{pmatrix}0&0\\0&k^{[i]}\beta'_n(b)\end{pmatrix}U_{i,n}^*.\] Each map is c.p.c. order zero: its weight is a positive contraction in the third tensor factor, commuting with the range of \(\beta'_n\); corner inclusion and unitary conjugation preserve these properties. The conjugacies in Proposition 32 give, for every \(a\in A\), \[\lim_{n\to\omega} \|\varphi_{0,n}\alpha_n(a)+\varphi_{1,n}\alpha_n(a)-\kappa_A(a)\|=0,\] since \(k^{[0]}+k^{[1]}=1\). Choose one coordinate in the intersection of the finitely many \(\omega\)-large approximation sets for \(a\in\mathcal F\). This proves the assertion. The first map into \(F_n\oplus F_n\) is \(a\mapsto(\alpha_n(a),\alpha_n(a))\); it is contractive because the direct-sum norm is the maximum norm. ◻

Proof. Fix \(m\ge1\). For a finite set of elements of \(A\otimes M_m\), choose a two-colour approximation of \(A\to M_2(A)\) accurate on all their matrix entries. Amplify every map by \(\mathrm{id}_{M_m}\). The intermediate algebra is still the direct sum of two finite-dimensional algebras, namely \(E_0\otimes M_m\) and \(E_1\otimes M_m\). The first map remains c.p.c. Each amplified return map remains c.p.c. order zero: if \(\varphi=h\varphi_{\mathrm{hom}}\), then \[(\varphi\otimes\mathrm{id}_{M_m})(x) =(h\otimes1)(\varphi_{\mathrm{hom}}\otimes\mathrm{id}_{M_m})(x),\] with a commuting positive contractive weight.

Embed \(M_2(A)\otimes M_m\) into \(A\otimes\mathcal K\) so that its first diagonal \(m\times m\) block is the prescribed first \(m\)-corner. The amplified approximation then approaches the inclusion of \(A\otimes M_m\) into \(A\otimes\mathcal K\). Entrywise accuracy suffices: an \(m\times m\) matrix with entries of norm at most \(\delta\) has norm at most \(m\delta\).

Let \(p_m=1_A\otimes\sum_{r=1}^m e_{rr}\). Precompose the amplified first map with the c.p.c. compression \(x\mapsto p_mxp_m\) on \(A\otimes\mathcal K\). For an arbitrary prescribed finite set in the stabilization, first choose \(m\) so that these compressions approximate every element, then approximate the compressed finite set as above. The resulting composite has the same value on \(x\) and \(p_mxp_m\), so the total error is bounded by the compression error plus the finite-corner approximation error. This gives the required two-colour approximation on the stabilization. ◻

Proof of Theorem 3. First let \(A_0\) be nonzero and unital, and set \(A=A_0\otimes\mathcal Z\). Strong self-absorption gives isomorphisms \[\sigma_l:\mathcal Z\otimes\mathcal Z\longrightarrow\mathcal Z, \qquad \sigma_l(z\otimes1)\longrightarrow z.\] Identify the three extra copies of \(\mathcal Z\) with one copy by a fixed unital isomorphism. The maps \(\mathrm{id}_{A_0}\otimes\sigma_l\) then give isomorphisms \[\Gamma_l:A\otimes\mathcal Z^{\otimes3}\longrightarrow A, \qquad \Gamma_l(a\otimes1)\longrightarrow a\quad(a\in A).\] The convergence holds first on elementary tensors and then on all of \(A\) by contractivity.

Given a finite subset of \(A\), choose \(l\) so that \(\Gamma_l\) has the required accuracy on its first-factor images. Apply Proposition 33 with a sufficiently small error, and postcompose its return maps with \(\mathrm{id}_{M_2}\otimes\Gamma_l\). This gives two-colour approximations to \(A\to M_2(A)\). Lemma 35 shows that \(\dim_{\mathrm{nuc}}(A\otimes\mathcal K)\le1\).

Nuclear dimension decreases on hereditary subalgebras [34]. Applying this to the first corner \(A\otimes e_{11}\) gives \(\dim_{\mathrm{nuc}}(A_0\otimes\mathcal Z)\le1\). For nonunital separable nuclear \(A_0\), its unitization \(\widetilde A_0\) is separable and nuclear. The unital case applies to \(\widetilde A_0\otimes\mathcal Z\), and \(A_0\otimes\mathcal Z\) is a closed two-sided ideal, hence a hereditary subalgebra, of that algebra. The same permanence result gives the bound. The zero algebra is immediate.

Finally, if \(A_0\cong A_0\otimes\mathcal Z\), transport these finite-dimensional approximations across the isomorphism. The first map is c.p.c., the two return maps are individually c.p.c. order zero, and their sum approximates the identity, exactly as in Definition 5. ◻

Proof. Choose \[0<a<\min\left\{\log2-\log(4/3)-\tfrac14\log3,\ -\tfrac14\log r\right\}.\] The first expression is positive. A binary Hamming ball of radius \(\lfloor n/4\rfloor\) has cardinality at most \[\sum_{b=0}^{\lfloor n/4\rfloor}\binom nb \leq 3^{n/4}\sum_{b=0}^n\binom nb3^{-b} =3^{n/4}(4/3)^n.\] Gilbert’s greedy deletion argument [12] therefore produces, for all sufficiently large \(n\), a set \(S_n\subset\{0,1\}^n\) of cardinality \(N=\lfloor e^{an}\rfloor\) whose distinct words have Hamming distance at least \(n/4\).

Let the coordinates of \(G\) be independent standard real Gaussian variables. Fix an admissible list and set \[X_\alpha=\langle G,u_1^{\alpha_1}\otimes\cdots\otimes u_n^{\alpha_n}\rangle \quad(\alpha\in S_n).\] These variables are jointly centered Gaussian. Their covariance matrix \(\Sigma\) has diagonal entries one and off-diagonal entries of absolute value at most \(r^{n/4}\). Consequently its eigenvalues lie in \([1-\lambda_n,1+\lambda_n]\), where \[\lambda_n=N r^{n/4}\longrightarrow0.\] For large \(n\), the density of this Gaussian vector is bounded pointwise by \[\left(\frac{1+\lambda_n}{1-\lambda_n}\right)^{N/2}\] times the density of \(N\) independent centered Gaussians of variance \(1+\lambda_n\). Indeed, \(\det\Sigma\geq(1-\lambda_n)^N\) and \(\Sigma^{-1}\geq(1+\lambda_n)^{-1}I\).

Writing \(\Phi\) for the standard normal distribution function gives \[ \Pr(X_\alpha\leq1\ \forall\alpha\in S_n) \leq\left[\sqrt{\frac{1+\lambda_n}{1-\lambda_n}}\, \Phi\big((1+\lambda_n)^{-1/2}\big)\right]^N \leq q^N \tag{77}\] for one \(q<1\) and every sufficiently large \(n\). The last inequality is uniform over admissible lists because the bracket tends to \(\Phi(1)<1\). By symmetry the same bound holds for the event \(X_\alpha\geq-1\) for all \(\alpha\).

Set \(\eta_n=(4n k^{n/2})^{-1}\). The compact space of admissible lists has an \(\eta_n\)-net, in the maximum Euclidean distance between corresponding vectors, of size at most \[(C_k/\eta_n)^{2kn}.\] For example, mesh the coordinate cubes and select an admissible representative from each occupied box. The logarithm of this bound is \(O(n^2)\). A union bound using (77) shows that the probability that some net list lacks a value above \(1\) or below \(-1\) is at most \[2(C_k/\eta_n)^{2kn}q^N\longrightarrow0.\] Also \(\mathbb E\|G\|^2=k^n\), so \(\Pr(\|G\|\leq2k^{n/2})\geq3/4\). For large \(n\), both the successful-net event and this norm bound therefore hold for some \(G\).

If an admissible list is approximated by a net list, telescoping the elementary tensors of unit vectors bounds the change of each evaluation by \[\|G\|\,n\eta_n\leq\tfrac12.\] The two evaluations exceeding \(1\) and below \(-1\) on the net list retain values above \(1/2\) and below \(-1/2\), respectively. This proves the assertion for every admissible list. ◻

Proof. First, a product of nonzero projections \(p_j\in M_j\) is nonzero. Induct on \(n\). By induction \(q=p_2\cdots p_n\) is a nonzero projection commuting with \(M_1\). The restriction \(M_1\to\mathcal B(qH)\) is a nonzero normal representation. Its kernel is a weak-* closed ideal in a factor and is therefore zero. Thus \(p_1q\ne0\).

For \(0<\varepsilon<1\), the spectral projections \(p_j=\mathbf1_{(1-\varepsilon,1]}(e_j)\) are nonzero. Their product is nonzero, and on its range the positive product \(\prod_j e_j\) is at least \((1-\varepsilon)^n\). Letting \(\varepsilon\downarrow0\) proves the lower bound one; the reverse bound follows from contractivity. ◻

Proof. Let \((\gamma,H,\xi)\) be the irreducible GNS representation of \(\rho\), with \(\|\xi\|=1\), and put \[L=\{z\in E:\gamma(z)\xi=0\}.\] We first show that \(\ker\rho=L+L^*\). The inclusion from right to left is immediate. If \(\rho(x)=0\), write \(\eta=\gamma(x)\xi\perp\xi\). If \(\eta=0\), then \(x\in L\). Otherwise, on \(\operatorname{span}\{\xi,\eta\}\) let \(T\) be the rank-one operator with \(T\xi=\eta\) and \(T\eta=0\); thus \(T^*\xi=0\). Apply the selfadjoint form of Kadison transitivity [15] separately to the real and imaginary parts of \(T\). This gives \(y\in E\) such that \[\gamma(y)\xi=\eta,\qquad\gamma(y^*)\xi=0.\] Then \(x-y\in L\) and \(y\in L^*\), as desired.

The algebra \(L\cap L^*\) is hereditary. Let \((g_\lambda)\) be a positive contractive approximate identity for it. If \(z\in L\), then \(z^*z\in L\cap L^*\), whence \[\|z(1-g_\lambda)\|^2 =\|(1-g_\lambda)z^*z(1-g_\lambda)\|\longrightarrow0.\] The adjoint assertion holds for elements of \(L^*\). Therefore, with \(f_\lambda=1-g_\lambda\), the equality \(\ker\rho=L+L^*\) gives \[\|f_\lambda(x-\rho(x)1)f_\lambda\|\longrightarrow0 \qquad(x\in E).\] Each \(f_\lambda\) is a positive contraction of norm one because \(\gamma(f_\lambda)\xi=\xi\). One sufficiently large \(\lambda\) works for \(\mathcal F\). ◻

Proof of Theorem 4. The closed ideal generated by commutators is all of \(D\); otherwise a nonzero unital commutative quotient would supply a character. Decomposing elements into selfadjoint parts and rescaling, choose selfadjoint contractions \(s_i,t_i\) and elements \(v_i,w_i\), for \(1\leq i\leq m\), such that \[ \left\|1-\sum_{i=1}^m v_i[s_i,t_i]w_i\right\|<\tfrac12. \tag{78}\] Put \(s_0=1\), \(k=m+1\), and \(M=\sum_i\|v_i\|\|w_i\|>0\). Under any nonzero unital representation \(\sigma\), some represented commutator has norm greater than \(\delta=(2M)^{-1}\). Since \[\|[S,T]\|\leq2\|S-c1\|\|T\| \leq\operatorname{diam}(\operatorname{sp}S)\] when \(S=S^*\), \(\|T\|\leq1\), and \(c\) is the midpoint of \(\operatorname{sp}S\), the spectrum of some \(\sigma(s_i)\) has diameter at least \(\delta\).

There are pure states \(\rho^0,\rho^1\) of \(\sigma(D)\) taking that element to its two spectral endpoints. Indeed, a state attaining an endpoint extends from the unital abelian algebra \(C^*(1,\sigma(s_i))\), and the states attaining that endpoint form a nonempty weak-* compact face of the state space. An extreme point of this face is pure. The associated real expectation vectors \[v^\alpha=\big(\rho^\alpha(\sigma(s_b))\big)_{b=0}^m \in\{1\}\times[-1,1]^m\qquad(\alpha=0,1)\] differ by at least \(\delta\) in one coordinate. The set of pairs in \(\big(\{1\}\times[-1,1]^m\big)^2\) with \(\|v^0-v^1\|_\infty\geq\delta\) is compact. Consequently there is \(0<r<1\), independent of \(\sigma\), such that \[ \left|\left\langle\frac{v^0}{\|v^0\|},\frac{v^1}{\|v^1\|}\right\rangle\right|\leq r. \tag{79}\] To justify the strict bound, two vectors in this compact set cannot be negatively collinear because both first coordinates are one; positive collinearity forces them to coincide, contrary to their separation.

Choose \(G,n\) from Lemma 36 for \(k,r\) and form the selfadjoint element \[ h=\sum_{b_1,\ldots,b_n=0}^m G_{b_1\ldots b_n} s_{b_1}\otimes\cdots\otimes s_{b_n} \in D^{\otimes_{\max}n}. \tag{80}\] Let \(\pi\) be any nonzero irreducible representation of this tensor product, with factor restrictions \(\pi_j:D\to\mathcal B(H)\). For each \(j\), choose the two pure states of \(\pi_j(D)\) described above. By (79) and Lemma 36, choosing one state \(\rho_j\) from each pair makes \[ p=\sum_{b_1,\ldots,b_n=0}^mG_{b_1\ldots b_n} \prod_{j=1}^n\rho_j(\pi_j(s_{b_j})) \tag{81}\] strictly positive for one choice and strictly negative for another. Removing the normalizations multiplies each tensor evaluation by a positive number and so preserves its sign.

The von Neumann algebras \(M_j=\pi_j(D)''\) are mutually commuting factors: a central element of any \(M_j\) commutes with every factor image and hence with the irreducible joint representation, so is scalar. For one of the choices in (81), use Lemma 38 to choose positive norm-one elements \(e_j\in\pi_j(D)\) with \[\|e_j\pi_j(s_b)e_j-\rho_j(\pi_j(s_b))e_j^2\|<\varepsilon \quad(0\leq b\leq m).\] The product \(e=\prod_j e_j\) is positive and has norm one by Lemma 37. Commuting distinct factors and telescoping their products in (80) yields \[ \|e\pi(h)e-pe^2\| \leq n\varepsilon\sum_{b_1,\ldots,b_n}|G_{b_1\ldots b_n}|. \tag{82}\] All factors in this estimate are contractions.

If \(p>0\) and \(\pi(h)\leq0\), then \(pe^2-e\pi(h)e\geq pe^2\), so the left side of (82) is at least \(p\). Choosing \(\varepsilon\) sufficiently small is a contradiction. The negative choice of \(p\), with its own excisers, similarly rules out \(\pi(h)\geq0\). Thus \[\pi(h_+)\ne0,\qquad\pi((-h)_+)\ne0\] for every nonzero irreducible \(\pi\). Each positive part is full: a proper closed ideal containing it would yield a nonzero quotient and hence an irreducible representation killing it. The two positive parts are orthogonal by functional calculus. ◻

Proof. A hereditary subalgebra \(H\) is strongly Morita equivalent to the ideal \(I\) it generates. If \(H\) had a matrix quotient, ideal correspondence would give an ideal \(J\subseteq I\) for which \(I/J\) is strongly Morita equivalent to a matrix algebra, hence to \(\mathbb C\). By imprimitivity, \(I/J\) would be elementary. Ideals of \(I\) are ideals of \(A\), contradicting nowhere scatteredness. This is also the equivalence in [28].

Apply Theorem 39(1) to \(A\), put \(B=F(A)\), and retain its maps \(\varphi_j:E_j\to B\), elements \(c_j\), and displayed identity.

If \(\chi:B\to\mathbb C\) were a character, its compositions with these maps would be order-zero positive functionals. Such a functional on \(M_r\), \(r\geq2\), is zero. To see this, a nonzero positive functional has a positive density matrix with an eigenvector \(\xi\) of positive eigenvalue. For a unit eigenvector \(\eta\perp\xi\), the orthogonal rank-one projections onto \((\xi+\eta)/\sqrt2\) and \((\xi-\eta)/\sqrt2\) both have strictly positive values, contrary to order zero. Thus \(\chi\circ\varphi_j=0\) for every \(j\), contradicting the displayed identity and \(\chi(1_B)=1\). ◻

Proof. It suffices to prove that for every separable subalgebra \(C\subseteq F(A)\), there is a unital \(*\)-homomorphism from \(D\) into \(F(A)\) with range commuting with \(C\). This is Kirchberg’s reindexing mechanism, in the form recorded in [17] for unital \(A\). We include the annihilator details needed for nonunital \(A\).

Choose a countable dense unital \(*\)-algebra \(D_0\subset D\) over \(\mathbb Q+i\mathbb Q\). For each \(x\in D_0\), choose a bounded lift \((a_l^x)_l\) in \(A\) that represents a central element mapping to \(x\), with \[ \sup_l\|a_l^x\|\leq2\|x\|. \tag{83}\] Take the zero lift for zero and the approximate identity for \(1\). The quotient norm permits these choices: first lift within a bound strictly below \(2\|x\|\), then truncate coordinate norms without changing the ultrapower class. The additive, scalar, multiplicative, and adjoint relation defects between these lifts annihilate \(A\) on both sides along \(\omega\).

Choose a countable dense family in \(C\) with bounded lifts \((b_s^u)_s\) in \(A\). Also fix a countable dense subset of \(A\). List the following countably many requirements on an old index \(l\): centrality of each \(a_l^x\) against that subset; left and right products of every relation defect with that subset tending to zero; and \[a_l^1a\longrightarrow a,\qquad aa_l^1\longrightarrow a\] for its elements \(a\). All these requirements hold along \(\omega\).

For each output index \(s\), choose \(l(s)\) so that the first \(s\) requirements hold with error less than \(1/s\), and also \[\|[a_{l(s)}^x,b_s^u]\|<1/s\] for the first \(s\) pairs \((x,u)\) in an enumeration. For this fixed \(s\), the finitely many \(b_s^u\) are fixed elements of \(A\). Original centrality therefore makes all these commutator conditions \(\omega\)-large as conditions on \(l\). A finite intersection with the other \(\omega\)-large conditions is nonempty, so the choice is possible.

The new sequences \((a_{l(s)}^x)_s\) are central, and all their relation defects annihilate \(A\) on both sides: the tests hold in the ordinary limit, and the uniform bounds extend them from the dense subset to all of \(A\). Thus they define a unital rational-complex \(*\)-homomorphism on \(D_0\) into \(F(A)\). The unit assertion follows from the approximate-identity tests. Bound (83) permits continuous extension to a complex unital \(*\)-homomorphism on \(D\). Its range commutes with the dense family in \(C\), hence with \(C\). Injectivity is not needed.

Starting with one copy and repeatedly taking \(C\) to be generated by the previous images gives \(n\) commuting unital images of \(D\). The universal property of the maximal tensor product gives the stated homomorphism. ◻

Proof of Theorem 1. The assertion is immediate for \(A=0\), so assume \(A\ne0\). Assume first that \(\dim_{\mathrm{nuc}}(A)<\infty\); we prove (1)\(\Rightarrow\)(2). By Lemma 40, \(B=F(A)\) has no characters. Choose in \(B\) a finite commutator witness as in (78), and let \(D\) be the unital \(C^*\)-subalgebra generated by all its entries. Then \(D\) is separable and has no characters, since applying a character of \(D\) to that inequality would give \(1<1/2\).

By Theorem 4, some \(D^{\otimes_{\max}n}\) contains full orthogonal positive elements \(a,b\). Lemma 41 supplies a unital homomorphism into \(B\). Their images remain orthogonal and full: applying a unital homomorphism to finite ideal combinations approximating the unit proves that fullness passes to the target. Theorem 39(2) now gives \(A\cong A\otimes_{\min}\mathcal Z\). Conversely, for separable nuclear \(A\), absorption and Theorem 3 imply \(\dim_{\mathrm{nuc}}(A)\le1\). This bound implies finite nuclear dimension, completing all three equivalences in Theorem 1. ◻