Proof. Take a nonexact \(C^*\)-algebra \(W\), whose existence is recorded in (Pisier 1995, Remark (i) following Theorem 1, pp. 163–164). There are \(I\triangleleft J\) such that the canonical surjection \[(W\otimes_{\min}J)/(W\otimes_{\min}I) \longrightarrow W\otimes_{\min}(J/I)\] is not injective. The map is surjective because its algebraic image is dense and the range of a \(C^*\)-homomorphism is closed. Choose a representative \(T_0\in W\otimes_{\min}J\) of a nonzero class in its kernel. Its norm modulo \(W\otimes_{\min}I\) is positive, whereas its image in \(W\otimes_{\min}(J/I)\) is zero. A sufficiently close finite algebraic approximation to \(T_0\) retains a strict gap between these two norms. After normalizing its nonzero first factors and absorbing their norms into the second factors, we obtain contractions \(z_j\in W\) and elements \(h_j\in J\) for which \[ \left\|\sum_j z_j\otimes h_j+W\otimes_{\min}I\right\| >\left\|\sum_j z_j\otimes(h_j+I)\right\|_{\min}. \tag{53}\] For a tuple \(\mathbf v=(v_j)_{j=1}^{\ell}\) in a \(C^*\)-algebra \(D\), write \[\mu_{\mathbf v,m}(A_1,\ldots,A_\ell) =\left\|\sum_{j=1}^{\ell}A_j\otimes v_j\right\|_{M_m(D)}.\] We show that these functions determine both norms in (53), with the coefficient algebra and its ideal held fixed.

For arbitrary coefficients \(d_j\in J\), represent the fixed nonzero algebra \(J\) faithfully on a Hilbert space \(\mathcal H\). Then \[ \left\|\sum_j v_j\otimes d_j\right\|_{\min} =\sup_{m,V} \mu_{\mathbf v,m}(V^*d_1V,\ldots,V^*d_\ell V), \tag{54}\] where \(V:\mathbb C^m\to\mathcal H\) ranges over finite-dimensional isometries. Indeed, also represent \(D\) faithfully. The operators on the right are the two-sided compressions in the second Hilbert factor, so their norms are at most the left-hand norm. Finite-rank projections, directed by inclusion, converge strongly to the identity. Their two-sided compressions converge strongly to the tensor sum; evaluation at a fixed vector whose image nearly attains its norm gives the reverse inequality. This applies to arbitrary, including nonselfadjoint, tensor sums.

For the quotient norm, choose a positive contractive approximate identity \((f_\lambda)\) of the fixed ideal \(I\). Put \(J_D=D\otimes_{\min}I\). For \(T\in D\otimes_{\min}J\), define \[R_\lambda(T) =(1\otimes(1-f_\lambda))T(1\otimes(1-f_\lambda)),\] using the unitized multiplier operations. Elementary tensors and density give \[T-R_\lambda(T)\in J_D,\qquad \|R_\lambda(K)\|\longrightarrow0\quad\text{for each fixed }K\in J_D.\] The cutdowns are contractions. Hence, for every fixed \(K\in J_D\), \[\operatorname{dist}(T,J_D)\le\|R_\lambda(T)\| \le\|T-K\|+\|R_\lambda(K)\|.\] Taking lower and upper limits and then the infimum over \(K\) yields the net limit \[ \left\|\sum_j v_j\otimes d_j+J_D\right\| =\lim_\lambda \left\|\sum_j v_j\otimes (1-f_\lambda)d_j(1-f_\lambda)\right\|_{\min}. \tag{55}\] The cutdown coefficients lie in \(J\), and no quasicentrality is needed. This argument uses convergence on each fixed element of \(J_D\). A related fixed-element cutdown mechanism appears in (Pisier 1995, proof of Lemma 3, p. 165); the calculation above establishes the stated formula for general \(I\triangleleft J\) and its approximate-identity net. Equations (54) and (55) show that the displayed quotient norm depends only on the functions \(\mu_{\mathbf v,m}\). Applying the same compression argument in \(J/I\) shows the same for \(\|\sum_j v_j\otimes(d_j+I)\|_{\min}\).

If a tuple in an exact algebra had the same functions \(\mu_{\mathbf z,m}\) at every matrix level, the two formulas above would transfer this strict gap to that algebra. Exactness makes its canonical quotient map isometric, a contradiction. This proves (i). The conclusion excludes simultaneous equality for all finite matrix tests; it does not assert that a fixed finite list of tests suffices.

For (ii), set \(W_0=C^*(z_1,\ldots,z_\ell)\). This algebra is separable, and its inclusion in \(W\) preserves matrix norms. It has a faithful representation on a separable Hilbert space. Indeed, choose a dense sequence in the positive unit sphere and, for each member, a state attaining its norm. Their GNS representations have faithful direct sum: for every nonzero positive element, approximation by that sequence shows that some chosen state is positive on it. Choose increasing nonzero finite-rank projections \(P_r\to1\) strongly in that representation, and set \[N_r=\dim P_r\mathcal H,\qquad z_j^{(r)}=P_rz_jP_r\big|_{P_r\mathcal H}.\] These are contractions. For each fixed \(m\) and \((A_j)\), the corresponding two-sided compressions of \(\sum_j A_j\otimes z_j\) converge strongly to that operator and have norms at most its norm. Evaluation at a fixed near-norming vector gives the matching lower limit, proving (52). The same sequence works for every fixed test. The assertion is pointwise in the test, and uses no multiplicativity of the compressions. ◻

Proof. Put \(g=(X-\alpha_0)_+\), choose \(0<\nu<s-\alpha_0\), and put \(h_g=(g-\nu)_+\). Choose a ramp \(t'=t'(X)\) that is zero on \([0,\alpha_0+\nu]\) and one on \([s,\infty)\). Then \(t'\in\operatorname{Her}(h_g)\), and \[ \|\pi(X)\|>s\quad\Longrightarrow\quad\|\pi(t')\|=1. \tag{57}\] The element \(v=g^{\oplus n}\in M_n(C)\) is properly infinite by \(\textup{(F\ensuremath{_n})}\), applied to the scalar positive \(g\). For the dimensions \(N_r\) of Lemma 18, \[d_r=g^{\oplus N_r}\precsim v^{\oplus N_r}\precsim v.\] The first comparison uses \(g\) as a summand of \(v\); the second uses proper infiniteness of \(v\). Lemma 5, with cutoff \(\nu\), gives one rectangular carrier \[w_r\in M_{n,N_r}(C^{**}),\qquad w_r^*w_r=1_{N_r}\otimes e_{h_g},\qquad w_rw_r^*\le1_n\otimes e_g.\] Its conjugation sends \(M_{N_r}(\operatorname{Her}(h_g))\) into \(M_n(\operatorname{Her}(g))=\operatorname{Her}_{M_n(C)}(v)\). Use this same carrier for every \(j\), defining \[ S_j^{(r)}=w_r(z_j^{(r)}\otimes t')w_r^* \in M_n(\operatorname{Her}(g)),\qquad S_{j,pq}^{(r)}=(S_j^{(r)})_{pq}. \tag{58}\] Each matrix has norm at most one, so its entries are contractions in the required hereditary subalgebra.

Suppose that every \(r\) admits a representation \(\pi_r\) satisfying (56) for these entries. Restricting to a reducing cyclic subrepresentation that still detects \(\|\pi_r(X)\|>s\), we may make these choices from a set of representations. Write \(B_j=[B_{j,pq}]\in M_n(C)\); these elements are fixed independently of \(r\). At every matrix level \(m\), the normal amplification of the single \(w_r\) is isometric on the supported source algebra, by Lemma 5. The represented entry equalities and (57) therefore give, for every \(A_j\in M_m(\mathbb C)\), \[\begin{align*} \left\|\sum_j A_j\otimes\pi_r^{(n)}(B_j)\right\| &=\left\|\left(\sum_j A_j\otimes z_j^{(r)}\right) \otimes\pi_r(t')\right\|\\ &=\left\|\sum_j A_j\otimes z_j^{(r)}\right\|. \tag{59}\end{align*}\] This equality is exact at each \(r\). Using a different carrier for each \(j\) would not give the joint equality.

Define the \(*\)-homomorphism \[\Psi:M_n(C)\longrightarrow \mathcal Q= \frac{\prod_r B(\mathcal H_r^{\oplus n})} {\bigoplus_r B(\mathcal H_r^{\oplus n})}, \qquad \Psi(b)=[(\pi_r^{(n)}(b))_r],\] where the numerator is the algebra of bounded sequences and the denominator is its ideal of norm-null sequences. Contractivity of the representations makes each displayed sequence bounded. The image \(E=\Psi(M_n(C))\) is closed and isomorphic to \(M_n(C)/\ker\Psi\). Since \(M_n(C)\cong M_n\otimes C\), nuclearity of matrix algebras and permanence under tensor products make \(M_n(C)\) nuclear (Blackadar 2006, II.9.4.2 and IV.3.1.1). Its quotient \(E\) is nuclear by (Choi and Effros 1977, Corollary 4), and hence exact by (Blackadar 2006, II.9.6.6). These permanence facts require neither units nor separability. It is this quotient of a nuclear algebra that must be exact; the argument requires no exactness of the ambient \(\mathcal Q\).

The norm of a class modulo norm-null sequences is the \(\limsup\) of its coordinate norms. Subtracting a norm-null sequence gives the lower bound, and subtracting finitely many initial entries gives the upper bound. The same formula holds at each fixed matrix level under the canonical identifications of finite matrices over a product and its norm-null ideal. Since \(E\subset\mathcal Q\) is completely isometric, (59) and (52) give \[\left\|\sum_j A_j\otimes\Psi(B_j)\right\|_{M_m(E)} =\limsup_r\left\|\sum_j A_j\otimes\pi_r^{(n)}(B_j)\right\| =\left\|\sum_j A_j\otimes z_j\right\|\] for every \(m\) and every \((A_j)\). Only the fixed elements \(B_j\) have been sent into \(E\); the varying choices in (58) were used in the coordinatewise equalities. The displayed identity contradicts Lemma 18(i). Thus some \(r\) admits no such representation, and its entries in (58) prove the proposition. ◻

Proof. Choose \(\ell\) from Lemma 18, set \(k=\ell n^2\), and choose graph coefficients \(\gamma_g,\sigma_g>0\) with \(\gamma_g^2+k\sigma_g^2<1\). With these choices fixed, apply Proposition 8 to obtain its measurements \(X,B_i\). Group the \(B_i\) into \(\ell\) matrices \(B_j=[B_{j,pq}]\in M_n(C)\), and choose the corresponding \(S_{j,pq}\) by Proposition 19. The graph gives a rich positive contraction \(p\in\operatorname{Her}(a)\) with \(\|\pi(p)\|=1\) whenever \(\|\pi(a)\|>1/2\). If \(d=y(1-c^*p^2c)y\) vanishes in a representation where \(\|\pi(y)\|>s\), the graph would give all of (56), contrary to the choice of the entries. Thus \[ \pi\bigl(y(1-c^*p^2c)y\bigr)=0 \quad\Longrightarrow\quad\|\pi(y)\|\le s=1/4 \tag{61}\] for every representation. Lemma 9 applies to this \(p\), its norm signal, and the displayed implication, and gives the asserted \(p_1,q_1\). The one fixed \(n\) controlled both the number of entries and the algebra \(M_n(C)\) in the quotient argument; only the finite dimensions \(N_r\) varied. ◻

Proof. We will find \(n\) orthogonal targets in \(\operatorname{Her}(u)\), each detected by \(\pi\), and one positive element comparing into every target. Transporting \(n\) copies of its cutoff will allow \(\textup{(F\ensuremath{_n})}\) to produce richness inside \(\operatorname{Her}(u)\).

Put \(\mathcal H_u=\overline{\pi(u)\mathcal H}\). This space is infinite dimensional. Otherwise \(\pi(u)\) would have finite rank \(r>0\). Applying the stabilized representation to proper infiniteness of \(u^{\oplus n}\) would approximate \(\pi(u)^{\oplus2n}\), of rank \(2nr\), by positive operators of rank at most \(nr\). On the \(2nr\)-dimensional range of the target, each such approximant has a unit vector in its kernel. Its distance from the target is therefore at least the least positive eigenvalue of \(\pi(u)\), contradicting norm approximation.

The action of \(D=\operatorname{Her}(u)\) on \(\mathcal H_u\) is irreducible. Indeed, \(\pi(u)\) has no kernel on its support, so \(\pi(u)\xi\ne0\) for \(0\ne\xi\in\mathcal H_u\). Irreducibility of \(\pi\) gives \(\overline{\pi(C)\pi(u)\xi}=\mathcal H\). Applying \(\pi(u)\) and taking closure yields \[\overline{\pi(uCu)\xi}=\mathcal H_u.\] Thus every nonzero vector is cyclic for the restricted hereditary algebra, which proves irreducibility.

Apply the selfadjoint form of Kadison transitivity (Kadison 1957, Theorem 1) to the irreducible unitization of this action. Choose \(n+1\) orthonormal vectors and prescribe \(n+1\) distinct real eigenvalues. A selfadjoint interpolant has the form \(h_0+\lambda1\), with \(h_0\in D_{\rm sa}\). The corresponding eigenvalues of \(\pi(h_0)\) remain distinct, so at most one is zero. Choose \(n\) nonzero ones. Nonnegative continuous functions supported in pairwise disjoint neighborhoods of these values, avoiding zero and positive at the chosen values, give orthogonal elements \[b_1,\ldots,b_n\in D_+,\qquad \pi(b_i)\ne0.\] The functions vanish at zero, so these elements belong to \(D\) also when \(D\) is nonunital.

There is an element \(b'\in D_+\), still detected by \(\pi\), with \(b'\precsim b_i\) for every \(i\). Start with \(e'=b_1\). Given the current \(e'\) and the next \(b_i\), choose a vector on which \(\pi(e')\) is nonzero. Cyclicity for \(D\) and nonzero \(\pi(b_i)\) give \(h\in D\) such that \(\pi(z)\ne0\), where \(z=b_ihe'\). Then \[z^*z\in\operatorname{Her}(e'),\qquad zz^*\in\operatorname{Her}(b_i),\qquad z^*z\sim_{\mathrm{Cu}}zz^*.\] Replacing \(e'\) by \(z^*z\) preserves nonzero image and every previous comparison, by transitivity. At the end we obtain \(b'\) and \[ (b')^{\oplus n}\precsim b_1\oplus\cdots\oplus b_n \sim_{\mathrm{Cu}}\sum_{i=1}^{n}b_i\precsim u. \tag{62}\] The middle equivalence uses orthogonality, and the last comparison uses \(\sum_i b_i\in\operatorname{Her}(u)\).

Choose \(\tau>0\) with \(2\tau<\|\pi(b')\|\). Transport the \(n\) source copies in (62), after cutoff \(\tau\), into \(\operatorname{Her}(u)\). Let \(t\) be a ramp of \(b'\), zero on \([0,\tau]\) and one on \([2\tau,\infty)\). It belongs to the common cutoff hereditary algebra and has represented norm one. The sum \(p\) of its \(n\) orthogonal transported copies is a positive contraction in \(\operatorname{Her}(u)\) with \(\|\pi(p)\|=1\). By (9), for every \(\varepsilon\ge0\), \[(p-\varepsilon)_+\sim_{\mathrm{Cu}}((t-\varepsilon)_+)^{\oplus n}.\] The right side is properly infinite by the hypothesis applied to the scalar positive \((t-\varepsilon)_+\). Thus \(p\) is rich. ◻

Proof. All comparisons remain in \(C\otimes\mathcal K\), and all uses of shrinking are in \(C\). Since \(b\) is properly infinite, \(b^{\oplus n}\precsim b\). Apply Lemma 5 with cutoff \(1/8\) to obtain orthogonal copy maps \[\theta_i(h)=w_i h w_i^*: \operatorname{Her}((b-1/8)_+)\longrightarrow\operatorname{Her}(b)\subset\operatorname{Her}(u) \quad(1\le i\le n).\] Let \(y_0\) be a ramp of \(b\), zero on \([0,1/4]\) and one on \([1/2,1]\), and set \(a_0=a\). For \(1\le j\le n\) we construct positive contractions \(a_j,y_j\) such that \(a_j\) is rich and belongs to \(\operatorname{Her}(u)\), \(y_j\) belongs to the common domain of the copy maps, and the following invariants hold: \[ \begin{gathered} a_j\theta_i(y_j)=0\qquad(1\le i\le j),\\ \|\pi(a)\|>1/2\ \Longrightarrow\ \|\pi(a_j)\|=1,\\ \|\pi(b)\|>1/2\ \Longrightarrow\ \|\pi(y_j)\|=1 \quad\text{for every representation }\pi. \end{gathered} \tag{64}\]

Once the induction reaches \(j=n\), we will set \(c_0=a_n+\sum_{i=1}^n\theta_i(y_n)\). The induction will make the summands orthogonal, and the final use of \(\textup{(F\ensuremath{_n})}\) will prove the sum of the \(n\) copies rich without requiring \(y_n\) itself to be rich.

Suppose \(a_{j-1},y_{j-1}\) have been constructed. At \(j=1\), use the initial pair \(a_0,y_0\); at later stages use the preceding invariants. Apply Proposition 20 to \(a_{j-1},\theta_j(y_{j-1})\), and \(1\in\widetilde C\), and call its outputs \(p_j,q_j\). Thus \(p_j\) is rich and belongs to \(\operatorname{Her}(a_{j-1})\), while \[\kappa_j=\|e_{p_j}e_{q_j}\|<1.\] The hereditary-image equality (6) supplies a positive contraction \(z_j\in\operatorname{Her}(y_{j-1})\) such that \[ q_j=\theta_j(z_j). \tag{65}\] This pullback is an algebra element. Its represented norms equal those of \(q_j\) by (7).

For \(0\le t\le1\), put \(f(t)=(2t-1)_+\) and \(g(t)=\min\{1,2t\}\). We next move \(p_j\) away from a cutoff of \(q_j\). The support bound ensures that multiplication by \(1-g(q_j)\) is bounded below on the support of \(p_j\). The polar part of \((1-g(q_j))p_j\) will therefore transport \(p_j\), preserving its cutoff Cuntz classes and represented norms. Define \[y_j=f(z_j),\qquad H_j=1-g(q_j),\qquad T_j=H_jp_j.\] Here \(T_j=p_j-g(q_j)p_j\in\operatorname{Her}(u)\). In the universal representation, \(g(q_j)\) is supported under \(e_{q_j}\). For \(\xi\in e_{p_j}\mathcal H\), this gives \[\|H_j\xi\|^2 \ge\|(1-e_{q_j})H_j\xi\|^2 =\|(1-e_{q_j})\xi\|^2 \ge(1-\kappa_j^2)\|\xi\|^2.\] Since \(H_j\) is a contraction, we obtain the order bounds \[ (1-\kappa_j^2)p_j^2\le T_j^*T_j\le p_j^2. \tag{66}\] Let \(T_j=l_j|T_j|\) be its polar decomposition in the bidual. The order bounds give \(\operatorname{Her}(|T_j|)=\operatorname{Her}(p_j)\) and \(l_j^*l_j=e_{p_j}\). The polar membership argument in the proof of Lemma 5, applied now to \(T_j\), gives \[a_j=l_jp_jl_j^*\in\operatorname{Her}(T_jT_j^*)\subset\operatorname{Her}(u).\] It is a positive contraction. The same argument makes each transported cutoff Cuntz equivalent to the corresponding cutoff of \(p_j\), so \(a_j\) is rich. The normal extensions of the representations preserve \(l_j^*l_j=e_{p_j}\); conjugating back by \(l_j^*\) shows that \(a_j\) and \(p_j\) have equal norm in every representation.

We check that this change preserves every earlier orthogonality relation. Since \(gf=f\), functional calculus and (65) give \[\theta_j(y_j)H_j=f(q_j)(1-g(q_j))=0,\] and hence \(\theta_j(y_j)T_j=0\). For \(i<j\), the previous orthogonality and \(p_j\in\operatorname{Her}(a_{j-1})\) give \(\theta_i(y_{j-1})p_j=0\), while orthogonality of the copy ranges gives \(\theta_i(y_{j-1})g(q_j)=0\). Thus \(\theta_i(y_{j-1})T_j=0\). Each of these positive elements annihilates \(\overline{\operatorname{ran}(T_j)}\), whose projection is \(l_jl_j^*\), and so annihilates \(a_j\). Because \(y_j\in\operatorname{Her}(y_{j-1})\), the same is true of \(\theta_i(y_j)\) for \(i<j\). This proves the orthogonality in (64).

The norm signals also persist simultaneously. Wherever \(\|\pi(a)\|>1/2\), the first shrinking step gives \(\|\pi(p_1)\|=1\), hence \(\|\pi(a_1)\|=1\); that signal is above \(1/2\) and survives every subsequent step. Wherever \(\|\pi(b)\|>1/2\), the initial ramp has \(\|\pi(y_0)\|=1\). If \(\|\pi(y_{j-1})\|=1\), its copy under \(\theta_j\) has norm one, shrinking gives \(\|\pi(q_j)\|=1\), and transport gives \(\|\pi(z_j)\|=1\). Since \(f(1)=1\), spectral mapping gives \(\|\pi(y_j)\|=1\).

Finally set \[c_0=a_n+\sum_{i=1}^{n}\theta_i(y_n).\] All summands are orthogonal positive contractions in \(\operatorname{Her}(u)\). Their sum is a contraction, and in each representation its norm is the maximum of the summand norms. The preceding signals prove (63). For every \(\tau\ge0\), orthogonal functional calculus and (9) give \[(c_0-\tau)_+ \sim_{\mathrm{Cu}}(a_n-\tau)_+\oplus((y_n-\tau)_+)^{\oplus n}.\] The first summand is properly infinite because \(a_n\) is rich. The second is properly infinite by \(\textup{(F\ensuremath{_n})}\), applied to the scalar positive \((y_n-\tau)_+\). Their block sum is properly infinite, so \(c_0\) is rich. The construction includes zero inputs through zero copy maps and zero cutoffs. ◻

Proof of Theorem 17. Fix \(u\in C_+\), and let \(\mathcal R_u\) be the set of rich positive contractions in \(\operatorname{Her}(u)\). Let \(J\) be the closed ideal generated by all \((p-1/2)_+\) with \(p\in\mathcal R_u\). We claim that \(u\in J\). Otherwise an irreducible representation of \(C/J\) would detect \(u+J\). Its composition with the quotient map is an irreducible representation of \(C\), so Lemma 21 gives \(p\in\mathcal R_u\) with represented norm one. Then \((p-1/2)_+\) has nonzero image, contrary to its membership in \(J\).

The ideals generated by finite subfamilies of these cutoffs form an increasing directed family with dense union in \(J\). For \(\varepsilon>0\), Lemma 6(i) therefore puts \((u-\varepsilon)_+\) in the ideal generated by \((p_i-1/2)_+\) for finitely many \(p_1,\ldots,p_r\in\mathcal R_u\). Starting from the rich zero element, merge them successively by Lemma 22. The result is a rich \(p\in\operatorname{Her}(u)\) for which every representation satisfies \[\|\pi(p_i)\|>1/2\text{ for some }i \quad\Longrightarrow\quad\|\pi(p)\|=1.\] A norm-one signal survives the next merger because it is above \(1/2\). The primitive ideal test in Lemma 6(ii) now gives \((p_i-1/2)_+\in\operatorname{Id}(p)\) for every \(i\). Consequently \[ (u-\varepsilon)_+\in\operatorname{Id}(p). \tag{67}\] If the cutoff is zero, the zero element already suffices.

Since \(p\) is properly infinite, ideal comparison with this target in the stabilization, followed by hereditary comparison, yields \[ ((u-\varepsilon)_+)^{\oplus2}\precsim p\precsim u. \tag{68}\] The target \(u\) is fixed as \(\varepsilon\downarrow0\). Explicitly, choose \(\varepsilon_j\downarrow0\) and, for each \(j\), a comparison coefficient in (68) with error less than \(\varepsilon_j\) to the doubled cutoff. That cutoff differs from \(u\oplus u\) by at most \(\varepsilon_j\). The resulting coefficients therefore compare \(u\oplus u\) to \(u\) in norm. Thus \(u\) is properly infinite. The fixed \(n\) was applied only to scalar positives and their scalar cutoffs throughout the proof. ◻

Proof of Corollary 3. Set \(C=A\), the separable nuclear algebra in the corollary, with its stated fixed-amplification hypothesis. Theorem 17 gives \(\textup{(P)}\), and Theorem 1 gives strong pure infiniteness. The general nonunital absorption implication is recorded in the paragraph following (Kirchberg and Sierakowski 2016, Definition 3.1). We spell out the central-sequence interface that includes the nonunital case.

Let \(C_\omega\) be the norm ultrapower of \(C\) along a free ultrafilter on \(\mathbb N\), with \(C\) embedded diagonally, and put \[F(C)=\frac{C_\omega\cap C'}{\operatorname{Ann}(C,C_\omega)}, \qquad \operatorname{Ann}(C,C_\omega) =\{x\in C_\omega:xC=Cx=0\}.\] Suppose first that \(C\ne0\). Separability makes \(C\) \(\sigma\)-unital, so \(F(C)\) is unital by (Kirchberg 2006, Corollary 1.10(1)). The central-sequence embedding statement immediately before (Kirchberg 2006, Proposition 4.4) gives a unital embedding \(\mathcal O_\infty\hookrightarrow F(C)\): its ambient original algebra is the strongly purely infinite \(C\), and the diagonal copy of \(C\) is the required separable nuclear subalgebra of \(C_\omega\). The algebra \(\mathcal O_\infty\) is unital and separable, and Remark 4.6 supplies self-absorption and connectedness of its unitary group. The “in particular” clause of Proposition 4.4(4) therefore gives stable \(\mathcal O_\infty\)-absorption. Since \(U(\mathcal O_\infty)=U_0(\mathcal O_\infty)\), where \(U_0\) is the identity component, the commutator subgroup of \(U(\mathcal O_\infty)\) lies in \(U_0(\mathcal O_\infty)\); part (5) therefore removes stabilization. Hence \(C\cong C\otimes_{\min}\mathcal O_\infty\). The Proposition permits \(C\) to be nonunital; the unital embedding is into the annihilator quotient \(F(C)\). If \(C=0\), the asserted isomorphism is immediate. ◻

Proof. We first embed the full group algebra \(Q=C^*(\mathbb F_2)\) into \(\mathcal U\). The two free generators are sent to \[\exp(2\pi i\operatorname{Re}u), \qquad \exp(2\pi i\operatorname{Im}u).\] To see that the resulting homomorphism is injective, take any pair of unitaries \(S,T\) on a Hilbert space. Bounded Borel functional calculus gives selfadjoint contractions \(H,K\) such that \(S=\exp(\pi iH)\) and \(T=\exp(\pi iK)\). The contraction \((H+iK)/2\) evaluates the two displayed unitaries to \(S,T\). Thus every unital representation of \(Q\) is a restriction of an evaluation of \(\mathcal U\), and the homomorphism is isometric. The logarithms need only belong to the bounded operators on the representation space. Courtney and Sherman give a related embedding with a different normalization in (Courtney and Sherman 2020, Theorem 6.10).

Nonexactness of \(C^*(\mathbb F_2)\) was established by Wassermann (Wassermann 1976). Here is an explicit proof for \(Q\). We construct a representation of \(Q\otimes_{\min}Q\) that annihilates \(Q\otimes_{\min}I\), where \(I\) is the kernel of the regular quotient. Exactness would make it descend to \(Q\otimes_{\min}(Q/I)\). A concrete tensor sum, however, has spatial norm in that algebra smaller than the norm the proposed descended representation would assign to it. This will exhibit the required failure of the tensor quotient map. Put \(G=\mathbb F_2\), and write \(u_s\) for its canonical unitaries in \(Q\). On \(\ell^2(G)\) let \[\lambda_s\delta_g=\delta_{sg}, \qquad \rho_s\delta_g=\delta_{gs^{-1}}.\] There is a state \(\varphi\) on \(Q\otimes_{\min}Q\) with \[ \varphi(u_s\otimes u_t)= \begin{cases}1,&s=t,\\0,&s\ne t.\end{cases} \tag{71}\] For completeness, finite quotients supplying this state can be constructed directly. On the finite word ball of radius \(m\) in \(G\), partially defined left translation by each of the two generators is an injection between subsets of equal size, hence extends to a permutation of the ball. Freeness gives a homomorphism \(q_m\) from \(G\) to the finite permutation group generated by these extensions. The inverse of each extension agrees with inverse left translation whenever both endpoints lie in the ball. If a reduced word has length at most \(m\), applying its permutations to the identity follows its successive suffixes inside the ball and ends at that word. Consequently \(q_m\) distinguishes distinct elements of the word ball of radius \(m\).

For the finite group \(F_m=q_m(G)\), its commuting left and right regular representations give a representation of \(C^*(F_m)\otimes_{\min}C^*(F_m)\): these algebras are finite direct sums of matrix algebras, for which the spatial and maximal tensor norms agree. Pulling its vector state at the identity back to \(Q\otimes_{\min}Q\) gives the coefficient \(\boldsymbol{1}_{q_m(s)=q_m(t)}\) on \(u_s\otimes u_t\). These coefficients tend to those in (71). The group tensors span a dense \(*\)-subalgebra, and all the states have norm one, so their limit on that span extends to a state on the spatial completion.

The GNS representation of \(\varphi\) is precisely \[ u_s\otimes u_t\longmapsto\lambda_s\rho_t \quad\hbox{on }\ell^2(G), \tag{72}\] with cyclic vector \(\delta_e\). Indeed the cyclic vector associated with \(u_s\otimes u_t\) corresponds to \(\delta_{st^{-1}}\); the inner products agree because \[s^{-1}a=t^{-1}b\quad\Longleftrightarrow\quad st^{-1}=ab^{-1},\] and left multiplication by \(u_x\otimes u_y\) becomes \(\lambda_x\rho_y\). These vectors span \(\ell^2(G)\).

Let \(I=\ker(Q\to C_r^*(G))\). The linear unitary \(R\delta_g=\delta_{g^{-1}}\) satisfies \(R\lambda_sR^*=\rho_s\). Thus the representation in (72) annihilates \(Q\otimes_{\min}I\). Exactness of \(Q\) would make it descend to a representation of \(Q\otimes_{\min}C_r^*(G)\), in which every \(u_s\otimes\lambda_s\) fixes \(\delta_e\).

This contradicts a spatial norm bound. In a faithful unital representation \(\pi:Q\to B(H)\), the unitary \[V(\xi\otimes\delta_g)=\pi(u_g)\xi\otimes\delta_g\] satisfies \(V(1\otimes\lambda_s)V^*=\pi(u_s)\otimes\lambda_s\). If \(a,b\) are the free generators, the norm of \[u_a\otimes\lambda_a+u_{a^{-1}}\otimes\lambda_{a^{-1}} +u_b\otimes\lambda_b+u_{b^{-1}}\otimes\lambda_{b^{-1}}\] in this faithful spatial representation is therefore the norm of the adjacency operator on the \(4\)-regular tree. That norm is at most \(2\sqrt3<4\). To check the bound, orient each edge away from the identity, and for a finitely supported vector \(\xi\) bound the contribution of a parent–child edge by \[2|\xi(p)\xi(c)| \leq 3^{-1/2}|\xi(p)|^2+3^{1/2}|\xi(c)|^2.\] Every nonroot vertex has three children and one parent, and hence total coefficient \(2\sqrt3\); the root has coefficient \(4/\sqrt3\leq2\sqrt3\). The quadratic-form bound extends by density and proves the asserted operator norm bound. In the purported descended representation the same sum has eigenvalue \(4\) on \(\delta_e\), which is impossible for a \(*\)-representation. Thus \(Q\) is not exact. Its unital inclusion in \(\mathcal U\) and the inheritance proved above show that \(\mathcal U\) is not exact either. ◻

Proof. Contractivity of evaluations proves one inequality. Every unital representation of \(\mathcal U\) is evaluation at the image of \(u\), so these evaluations also compute the matrix norm on the left. For the other inequality, evaluate at a contraction \(T\in B(H)\), and take a unit vector \(\xi=(\xi_1,\ldots,\xi_r)\in H^r\) arbitrarily close to attaining the norm of \(Z(T,T^*)\). If the entries of \(Z\) have degree at most \(m\), let \[H_0=\operatorname{span} \{w(T,T^*)\xi_j:|w|\leq m,\ 1\leq j\leq r\},\] where the empty word is included. This is a finite-dimensional nonzero space. Set \(D=P_{H_0}T|_{H_0}\), so that \(D^*=P_{H_0}T^*|_{H_0}\). Applying a word successively from the right never changes the relevant vectors under compression, since all its suffix vectors lie in \(H_0\). Hence \[w(D,D^*)\xi_j=w(T,T^*)\xi_j\quad (|w|\leq m),\] and \(Z(D,D^*)\xi=Z(T,T^*)\xi\). Taking the supremum over evaluations and over these vectors proves (73). The scalar norm-test formula is proved in (Courtney and Sherman 2020, Proposition 2.2); the finite-word compression used here is the argument in the proof of (Courtney and Sherman 2020, Theorem 3.1). Including all components of the test vector gives the matrix-valued assertion needed here. ◻

Proof. If \(E=0\), take \(C_E=0\). Otherwise choose a basis \(P_1,\ldots,P_k\) of \(E\), with coordinate functionals \(f_1,\ldots,f_k\), and choose polynomial representatives \(P_i=\sum_w a_{iw}w\). Define \[ C_E=\sum_{i=1}^k\|f_i\|\sum_w|a_{iw}|\,|w|. \tag{75}\] Constants are allowed in the representatives; the empty word has length zero. For \(Z=\sum_i B_i\otimes P_i\in M_r(E)\) and scalar unit vectors \(\alpha,\beta\in\mathbb C^r\), \[|\alpha^*B_i\beta| =|f_i(\alpha^*Z\beta)|\leq\|f_i\|\|Z\|.\] It follows that \(\|B_i\|\leq\|f_i\|\|Z\|\) for every matrix size \(r\).

Given a finite contraction \(D\), choose \(\pi,J\) as in the hypothesis. Both \(\pi(v)\) and \(D\) are contractions, and telescoping the two intertwining relations gives \[\|w(\pi(v),\pi(v)^*)J-Jw(D,D^*)\|\leq|w|\theta\] for every word \(w\), including words in both letters. More explicitly, the difference for a word \(A_1\cdots A_m\) is the sum of \(m\) terms, each consisting of contractions on either side of one difference \(\pi(v)J-JD\) or \(\pi(v)^*J-JD^*\). Put \(J_r=1_{\mathbb C^r}\otimes J\). The coefficient bound and (75) yield \[\|\pi^{(r)}(\Phi^{(r)}(Z))J_r-J_r Z(D,D^*)\| \leq C_E\theta\|Z\|.\] Since \(J_r\) is an isometry and \(\pi^{(r)}\) is contractive, \[\|Z(D,D^*)\|\leq\|\Phi^{(r)}(Z)\|+C_E\theta\|Z\|.\] Taking the supremum supplied by Lemma 26 proves (74). In particular, \(C_E\) depends on the fixed polynomial space and its chosen representatives, but on none of \(r,d,D,B,v,\pi,J\). ◻

Proof of Proposition 24. If the Proposition were false, for every integer \(j\geq1\) there would be a unital exact algebra \(B_j\) and a contraction \(v_j\in B_j\) satisfying (69) with \(1/j\) in place of \(\gamma\). Let \(\Phi_j:\mathcal U\to B_j\) send \(u\) to \(v_j\). Fix a finite-dimensional polynomial space \(E\). Before choosing any matrix amplification, tensor factor, compression, or ideal, choose one integer \(j=j(E)\geq\max\{1,2C_E\}\). Lemma 27 then gives \[ \|Z\|\leq2\|\Phi_j^{(r)}(Z)\| \qquad(r\geq1,\ Z\in M_r(E)). \tag{76}\]

This same \(j\) works after tensoring with an arbitrary \(C^*\)-algebra \(C\): for every \(z\in E\odot C\), \[ \|z\|\leq2\|(\Phi_j\otimes\operatorname{id}_C)z\|, \tag{77}\] where \(\odot\) denotes the algebraic tensor product and all norms are spatial. The assertion is immediate if \(C=0\). Otherwise, represent \(C\) faithfully on a nonzero Hilbert space \(K\), and use faithful representations of \(\mathcal U\) and \(B_j\) for their respective spatial tensor products. Compress the \(C\) representation to a nonzero finite-dimensional subspace \(K_0\subset K\). If \(z=\sum_i P_i\otimes c_i\), its compression has coefficients \(P_{K_0}c_i|_{K_0}\in M_d(\mathbb C)\) and, after interchanging the tensor factors, belongs to \(M_d(E)\). Inequality (76) bounds its norm by twice the norm of the corresponding compression on the \(B_j\) side, which is at most \(2\|(\Phi_j\otimes\operatorname{id}_C)z\|\). The norms of these compressions have supremum \(\|z\|\): as \(K_0\) increases, the projections \(1\otimes P_{K_0}\) converge strongly to the identity, and their two-sided compressions converge strongly to \(z\). This proves (77).

Now fix any ideal \(I\subset C\), with quotient map \(q:C\to C/I\) and a positive contractive approximate unit \((e_\lambda)\) of \(I\). For \(z\in E\odot C\), every \(z(1\otimes(1-e_\lambda))\) still belongs to \(E\odot C\). Consequently (77) applies to the entire net with the single index \(j(E)\). Formula (70) and exactness of \(B_j\) give \[\begin{align*} \|z+\mathcal U\otimes_{\min}I\| &=\lim_\lambda\|z(1\otimes(1-e_\lambda))\|\\ &\leq2\lim_\lambda \|(\Phi_j\otimes\operatorname{id}_C)(z)(1\otimes(1-e_\lambda))\|\\ &=2\|(\Phi_j\otimes\operatorname{id}_{C/I})((\operatorname{id}_\mathcal U\otimes q)z)\|\\ &\leq2\|(\operatorname{id}_\mathcal U\otimes q)z\|. \end{align*}\] Finite sums of polynomial tensors are dense in \(\mathcal U\otimes_{\min}C\). Although \(j(E)\) may vary with the polynomial space, the resulting constant \(2\) does not. The last inequality therefore extends by continuity to every \(z\in\mathcal U\otimes_{\min}C\). It says that the canonical surjective homomorphism \[\kappa:(\mathcal U\otimes_{\min}C)/(\mathcal U\otimes_{\min}I) \longrightarrow\mathcal U\otimes_{\min}(C/I)\] has zero kernel: if \(\kappa(z+\mathcal U\otimes_{\min}I)=0\), the quotient class has norm zero. An injective \(*\)-homomorphism of \(C^*\)-algebras is isometric, so the fixed factor \(2\) suffices to prove that \(\kappa\) is an isomorphism. Since \(C,I\) were arbitrary, \(\mathcal U\) would be exact, contrary to Lemma 25. ◻

Proof. In the common stabilization, \[\operatorname{Id}_{C\otimes\mathcal K}(h^{\oplus n}) =\operatorname{Id}_C(h)\otimes\mathcal K.\] Thus \(x^{\oplus k}\) belongs to the ideal generated by \(h^{\oplus n}\). The proper-target ideal comparison in Lemma 4 gives (78). This argument takes place in the stabilization and therefore covers every finite rectangular corner and nonunital \(C\). It also covers \(h=0\), when \(x=0\). ◻

Proof. If \(f(x)=0\), take \(R=0\). Otherwise, choose \(\delta>0\) so that \(f\) vanishes on \([0,\delta]\), and choose \(0<\varepsilon<\delta\). Then \(f(x)^{1/2}\in\operatorname{Her}((x-\varepsilon)_+)\). Apply Lemma 5 to \(x\precsim h\) at this cutoff, and let \(w\in M_{s,m}(C^{**})\) be its rectangular partial isometry. Its initial projection is the support of \((x-\varepsilon)_+\). The membership and support conclusions of that lemma therefore give \[R=w f(x)^{1/2}\in M_{s,m}(C),\qquad R^*R=f(x),\qquad RR^*=w f(x)w^*\in\operatorname{Her}(h).\] The norm identity follows from \(\|R\|^2=\|R^*R\|\). Only the product \(R\), rather than \(w\) itself, is asserted to belong to the rectangular algebra. ◻

Proof. Finite matrix exactness follows, for every \(C^*\)-algebra \(D_0\), from the identifications \[M_n(A)\otimes_{\min}D_0=M_n(A\otimes_{\min}D_0),\] which preserve the ideal kernels in the definition of exactness. For the unitization step, let \(A_0\) be a nonunital exact algebra, let \(I\triangleleft D_0\) be a closed two-sided ideal, and write \(q_0\colon D_0\to D_0/I\). The scalar character \(\chi\colon\widetilde A_0\to\mathbb C\) gives, for every \(z\in\widetilde A_0\otimes_{\min}D_0\), the bounded decomposition \[z=z_0+1\otimes d,\qquad d=(\chi\otimes\operatorname{id})(z),\quad z_0\in A_0\otimes_{\min}D_0.\] Indeed, both projections are bounded, and the decomposition holds first on algebraic tensors and then on their completion. If \((\operatorname{id}\otimes q_0)(z)=0\), applying \(\chi\otimes\operatorname{id}\) gives \(d\in I\). The isometric inclusion \[A_0\otimes_{\min}(D_0/I) \subset\widetilde A_0\otimes_{\min}(D_0/I)\] then shows \((\operatorname{id}\otimes q_0)(z_0)=0\). Exactness of \(A_0\) gives \(z_0\in A_0\otimes_{\min}I\), hence \(z\in\widetilde A_0\otimes_{\min}I\). If \(M_n(A)\) is already unital, this second step is unnecessary. ◻

Proof. Write \(K=\{P\in\operatorname{Prim}(C):\|c_P\|\geq\alpha\}\). If \(K\) is empty, \(y=0\) satisfies all conclusions. Suppose henceforth that \(K\) is nonempty. The contradiction will have three levels of choices. First we fix comparison rows and a single contraction in \(M_n(C)\). Each finite matrix contraction then determines an admissible perturbation. If it fails, a primitive quotient witnessing that failure supplies an approximate reducing copy for that fixed contraction. All these copies therefore come from one element of the same exact algebra \(\widetilde{M_n(C)}\).

Fixed comparison rows. Choose a continuous function \(e\colon[0,\infty)\to[0,1]\) that vanishes near zero and is one on \([\alpha/2,\infty)\). Apply Lemma 29 to the last two comparisons in (81). This gives rows \(R,S\in M_{1,n}(C)\) with \[ R^*R=S^*S=E=e(c)^{\oplus n},\qquad RR^*\in\operatorname{Her}(h),\quad SS^*\in\operatorname{Her}(k),\qquad \|R\|,\|S\|\leq1. \tag{83}\] The relation \(p^*ph=h\), together with its adjoint, shows that left and right multiplication by \(p^*p\) are the identity on \(\operatorname{Her}(h)\). Hence \((1-p^*p)RR^*(1-p^*p)=0\), which implies \((1-p^*p)R=0\). The same argument for \(S\) gives \[ p^*pR=R,\qquad qq^*S=S. \tag{84}\] At this point fix the single element \[ V=\eta^{-1}R^*p^*Tq^*S\in M_n(C). \tag{85}\] The rows \(R,S\) and the element \(V\) will not depend on any finite matrix model. Fix now also the global contraction \[\widehat V=V(\max\{1,|V|\})^{-1}\in M_n(C).\] The inverse is evaluated in the unitization, so its product with \(V\) remains in \(M_n(C)\).

Reducing copies of arbitrary finite contractions. Let \(d\geq1\) and let \(D\in M_d(\mathbb C)\) be a contraction. Put \(c_1=(c-\alpha/2)_+\). Condition (F\(_n\)) and Lemma 28, applied to the scalar element \(c_1\), give \[c_1^{\oplus d}\precsim c_1^{\oplus n}.\] Choose a continuous \(\varphi\colon[0,\infty)\to[0,1]\), vanishing near zero and equal to one on \([\alpha/4,\infty)\), and put \(e_1(c)=\varphi(c_1)\). Lemma 29 yields \(L\in M_{n,d}(C)\) such that \[ L^*L=e_1(c)^{\oplus d},\qquad LL^*\in\operatorname{Her}(c_1^{\oplus n}),\qquad \|L\|\leq1. \tag{86}\] Because \(e(c)c_1=c_1\), multiplication by \(E\) is the identity on \(\operatorname{Her}(c_1^{\oplus n})\). The range-square argument used for (84) gives \(EL=L\). Define \[ w=LDL^*\in M_n(C),\qquad \|w\|\leq1,\qquad EwE=w. \tag{87}\] Scalar matrix multiplication here requires no unit in \(C\).

For each \(P\in K\), choose an irreducible representation \(\pi_P\colon C\to B(H_P)\) with kernel \(P\), and use its matrix amplifications. Identify \(C/P\) with the image of the induced faithful representation on \(H_P\), and make the same identification at rectangular matrix levels. The spectral subspace \(1_{(3\alpha/4,\infty)}(\pi_P(c))H_P\) is nonzero, since \(\|c_P\|\geq\alpha\). Choose a unit vector \(\xi_P\) in that subspace, so that \(\pi_P(e_1(c))\xi_P=\xi_P\). Define \[J_P\colon\mathbb C^d\longrightarrow H_P^{\oplus n},\qquad J_P\lambda=L_P(\lambda\otimes\xi_P).\] Equation (86) proves \(J_P^*J_P=1_d\). Both \(D\) and \(D^*\) preserve \(\mathbb C^d\otimes\xi_P\), and the same equation gives \[ w_PJ_P=J_PD,\qquad w_P^*J_P=J_PD^*. \tag{88}\] Thus one copy reduces both \(w_P\) and its adjoint. No eigenvector at the norm of \(c_P\), and no continuous choice of the vectors \(\xi_P\), is required.

Admissible tests and the two optional conclusions. For the chosen \(d,D\), take \[y_0=-RwS^*\in C,\qquad\|y_0\|\leq1.\] If \(h,k\in\mathcal D\), then \(RR^*,SS^*\in\mathcal D\). For a positive contractive approximate unit \((a_\lambda)\) of \(\mathcal D\), the identity \[\|(1-a_\lambda)R\|^2 =\|(1-a_\lambda)RR^*(1-a_\lambda)\|\longrightarrow0\] and its analogue for \(S\) give \(a_\lambda RwS^*a_\lambda\to RwS^*\) in norm. Each approximant belongs to \(\mathcal D C\mathcal D\subset\mathcal D\), so \(y_0\in\mathcal D\). This uses the range squares of the rows and makes no assertion that their individual entries belong to \(\mathcal D\).

If \(p,q\geq0\) and \(pq=0\), then \(p^2R=R\) implies \(pR=R\): multiply \((1-p^2)R=0\) by \((1+p)^{-1}\) in the unitization. Likewise \(qS=S\). It follows that \(pS=qR=0\) and \(S^*R=0\). Consequently \(y_0^2=0\) and \(py_0^*q=0\). The positive elements \(y_0y_0^*\) and \(y_0^*y_0\) are orthogonal, whence \[\|y_0+y_0^*\|^2 =\|y_0y_0^*+y_0^*y_0\|=\|y_0\|^2\leq1.\] We may therefore use \(y=y_0+y_0^*\) when selfadjointness is requested, and \(y=y_0\) otherwise. In both cases \(pyq=py_0q\); hereditary membership is preserved if it is also requested.

One global contraction and the uniform gap. Suppose that no admissible \(y\) satisfies (82). For each \(d,D\), choose a witness \(P\in K\) for the test just constructed. Then \[\|(T-\eta pRwS^*q)_P\|<\sigma\eta.\] Multiplying by the contractive outer factors \(R^*p^*\) and \(q^*S\), and using (84), gives \[ \|(V-EwE)_P\|=\|(V-w)_P\|<\sigma. \tag{89}\] Although \(P\) may depend on \(d,D\), the element \(V\) is the one fixed in (85).

Functional calculus commutes with quotient maps for the fixed global clipping \(\widehat V\). Hence in each witness quotient \[ \|V_P-\widehat V_P\| =\|(|V_P|-1)_+\| =(\|V_P\|-1)_+<\sigma. \tag{90}\] The last strict inequality follows from (89) and \(\|w_P\|\leq1\). The clipping identity follows by squaring the absolute value of \(V_P-V_P\max\{1,|V_P|\}^{-1}\) and applying scalar functional calculus. Combining (89), (90), and (88) gives \[ \|\widehat V_PJ_P-J_PD\|<2\sigma<\gamma,\qquad \|\widehat V_P^*J_P-J_PD^*\|<2\sigma<\gamma. \tag{91}\]

By Lemma 30, \(B=\widetilde{M_n(C)}\) is a unital exact algebra. The amplified representations \(\pi_P^{(n)}\) extend unitally to \(B\). Although the witnessing \(P\) varies with \(d,D\), the contraction \(\widehat V\) and the exact algebra \(B\) remain fixed. Thus \(\widehat V\in B\) has, for every \(d,D\), a unital representation and an isometry satisfying both inequalities in (91). This contradicts Proposition 24 and proves the Proposition with both optional conclusions and the same \(\sigma\). ◻

Proof. Fix an irreducible representation \(\pi:C\to B(H_\pi)\) with kernel \(P\). First, no nonzero \(\pi(a)\), \(a\in C_+\), has finite rank. Indeed, if its rank were \(d\in\mathbb N\), applying the amplified representation to \(a^{\oplus 2n}\precsim a^{\oplus n}\) would express an operator of rank \(2nd\) as a norm limit of operators of rank at most \(nd\). This is impossible, since the set of operators of rank at most a fixed finite integer is norm closed.

Put \(H_h=\overline{\pi(h)H_\pi}\). We claim that the restriction of \(\pi(\operatorname{Her}(h))\) to \(H_h\) is strongly dense in \(B(H_h)\). Irreducibility and the bicommutant theorem (Blackadar 2006, I.9.1.1) give strong density of \(\pi(C)\) in \(B(H_\pi)\). Consequently the strong closure of \(\pi(\operatorname{Her}(h))\) contains \(\pi(h)B(H_\pi)\pi(h)\). For \(t>0\), let \(E_t=1_{[t,\infty)}(\pi(h))\). Every operator \(E_tTE_t\), with \(T\in B(H_h)\) extended by zero on \(H_h^\perp\), has the form \(\pi(h)S\pi(h)\): the inverse of \(\pi(h)\) on \(E_tH_\pi\) is bounded. As \(t\downarrow0\), \(E_tTE_t\to T\) strongly on \(H_h\). This proves the claim.

The space \(H_h\) is infinite dimensional by the finite-rank obstruction. Choose orthonormal vectors \(\xi_1,\ldots,\xi_m\) in \(H_h\) and distinct numbers \(\lambda_1,\ldots,\lambda_m\in(0,1)\). Let \(T\in B(H_h)\) be the positive contraction with \(T\xi_j=\lambda_j\xi_j\) and zero on their orthogonal complement. Choose nonnegative continuous functions \(f_j\) on \(\mathbb R\), with disjoint compact supports in \((0,1)\) and \(f_j(\lambda_j)=1\). Kaplansky density (Blackadar 2006, I.9.1.3) supplies a bounded net of selfadjoint elements \(b_\lambda\) of the represented hereditary algebra converging strongly to \(T\). Continuous functional calculus on bounded selfadjoint sets is strongly continuous; this follows by uniform polynomial approximation on a common bounded interval. Hence \(f_j(b_\lambda)\xi_j\to\xi_j\). One \(b_\lambda\) makes all these \(m\) vectors nonzero. Lift it to a selfadjoint \(b\in\operatorname{Her}(h)\). The elements \(f_j(b)\) lie in \(\operatorname{Her}(h)\), are pairwise orthogonal, and are nonzero at \(P\). The functions vanish at zero, so this argument also applies to a nonunital hereditary algebra. This proves (i).

For two positives \(u,v\) nonzero at \(P\), strong density of \(\pi(C)\) gives \(x\in C\) with \(\pi(uxv)\ne0\). For example, choose a vector on which \(\pi(v)\) is nonzero and a rank-one operator carrying its image to a vector not annihilated by \(\pi(u)\); approximate that operator strongly on the image under \(\pi(v)\). With \(z=uxv\), the positive \(d=zz^*\) satisfies \[d_P\ne0,\qquad d\in\operatorname{Her}(u),\qquad d\sim_{\mathrm{Cu}}z^*z\in\operatorname{Her}(v).\] Here and below \(\sim_{\mathrm{Cu}}\) denotes Cuntz equivalence, and the comparison rules are those of Lemma 4. For a finite list of positives nonzero at \(P\), repeating this operation produces one positive, nonzero at \(P\), subequivalent to every element of the list. At each step the new element lies in the hereditary algebra of the preceding one, so all previous comparisons and any prescribed hereditary membership persist.

Apply (i) with \(m=n\) separately to \(h\) and \(k\), obtaining orthogonal lists \(h_1,\ldots,h_n\) and \(k_1,\ldots,k_n\). Take a common element \(c\) for the resulting list of \(2n\) positives. Then \[c^{\oplus n}\precsim\bigoplus_{j=1}^n h_j \sim_{\mathrm{Cu}}\sum_{j=1}^n h_j\precsim h,\] and the same argument gives \(c^{\oplus n}\precsim k\). The asserted choice in \(\mathcal D\) follows from the hereditary membership just observed. ◻

Proof. In the state space of the unitization \(\widetilde B\), the set \[F_b=\{\varphi:\varphi(b)=\|b\|\}\] is nonempty: evaluation at the largest spectral value of \(b\) on \(C^*(1,b)\) extends to a state of \(\widetilde B\) by the positive Hahn–Banach extension theorem (Blackadar 2006, II.6.3.1). It is a weak-star compact face, since every state has value at most \(\|b\|\) on \(b\). By the Krein–Milman theorem it has an extreme point, which is therefore a pure state of \(\widetilde B\); see (Blackadar 2006, II.6.3.2–3). Let \(\widetilde\pi\) be its irreducible GNS representation (Blackadar 2006, II.6.4.8). It does not annihilate \(B\), because its value on \(b\) is positive. The closure of \(\widetilde\pi(B)H_{\widetilde\pi}\) is a nonzero reducing subspace for \(\widetilde\pi(\widetilde B)\), hence the whole space. If an operator commutes with \(\widetilde\pi(B)\), then its commutator with \(\widetilde\pi(x)\) annihilates \(\widetilde\pi(B)H_{\widetilde\pi}\) for every \(x\in\widetilde B\), since \(xB\subset B\). It therefore commutes with \(\widetilde\pi(\widetilde B)\). Thus the restriction \(\pi=\widetilde\pi|_B\) is nondegenerate and irreducible. The vector state gives \(\|\pi(b)\|\ge\|b\|\), and contractivity gives equality. ◻

Proof. The first set is \(\{P:(c-\alpha)_+\notin P\}\), which is open in the hull-kernel topology. For the second assertion, extend any relative open cover to ambient open sets. Write those sets as \[\mathcal O_\lambda=\{P:J_\lambda\not\subset P\}, \qquad I=\overline{\sum_\lambda J_\lambda},\] where each \(J_\lambda\) is a closed ideal of \(C\). If \(\|c+I\|\ge\alpha\), Lemma 33 applied in \(C/I\) gives an irreducible representation attaining that norm. Its pullback has a primitive kernel containing \(I\), belongs to \(K(c,\alpha)\), and lies outside every \(\mathcal O_\lambda\). This contradicts the covering property. Thus \(\|c+I\|<\alpha\); if \(C/I=0\), the same inequality is immediate. Choose \(d\in I\) with \(\|c-d\|<\alpha\), and then approximate \(d\) by an element \(d_F\in\sum_{\lambda\in F}J_\lambda\) for a finite index set \(F\), closely enough that \(\|c-d_F\|<\alpha\) still holds. Any primitive ideal containing every \(J_\lambda\), \(\lambda\in F\), has \(\|c_P\|<\alpha\). Hence the finitely many \(\mathcal O_\lambda\), \(\lambda\in F\), cover \(K(c,\alpha)\). The proof uses quasi-compactness only and does not assert that \(K(c,\alpha)\) is closed. ◻

Proof. If \(K_0\) is empty, choose \(U=0\) and the empty family. Otherwise, Lemma 32 associates to every original choice a positive \(c\) nonzero at its specified point, with \(c^{\oplus n}\precsim h,k\). Choose \(0<\alpha<\|c_P\|\). The resulting open sets \(\Omega(c,\alpha)\) cover \(K_0\), so choose a finite subcover \(\Omega_i=\Omega(c_i,\alpha_i)\), \(1\le i\le N\).

Set \(\theta=\sigma/4\), \(\eta_i=\theta^{i-1}\), and \(U_0=0\). At stage \(i\), apply Proposition 31 with \(T=p_iU_{i-1}q_i\), scale \(\eta_i\), and the selected data. The identities \(p_ih_i=h_i\), \(q_ik_i=k_i\) imply \(p_i^2h_i=h_i\), \(q_i^2k_i=k_i\). The simultaneous hereditary and selfadjoint options give a selfadjoint contraction \(y_i\in\mathcal D\) such that \(U_i=U_{i-1}+\eta_i y_i\) satisfies \[\|(p_iU_iq_i)_P\|\ge\sigma\eta_i \qquad(P\in K(c_i,\alpha_i)).\] Let \(U=U_N\). Later perturbations change the \(i\)th corner by at most \[\sum_{j>i}\eta_j \le\eta_i\frac{\theta}{1-\theta} =\eta_i\frac{\sigma}{4-\sigma} <\frac{\sigma\eta_i}{2}.\] Consequently \[ \|(p_iUq_i)_P\|>\frac{\sigma\eta_i}{2} \qquad(P\in K(c_i,\alpha_i)). \tag{92}\] In particular each selected corner is nonzero throughout its entire open set \(\Omega_i\), proving the assertion. ◻

Proof. If \(K_0=\varnothing\), take \(x=0\). Assume henceforth that \(K_0\) is nonempty. We first construct a selfadjoint element that is nonscalar on every relevant hereditary support space. Coupling its separated spectral windows and deleting the nearby spectral blocks will produce a selfadjoint element whose positive and negative parts are both nonzero at every point of \(K_0\). A final coupling of those two orthogonal parts gives the required square-zero element.

Proof of Theorem 2. Fix \(a\in C_+\) and \(r>0\). The set \(K_0=K(a,r)\) is quasi-compact by Lemma 34. If it is empty, every primitive image of \((a-r)_+\) is zero, so \((a-r)_+=0\), and \(((a-r)_+)^{\oplus2}\precsim a\) is immediate. Suppose that \(K_0\) is nonempty. Proposition 36 applied to \(\operatorname{Her}(a)\) gives \(x\in\operatorname{Her}(a)\) with \(x^2=0\) and \(x_P\ne0\) throughout \(K_0\). Set \(z_1=x^*x\). Its Cuntz-equivalent positive \(xx^*\) is orthogonal to it, since \((x^*x)(xx^*)=x^*x^2x^*=0\). Therefore \[z_1^{\oplus2}\sim_{\mathrm{Cu}}z_1\oplus xx^* \sim_{\mathrm{Cu}}z_1+xx^*\precsim a, \qquad (z_1)_P\ne0\quad(P\in K_0).\] Repeat the construction inside \(\operatorname{Her}(z_1)\), and then inside successive hereditary subalgebras, always over the same \(K_0\). It yields \(z_j\in C_+\) nonzero throughout \(K_0\) and \(z_{j+1}^{\oplus2}\precsim z_j\). Finite direct sums and transitivity give \[z_j^{\oplus 2^j}\precsim a.\] Choose a finite \(j\ge1\) with \(2^j\ge n\). Dropping extra summands gives \(z_j^{\oplus n}\precsim a\). If \((z_j)_P=0\), then \(P\notin K_0\), so \(\|a_P\|<r\) and \(((a-r)_+)_P=0\). The primitive ideal test in Lemma 6 therefore gives \((a-r)_+\in\operatorname{Id}(z_j)\). Condition (F\(_n\)) makes \(z_j^{\oplus n}\) properly infinite because \(z_j\) is a scalar positive element. Lemma 28, used with two copies on the left, then implies \[((a-r)_+)^{\oplus2}\precsim z_j^{\oplus n}\precsim a.\] Only scalar positives \(z_j\in C\) have been tested against (F\(_n\)). Finally, \(\|a-(a-r)_+\|\le r\). As \(r\downarrow0\), norm-limit closure of comparison with the fixed target \(a\) gives \[ a\oplus a\precsim a\qquad(a\in C_+). \tag{98}\] This includes \(a=0\); when \(n=1\), it also follows directly from the hypothesis. Therefore \(C\) satisfies (P). Theorem 1 now gives strong pure infiniteness, completing the proof. ◻

Proof. Property (P) makes \(h\) properly infinite, so Lemma 28 applies with multiplicity one. If \(h=0\), the ideal is zero and the assertion is immediate. For exact \(C\), property (P) is precisely the amplification hypothesis of Proposition 31 with \(n=1\) for every scalar positive element. Its constant was fixed independently of \(n\), so neither the constant nor the hereditary and selfadjoint options change. ◻