Predecessors and the contradiction

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

This lesson proves that the full C*-algebra of the group \(G\) has no unital embedding into the norm ultrapower \(B^\omega\) of a nuclear C*-algebra \(B\). Suppose there is one. Lifting the images of the group elements to sequences of unitaries of \(B\) gives, for every index \(n\), a tuple of unitaries of \(B\) and with it a finite set \(D_n\) of irreducible representations of the subgroup \(\Gamma\): those that the tuple matches approximately. The set is finite because \(B\) is nuclear. For most \(n\) it contains the representation \(\pi_*\). The element \(t\) of \(G\), which conjugates \(\Gamma\) onto a subgroup of index \(8\), forces every member \(\pi\) of \(D_n\) to have a predecessor \(\pi'\) in \(D_n\): a member with \(\pi\prec\pi'\circ\theta\). In a finite set, chains of predecessors must repeat, and this makes \(\pi_*\) equivalent to \(\pi_*\circ\theta^p\) for some \(p\ge1\), which is false.

The lesson follows [OpenAI-K, Sections 5 and 6]. It uses all of Norm ultrapowers and a group with property (T) and, from A spectral test and nuclear finiteness, the Laplacian, Lemmas 1.2, 2.1, 2.2, 3.1 and 3.2, and Proposition 8.1.

1. The setting

Assume the hypothesis (T). The group \(G\), its subgroup \(\Gamma\), the element \(t\), the endomorphism \(\theta\) with \(t\gamma t^{-1}=\theta(\gamma)\), the subgroup \(\Lambda=\theta(\Gamma)\) with the coset representatives \(r_1=e,r_2,\dots,r_8\), the representation \(\pi_*\), the generating set \(S\) and the numbers \(\kappa\), \(R=4|S|\), \(b=\kappa^2/(4|S|)\) and \(a<b\) are as in the first two lessons.

Fix a nuclear unital C*-algebra \(B\) and a free ultrafilter \(\omega\) on \(\mathbb N\), and suppose that \[ \Phi\colon C^*(G)\to B^\omega \] is a unital embedding. Sections 1 to 6 derive consequences of this assumption; Section 7 obtains a contradiction.

Lifts. For \(g\in G\) the element \(\Phi(u_g)\) is a unitary of \(B^\omega\). By the lemma on unitary lifts choose unitaries \(x_n(g)\in B\), \(n\in\mathbb N\), with \[ \Phi(u_g)=[(x_n(g))_n] , \] and \(x_n(e)=1\). Since \(\Phi(u_g)\Phi(u_h)=\Phi(u_{gh})\), the norm formula for ultraproducts gives, for every fixed pair \(g,h\in G\), \[ \lim_\omega\|x_n(g)\,x_n(h)-x_n(gh)\|=0 . \] We refer to this as the lift relation. For a single \(n\) the map \(g\mapsto x_n(g)\) need not be a homomorphism. Every use of the lift relation below concerns finitely many fixed group elements.

The sets \(D_n\). Let \(\mathbf x_n=(x_n(s))_{s\in S}\), a tuple of unitaries of \(B\), and \[ D_n=\{\,[\pi]:\ d(\mathbf x_n,\pi)<a\,\} , \] where \(\pi\) runs through the irreducible unitary representations of \(\Gamma\) of finite dimension and \([\pi]\) is the equivalence class. By the proposition on nuclear finiteness, every \(D_n\) is finite.

2. A gap for varying dimensions

Lemma 2.1. Let \(\pi_1,\pi_2,\dots\) be unitary representations of \(\Gamma\) of finite dimensions \(d_n\ge1\), not necessarily bounded. Then \[ \lim_\omega d(\mathbf x_n,\pi_n)\in\{0\}\cup[\kappa,R] . \] In particular, if \(d(\mathbf x_n,\pi_n)<a\) for all \(n\) in a large set, the limit is \(0\).

Proof. Let \(C_n=M_{d_n}(B)\) and let \(C_\omega\) be the ultraproduct of the \(C_n\). For \(g\in\Gamma\) put \(U_n(g)=x_n(g)\otimes\bar\pi_n(g)\), a unitary of \(C_n\). Since \(\bar\pi_n\) is a homomorphism, with \(y_n=x_n(g)x_n(h)-x_n(gh)\), \[ U_n(g)U_n(h)-U_n(gh)=y_n\otimes\bar\pi_n(gh) , \] and the norm of this element is \(\|y_n\|\). By the lift relation, \(g\mapsto[(U_n(g))_n]\) is a homomorphism from \(\Gamma\) into the unitary group of \(C_\omega\). By the lemma on the spectral gap, the Laplacian of this homomorphism has its spectrum in \(\{0\}\cup[\kappa,R]\). This Laplacian is the class of the sequence \[ \begin{gathered} P_n=L\bigl((x_n(s)\otimes\bar\pi_n(s))_{s\in S}\bigr),\\ 0\le P_n\le R , \end{gathered} \] and \(\min\operatorname{spec}P_n=d(\mathbf x_n,\pi_n)\). By the lemma on spectral minima in ultraproducts, the smallest point of the spectrum of the class is \(\lim_\omega d(\mathbf x_n,\pi_n)\). In the last statement the limit is at most \(a<\kappa\). \(\blacksquare\)

The lemma is where the approximate relations are turned into exact ones: the ultraproduct of the matrix algebras over \(B\) carries a true representation of \(\Gamma\), whatever the dimensions \(d_n\) are.

3. Induced representations

Let \(H\) be a subgroup of a group \(G_0\), and let \((q_i)_{i\in I}\) be representatives of the left cosets, so that every element of \(G_0\) is \(q_ih\) for exactly one \(i\in I\) and one \(h\in H\). Let the index \(1\in I\) have \(q_1=e\). For \(g\in G_0\) and \(i\in I\) define \(j(g,i)\in I\) and \(h(g,i)\in H\) by \[ g\,q_i=q_{j(g,i)}\,h(g,i) . \] Let \(\alpha\) be a unitary representation of \(H\) on \(V=\mathbb C^d\). Let \(\ell^2(I;V)\) be the Hilbert space of the families \(w=(w_i)_{i\in I}\) in \(V\) with \(\sum_i\|w_i\|^2<\infty\). The induced representation \(\varrho=\operatorname{Ind}_H^{G_0}\alpha\) is defined on it by \[ \bigl(\varrho(g)w\bigr)_{j(g,i)}=\alpha\bigl(h(g,i)\bigr)\,w_i\qquad(i\in I) . \]

Lemma 3.1. 1. \(\varrho\) is a unitary representation of \(G_0\).

  1. For \(h\in H\): \((\varrho(h)w)_1=\alpha(h)w_1\). So the families with \(w_i=0\) for \(i\ne1\) form a subspace that is invariant under \(H\), on which \(H\) acts by \(\alpha\). In particular \(\alpha\prec\varrho|_H\) if \(\alpha\) is irreducible.

  2. For every \(i\): \((\varrho(q_i^{-1})w)_1=w_i\).

  3. If \(I\) is finite, the conjugate \(\bar\varrho\) is given by the same rule with \(\bar\alpha\) in place of \(\alpha\).

Proof. Part 1. For fixed \(g\) the map \(i\mapsto j(g,i)\) is a bijection of \(I\): from \(g^{-1}q_{j(g,i)}=q_i\,h(g,i)^{-1}\) its inverse is \(j(g^{-1},\cdot)\). So \(\varrho(g)\) permutes the coordinates and applies unitaries; it is unitary. For \(g_1,g_2\) and \(i\), write \(i'=j(g_2,i)\). Then \[ \begin{aligned} g_1g_2\,q_i&=g_1\,q_{i'}\,h(g_2,i)\\ &=q_{j(g_1,i')}\,h(g_1,i')\,h(g_2,i) . \end{aligned} \] So \(j(g_1g_2,i)=j(g_1,i')\) and \(h(g_1g_2,i)=h(g_1,i')\,h(g_2,i)\), which gives \(\varrho(g_1g_2)=\varrho(g_1)\varrho(g_2)\).

Part 2. \(h\,q_1=q_1\,h\), so \(j(h,1)=1\) and \(h(h,1)=h\). Part 3. \(q_i^{-1}q_i=q_1\,e\), so \(j(q_i^{-1},i)=1\) and \(h(q_i^{-1},i)=e\). Part 4. In the standard basis of \(V^I\) the matrix of \(\varrho(g)\) consists of the blocks \(\alpha(h(g,i))\) at the places \((j(g,i),i)\). \(\blacksquare\)

Lemma 3.2 (Frobenius reciprocity, one direction). Let \(H\) have finite index in \(G_0\), let \(\alpha\) be an irreducible unitary representation of \(H\) of finite dimension, \(\varrho=\operatorname{Ind}_H^{G_0}\alpha\), and let \(\pi'\) be an irreducible representation of \(G_0\) with \(\pi'\prec\varrho\). Then \(\alpha\prec\pi'|_H\).

Proof. Let \(W\) be an invariant subspace of \(\varrho\) on which \(\varrho\) is equivalent to \(\pi'\), and let \(P_1\) be the map \(w\mapsto w_1\) from \(W\) to \(V\). By part 2 of Lemma 3.1, \(P_1\varrho(h)=\alpha(h)P_1\) on \(W\) for \(h\in H\). The map \(P_1\) is not zero: a vector \(w\ne0\) of \(W\) has a coordinate \(w_i\ne0\), and by part 3 the vector \(\varrho(q_i^{-1})w\in W\) has the first coordinate \(w_i\). Taking adjoints, \(P_1^*\) is a nonzero intertwiner from \(\alpha\) to the restriction of \(\varrho|_H\) to \(W\). By Schur's lemma \(\alpha\) occurs in this restriction, which is equivalent to \(\pi'|_H\). \(\blacksquare\)

We apply this to \(\Lambda=\theta(\Gamma)\) in \(\Gamma\). If \(\pi\) is an irreducible representation of \(\Gamma\), then \(\alpha=\pi\circ\theta^{-1}\) is an irreducible representation of \(\Lambda\). If \(\pi'\) is an irreducible representation of \(\Gamma\) with \(\pi'\prec\operatorname{Ind}_\Lambda^\Gamma\alpha\), the lemma gives an isometry \(T\) with \(\pi'(\lambda)T=T\alpha(\lambda)\) for \(\lambda\in\Lambda\); putting \(\lambda=\theta(g)\) gives \(\pi'(\theta(g))T=T\pi(g)\) for \(g\in\Gamma\). So \[ \pi'\prec\operatorname{Ind}_\Lambda^\Gamma(\pi\circ\theta^{-1})\quad\Longrightarrow\quad\pi\prec\pi'\circ\theta . \]

Definition. For irreducible representations \(\pi,\pi'\) of \(\Gamma\), \(\pi'\) is a predecessor of \(\pi\) if \(\pi\prec\pi'\circ\theta\).

4. The first representation is matched

Lemma 4.1. \(\lim_\omega d(\mathbf x_n,\pi_*)=0\). Consequently the set of the \(n\) with \([\pi_*]\in D_n\) is large.

Proof. Let \(d\) be the dimension of \(\pi_*\) and \(\varrho=\operatorname{Ind}_\Gamma^G\pi_*\), a unitary representation of \(G\) on a Hilbert space \(H\). By Lemma 3.1, \(\pi_*\prec\varrho|_\Gamma\). By the lemma on representations in the lesson on the spectral test, the number \(d\) for the tuple \((\varrho(s))_{s\in S}\) and \(\pi_*\) is \(0\).

By the universal property, \(\varrho\) extends to a unital \(*\)-homomorphism from \(C^*(G)\) to \(\mathcal B(H)\) with \(u_g\mapsto\varrho(g)\). Let \(\mathbf u=(u_s)_{s\in S}\) in \(C^*(G)\), and \(L_{\mathbf u}=L((u_s\otimes\bar\pi_*(s))_s)\) in \(M_d(C^*(G))\). If \(L_{\mathbf u}\ge m\cdot1\) with \(m>0\), its image \(L((\varrho(s)\otimes\bar\pi_*(s))_s)\) would be \(\ge m\) too, by (C1) and (C6). So \(d(\mathbf u,\pi_*)=0\).

The embedding \(\Phi\) is unital and injective, so \(d(\Phi(\mathbf u),\pi_*)=0\), where \(\Phi(\mathbf u)=([(x_n(s))_n])_{s\in S}\) in \(B^\omega\). By the lemma on matrices over ultraproducts, the Laplacian for \(\Phi(\mathbf u)\) and \(\pi_*\) is the class of the sequence \(P_n=L((x_n(s)\otimes\bar\pi_*(s))_s)\) in \(M_d(B)\). By the lemma on spectral minima, \(0=\lim_\omega\min\operatorname{spec}P_n=\lim_\omega d(\mathbf x_n,\pi_*)\). The last statement follows since \(a>0\). \(\blacksquare\)

This is the only place where the injectivity of \(\Phi\) is used.

5. Predecessors

Proposition 5.1. The set of the \(n\) for which every class in \(D_n\) has a predecessor in \(D_n\) is large.

In symbols: the set of the \(n\) such that for all \([\pi]\in D_n\) there is \([\pi']\in D_n\) with \(\pi\prec\pi'\circ\theta\) belongs to \(\omega\). The order of the quantifiers matters: one large set of indices serves for all members of \(D_n\), although the sets \(D_n\) vary with \(n\) and the dimensions of their members are not bounded.

Proof. Suppose the set is not large. Then its complement \(F\) is large. For \(n\in F\) choose an irreducible \(\pi_n\) with \([\pi_n]\in D_n\) that has no predecessor in \(D_n\). For \(n\notin F\) let \(\pi_n\) be the trivial representation of dimension \(1\). Let \(d_n\) be the dimension of \(\pi_n\).

Step 1: almost invariant vectors. Since \(d(\mathbf x_n,\pi_n)<a\) on \(F\), Lemma 2.1 gives \(\lim_\omega d(\mathbf x_n,\pi_n)=0\). Let \(B\) act faithfully with its unit on a Hilbert space \(\mathcal H\), and let \(U_n(g)=x_n(g)\otimes\bar\pi_n(g)\) on \(\mathcal H^{d_n}\) for \(g\in\Gamma\). Choose unit vectors \(\xi_n\in\mathcal H^{d_n}\) with \[ \sum_{s\in S}\|U_n(s)\xi_n-\xi_n\|^2<d(\mathbf x_n,\pi_n)+\frac1n . \] Since \(\omega\) is free, \(\lim_\omega\|U_n(s)\xi_n-\xi_n\|=0\) for every \(s\in S\). We claim that for every \(g\in\Gamma\) \[ \lim_\omega\|U_n(g)\xi_n-\xi_n\|=0 . \] Let \(\Gamma_0\) be the set of the \(g\in\Gamma\) for which this holds. As in the proof of Lemma 2.1, \(\|U_n(g)U_n(g')-U_n(gg')\|\to0\) along \(\omega\) for fixed \(g,g'\), and \(U_n(e)=1\). For \(g,g'\in\Gamma_0\), \[ \begin{aligned} &\|U_n(gg')\xi_n-\xi_n\|\\ &\quad\le\|U_n(gg')-U_n(g)U_n(g')\|\\ &\qquad+\|U_n(g')\xi_n-\xi_n\|\\ &\qquad+\|U_n(g)\xi_n-\xi_n\| , \end{aligned} \] so \(gg'\in\Gamma_0\). For \(g\in\Gamma_0\), \[ \begin{aligned} &\|U_n(g^{-1})\xi_n-\xi_n\|\\ &\quad\le\|\xi_n-U_n(g)\xi_n\|\\ &\qquad+\|U_n(g^{-1})U_n(g)-1\| , \end{aligned} \] so \(g^{-1}\in\Gamma_0\). So \(\Gamma_0\) is a subgroup that contains the generating set \(S\), and \(\Gamma_0=\Gamma\).

Step 2: transport to \(\Lambda\). Let \(\alpha_n=\pi_n\circ\theta^{-1}\), a representation of \(\Lambda\) on \(\mathbb C^{d_n}\), and let \[ \eta_n=(x_n(t)\otimes1)\,\xi_n , \] a unit vector of \(\mathcal H^{d_n}\). We claim that for every \(h\in\Lambda\) \[ \lim_\omega\|(x_n(h)\otimes\bar\alpha_n(h))\,\eta_n-\eta_n\|=0 . \] Let \(g=\theta^{-1}(h)\in\Gamma\). Then \(h=tgt^{-1}\), so \(ht=tg\) in \(G\), and \(\bar\alpha_n(h)=\bar\pi_n(g)\). So \[ \begin{aligned} &(x_n(h)\otimes\bar\alpha_n(h))\,\eta_n\\ &\qquad=(x_n(h)x_n(t)\otimes\bar\pi_n(g))\,\xi_n,\\ &(x_n(t)\otimes1)\,U_n(g)\,\xi_n\\ &\qquad=(x_n(t)x_n(g)\otimes\bar\pi_n(g))\,\xi_n . \end{aligned} \] The difference of the two vectors has a norm of at most \(\|x_n(h)x_n(t)-x_n(t)x_n(g)\|\), which tends to \(0\) along \(\omega\) by the lift relation, because both products are close to \(x_n(ht)=x_n(tg)\). And the second vector differs from \(\eta_n\) by \((x_n(t)\otimes1)(U_n(g)\xi_n-\xi_n)\), whose norm tends to \(0\) by Step 1.

Step 3: induction to \(\Gamma\). Let \(\varrho_n=\operatorname{Ind}_\Lambda^\Gamma\alpha_n\), with the coset representatives \(r_1=e,\dots,r_8\); it has the dimension \(8d_n\), and we write the vectors of \(\mathcal H^{8d_n}\) as families \((\zeta_i)_{i\le8}\) with \(\zeta_i\in\mathcal H^{d_n}\). For \(g\in\Gamma\) the operator \(x_n(g)\otimes\bar\varrho_n(g)\) maps the family \(\zeta\) to the family whose entry at \(j(g,i)\) is \((x_n(g)\otimes\bar\alpha_n(h(g,i)))\,\zeta_i\), by part 4 of Lemma 3.1. Define the unit vector \(\zeta_n\) by \[ (\zeta_n)_i=8^{-1/2}\,(x_n(r_i)\otimes1)\,\eta_n \] for \(1\le i\le8\). Fix \(s\in S\) and \(i\), and write \(j=j(s,i)\) and \(h=h(s,i)\), so that \(sr_i=r_jh\). Put \(c_n=x_n(s)x_n(r_i)-x_n(r_j)x_n(h)\) and \(\epsilon_n=(x_n(h)\otimes\bar\alpha_n(h))\eta_n-\eta_n\). The entry at \(j\) of \((x_n(s)\otimes\bar\varrho_n(s))\zeta_n-\zeta_n\) is \(8^{-1/2}\) times the difference of \((x_n(s)x_n(r_i)\otimes\bar\alpha_n(h))\,\eta_n\) and \((x_n(r_j)\otimes1)\,\eta_n\), and this difference is \[ (c_n\otimes\bar\alpha_n(h))\,\eta_n+(x_n(r_j)\otimes1)\,\epsilon_n . \] The first term has a norm of at most \(\|c_n\|\), which tends to \(0\) along \(\omega\) by the lift relation and \(sr_i=r_jh\). The second has the norm \(\|\epsilon_n\|\), which tends to \(0\) by Step 2. There are \(8|S|\) pairs \((s,i)\), all fixed. So \[ \lim_\omega\sum_{s\in S}\|(x_n(s)\otimes\bar\varrho_n(s))\zeta_n-\zeta_n\|^2=0 , \] and therefore \(\lim_\omega d(\mathbf x_n,\varrho_n)=0\). None of the estimates depends on \(d_n\).

Step 4: a predecessor. Let \(F'\) be the large set of the \(n\) with \(d(\mathbf x_n,\varrho_n)<a\). The representation \(\varrho_n\) is equivalent to a direct sum of irreducible representations. By parts 2 and 3 of the lemma on the properties of \(d\), for \(n\in F'\) one of them, \(\pi'_n\), has \(d(\mathbf x_n,\pi'_n)<a\); so \([\pi'_n]\in D_n\). Since \(\pi'_n\prec\varrho_n=\operatorname{Ind}_\Lambda^\Gamma(\pi_n\circ\theta^{-1})\), the consequence of Lemma 3.2 gives \(\pi_n\prec\pi'_n\circ\theta\). So for \(n\in F\cap F'\), which is not empty, \(\pi_n\) has a predecessor in \(D_n\), against its choice. \(\blacksquare\)

The roles of the pieces: the element \(t\) carries a vector that is almost invariant for \(\Gamma\) in the representation \(\pi_n\) to a vector that is almost invariant for the smaller group \(\Lambda\) (Step 2). The induction with eight cosets spreads it to a vector that is almost invariant for \(\Gamma\) in the larger representation \(\varrho_n\) (Step 3). The gap of Lemma 2.1 and the finiteness of the index turn this into a member of \(D_n\) (Step 4).

6. Finite chains

Lemma 6.1. Let \(\theta\) be a homomorphism from a group \(\Gamma\) to itself, and \(D\) a finite set of equivalence classes of irreducible unitary representations of \(\Gamma\) of finite dimension such that every \([\pi]\in D\) has a predecessor \([\pi']\in D\): \(\pi\prec\pi'\circ\theta\). Then for every \([\pi]\in D\) there is \(p\ge1\) with \(\pi\simeq\pi\circ\theta^p\).

Proof. We use three simple facts for representations of finite dimension: if \(\alpha\prec\beta\), then \(\alpha\circ\theta\prec\beta\circ\theta\); the relation \(\prec\) is transitive; and if \(\alpha\prec\beta\) and the dimensions are equal, then \(\alpha\simeq\beta\).

Choose \(\tau_0=\pi\) and then \(\tau_1,\tau_2,\dots\) with classes in \(D\) and \(\tau_i\prec\tau_{i+1}\circ\theta\). Since \(D\) is finite, there are \(l\ge0\) and \(p\ge1\) with \(\tau_l\simeq\tau_{l+p}\). Composing the relations along the chain, \[ \tau_l\prec\tau_{l+p}\circ\theta^p\simeq\tau_l\circ\theta^p . \] The dimensions are equal, so \(\tau_l\simeq\tau_l\circ\theta^p\), and by iteration \(\tau_l\simeq\tau_l\circ\theta^{kp}\) for all \(k\ge0\).

For every \(i\ge0\) the representation \(\tau_l\circ\theta^i\) is irreducible. Indeed, choose \(k\) with \(kp\ge i\). A subspace that is invariant under \(\tau_l(\theta^i(\Gamma))\) is invariant under \(\tau_l(\theta^{kp}(\Gamma))\), because \(\theta^{kp}(\Gamma)\subseteq\theta^i(\Gamma)\); and \(\tau_l\circ\theta^{kp}\simeq\tau_l\) is irreducible.

The beginning of the chain gives \(\pi\prec\tau_l\circ\theta^l\). The right side is irreducible, so \(\pi\simeq\tau_l\circ\theta^l\). Hence \[ \begin{aligned} \pi\circ\theta^p&\simeq\tau_l\circ\theta^{l+p}=(\tau_l\circ\theta^p)\circ\theta^l\\ &\simeq\tau_l\circ\theta^l\simeq\pi . \end{aligned} \] \(\blacksquare\)

Finiteness is essential (Exercise 9.2).

7. Proof of the theorem

Theorem 7.1 ([OpenAI-K, Section 1]). Assume (T). For every nuclear unital C*-algebra \(B\) and every free ultrafilter \(\omega\) on \(\mathbb N\) there is no unital embedding of \(C^*(G)\) into \(B^\omega\).

Proof. Suppose \(\Phi\) is one, and use the notation of Section 1. By Lemma 4.1 and Proposition 5.1 there is an index \(n\) such that \([\pi_*]\in D_n\) and every class in \(D_n\) has a predecessor in \(D_n\): two large sets have a common element. The set \(D_n\) is finite because \(B\) is nuclear. By Lemma 6.1 there is \(p\ge1\) with \(\pi_*\simeq\pi_*\circ\theta^p\). This contradicts the lemma on \(\pi_*\) in the first lesson: \(\pi_*\) is not trivial on the translations, and \(\pi_*\circ\theta^p\) is. \(\blacksquare\)

The proof used the following properties of \(G\) and nothing else: \(G\) contains a finitely generated subgroup \(\Gamma\) with property (T) and an element \(t\) such that \(\gamma\mapsto t\gamma t^{-1}\) maps \(\Gamma\) onto a subgroup of finite index of \(\Gamma\); and \(\Gamma\) has an irreducible unitary representation \(\pi_*\) of finite dimension that is not equivalent to \(\pi_*\circ\theta^p\) for any \(p\ge1\).

Where the assumptions on the target enter: that \(B\) is nuclear is used only for the finiteness of \(D_n\). The ultrapower is used through the lift relation and the norm formula. That \(\Phi\) is injective is used only in Lemma 4.1.

8. Kirchberg's problem and quoted consequences

With the hypothesis (O) of the first lesson, that \(\mathcal O_2\) is nuclear, Theorem 7.1 for \(B=\mathcal O_2\) gives: the separable unital C*-algebra \(C^*(G)\) has no unital embedding into \(\mathcal O_2^\omega\). This is the negative answer to Kirchberg's embedding problem (Corollary 4.2 of the first lesson).

[OpenAI-K, Section 6] derives three further statements from this answer by theorems of Goldbring and Sinclair [GS, Proposition 3.2, Theorems 3.3 and 3.7]. They are reported here without proof; the theorems of [GS] are not part of the programme, and their statements were not checked for this course.

9. Exercises

9.1. Let \(H=\{e\}\) and let \(\alpha\) be the trivial representation of dimension \(1\). Show that \(\operatorname{Ind}_H^{G_0}\alpha\), with \(I=G_0\) and \(q_g=g\), is the representation of \(G_0\) on \(\ell^2(G_0)\) by left translation.

9.2. Let \(\Gamma=\mathbb Z\) and \(\theta(k)=2k\). The irreducible unitary representations of finite dimension are the characters \(\chi_z(k)=z^k\), \(|z|=1\). Find an infinite set \(D\) of characters in which every member has a predecessor in \(D\), and a member \(\chi\) with \(\chi\not\simeq\chi\circ\theta^p\) for all \(p\ge1\).

9.3. For \(\Lambda\) in \(\Gamma\) with the representatives \(r_a=(a,I)\), \(a\in\{0,1\}^3\), compute \(j(g,a)\) and \(h(g,a)\) for \(g=(v,M)\).

9.4. Let \(\mathbf 1_\Lambda\) be the trivial representation of \(\Lambda\) of dimension \(1\). Show that \(\operatorname{Ind}_\Lambda^\Gamma\mathbf 1_\Lambda\) is the representation of \(\Gamma\) on the functions on \((\mathbb Z/2)^3\) of Exercise 7.5 of the first lesson, and conclude that the representation \(\pi_7\) there is a predecessor of the trivial representation of \(\Gamma\). Verify this directly.

9.5. Show that Theorem 7.1 also excludes embeddings that are not unital, in the following sense: there is no injective \(*\)-homomorphism \(\Phi\) from \(C^*(G)\) into \(B^\omega\) such that the projection \(\Phi(1)\) is the class of a sequence of projections \(e_n\in B\) for which the algebras \(e_nBe_n\) are nuclear.

10. Solutions

9.1. \(g\,q_{g'}=q_{gg'}\,e\), so \(j(g,g')=gg'\) and \(h(g,g')=e\). The rule reads \((\varrho(g)w)_{gg'}=w_{g'}\), which is left translation.

9.2. \(\chi_z\circ\theta=\chi_{z^2}\), and for characters \(\prec\) is equality. So \(\chi_{z'}\) is a predecessor of \(\chi_z\) exactly if \(z'^2=z\). Let \(z_i=e^{i\pi/2^i}\), \(i\ge0\); then \(z_{i+1}^2=z_i\) and \(z_0=-1\). In \(D=\{\chi_{z_i}:i\ge0\}\) every member has a predecessor, and \(\chi_{-1}\circ\theta^p=\chi_1\ne\chi_{-1}\) for all \(p\ge1\).

9.3. \(g\,r_a=(v+Ma,M)\). Let \(a'\in\{0,1\}^3\) be the residue of \(v+Ma\) modulo \(2\). Then \(g\,r_a=r_{a'}\,(v+Ma-a',M)\), so \(j(g,a)=a'\) and \(h(g,a)=(v+Ma-a',M)\in\Lambda\).

9.4. By Exercise 9.3 the induced representation permutes the eight coordinates by \(a\mapsto\bar M a+\bar v\), the action of Exercise 7.5 of the first lesson; a family \((w_a)\) is a function on \((\mathbb Z/2)^3\), and \(\varrho(g)\) is the translation of functions by \(g\). The functions of sum zero form an invariant subspace on which \(\varrho\) is \(\pi_7\). So \(\pi_7\prec\operatorname{Ind}_\Lambda^\Gamma\mathbf 1_\Lambda=\operatorname{Ind}_\Lambda^\Gamma(\mathbf 1_\Gamma\circ\theta^{-1})\), and the consequence of Lemma 3.2 gives \(\mathbf 1_\Gamma\prec\pi_7\circ\theta\). Directly: \(\pi_7(\theta(v,M))=\pi_7(2v,M)\) acts by the linear map \(\bar M\). The function that is \(7\) at \(0\) and \(-1\) at the seven other points has the sum zero and is invariant under all linear maps.

9.5. Let \(e=\Phi(1)=[(e_n)]\). Since \(\Phi\) is injective, \(e\ne0\), so the set of the \(n\) with \(e_n\ne0\) is large; for the other \(n\) replace \(e_nBe_n\) by \(\mathbb C\), which does not change the ultraproduct. The corner \(e\,B^\omega e\) is the ultraproduct of the unital C*-algebras \(e_nBe_n\), by the norm formula, and \(\Phi\) is a unital embedding of \(C^*(G)\) into it. The proof of Theorem 7.1 works for an ultraproduct of nuclear unital C*-algebras \(B_n\) in place of an ultrapower: the proposition on nuclear finiteness is applied to each \(B_n\) separately, and all other steps concern ultraproducts of varying algebras already.

References