Central sequences and the character criterion

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

A unital separable C*-algebra \(A\) absorbs the Jiang–Su algebra \(\mathcal Z\) tensorially exactly when its central sequence algebra \(F_\omega(A)\) contains, with the same unit, a separable subhomogeneous subalgebra without characters (Kirchberg and Rørdam). A character of \(F_\omega(A)\) would restrict to a character of such a subalgebra, so the absence of characters on \(F_\omega(A)\) is necessary; Kirchberg and Rørdam asked whether it is sufficient [KR, Question 3.1]. This course proves the theorem of [OpenAI-KRC]: it is. The new ingredient is a finite theorem: every nonzero unital C*-algebra without characters has a finite maximal tensor power that receives a unital homomorphism from the dimension-drop algebra \(I(2,3)\).

This lesson states the theorems, introduces the tools, and derives the main theorem and a corollary on infinite tensor powers from the finite theorem and two results from the literature, which are named hypotheses of the course.

The course follows [OpenAI-KRC]. All algebras are complex. Facts on C*-algebras are taken from the course Foundations of von Neumann algebras: continuous functional calculus (Theorem 5.1 there), positivity (Theorem 8.2 there), Gelfand's theorem for commutative algebras (Theorem 2.1 there) and quotients (Theorem 15.1 there).

1. Ultrapowers, central sequences and characters

Let \(\omega\) be a free ultrafilter on \(\mathbb N\) and \(A\) a nonzero unital C*-algebra. The ultrapower is \[ A_\omega=\ell^\infty(\mathbb N,A)\big/c_\omega(A), \] where \(c_\omega(A)\) is the closed ideal of bounded sequences with \(\lim_{n\to\omega}\|a_n\|=0\). The norm of the class of \((a_n)\) is \(\lim_{n\to\omega}\|a_n\|\) (Norm ultrapowers and a group with property (T), Lemma 2.2). Identify \(A\) with the classes of constant sequences. The central sequence algebra is \[ F_\omega(A)=A_\omega\cap A', \] the classes commuting with every constant sequence; it is a unital C*-subalgebra of \(A_\omega\), and it is nonzero since \(\|1\|=1\).

A character of a C*-algebra is a nonzero \(*\)-homomorphism to \(\mathbb C\); on a unital algebra it sends \(1\) to \(1\). A C*-algebra is subhomogeneous if the dimensions of its irreducible representations have a common finite bound. For integers \(p,q\ge1\), the dimension-drop algebra is \[ \begin{aligned} I(p,q)=\{f\in{}&C([0,1],M_p\otimes M_q):\\ &f(0)\in M_p\otimes1_q,\\ &f(1)\in1_p\otimes M_q\}. \end{aligned} \] The Jiang–Su algebra \(\mathcal Z\), constructed by Jiang and Su, is a simple unital separable C*-algebra; it is an inductive limit of dimension-drop algebras \(I(p,q)\) with \(p,q\) coprime, and it contains \(I(2,3)\) unitally [KR, Section 2]. This course uses \(\mathcal Z\) only through the two named hypotheses below. A C*-algebra \(A\) is \(\mathcal Z\)-stable if \(A\cong A\otimes_{\min}\mathcal Z\).

2. The theorems and the named hypotheses

Theorem 2.1 ([OpenAI-KRC, Theorem 1.1]). Let \(A\) be a nonzero unital separable C*-algebra and \(\omega\) a free ultrafilter on \(\mathbb N\). Then \(F_\omega(A)\) has no characters if and only if \(A\cong A\otimes_{\min}\mathcal Z\).

Theorem 2.2 (the finite theorem; [OpenAI-KRC, Theorem 1.2]). Let \(D\) be a nonzero unital C*-algebra without characters. There are an integer \(m\ge1\) and a unital \(*\)-homomorphism \[ I(2,3)\longrightarrow D^{\otimes_{\max}m}. \]

Corollary 2.3 (infinite tensor powers; [OpenAI-KRC, Corollary 1.3]). Let \(D\) be a nonzero unital separable C*-algebra without characters, and let \(B=D^{\otimes_{\min}\infty}\) be the inductive limit of the algebras \(D^{\otimes_{\min}n}\) under the maps \(x\mapsto x\otimes1_D\). Then \(B\cong B\otimes_{\min}\mathcal Z\), and \(B\) contains a unital copy of \(\mathcal Z\).

In Theorem 2.2 the exponent \(m\) depends on \(D\), and the conclusion is a unital homomorphism, not \(\mathcal Z\)-stability of a finite tensor power. Its image is a quotient of \(I(2,3)\), hence subhomogeneous and without characters (Lemma 4.1), and this is what the applications need. Theorem 2.2 is proved in the lessons The Rørdam–Winter criterion to From the cone to the dimension-drop algebra; its proof uses no separability, nuclearity or trace.

The two results from the literature, which this course does not prove:

Hypothesis (KR) (Kirchberg–Rørdam, [KR, Theorem 2.5]). For a unital separable C*-algebra \(A\) and a free ultrafilter \(\omega\), \(A\cong A\otimes_{\min}\mathcal Z\) if and only if \(F_\omega(A)\) contains, with the same unit, a separable subhomogeneous C*-algebra without characters.

Hypothesis (DT) (Dadarlat–Toms, as stated in [KR, Theorem 2.4]). For a unital separable C*-algebra \(A\), if \(A^{\otimes_{\min}\infty}\) contains, with the same unit, a subhomogeneous C*-algebra without characters, then \(A^{\otimes_{\min}\infty}\cong A^{\otimes_{\min}\infty}\otimes_{\min}\mathcal Z\).

Both rest on the theory of the Jiang–Su algebra and of strongly self-absorbing algebras, which the programme does not contain.

3. Maximal tensor products

Let \(A\) and \(B\) be unital C*-algebras with algebraic tensor product \(A\odot B\), a unital \(*\)-algebra. For \(x\in A\odot B\) put \[ \|x\|_{\max}=\sup_\pi\|\pi(x)\|, \] the supremum over all \(*\)-representations \(\pi\) of \(A\odot B\) on Hilbert spaces. For \(x=\sum_ia_i\otimes b_i\), each \(a\mapsto\pi(a\otimes1)\) and \(b\mapsto\pi(1\otimes b)\) is a \(*\)-homomorphism of a C*-algebra, hence contractive, so \(\|x\|_{\max}\le\sum_i\|a_i\|\|b_i\|\). The supremum is a C*-seminorm, and a norm because it dominates the minimal norm \(\|\cdot\|_{\min}\), which is realized by a representation (Tensor products of C*-algebras and the minimal norm, Theorem 2.2). The completion is the maximal tensor product \(A\otimes_{\max}B\). For several factors \(A_1,\dots,A_k\) the same definition gives \(A_1\otimes_{\max}\cdots\otimes_{\max}A_k\), and \(D^{\otimes_{\max}k}\) is the \(k\)-th maximal tensor power.

Lemma 3.1. Let \(A_1,\dots,A_k\) be unital C*-algebras and \(P=A_1\otimes_{\max}\cdots\otimes_{\max}A_k\).

  1. (Universal property.) If \(\varphi_j\colon A_j\to E\) are unital \(*\)-homomorphisms into a unital C*-algebra with pairwise commuting ranges, there is exactly one unital \(*\)-homomorphism \(P\to E\) with \(a_1\otimes\cdots\otimes a_k\mapsto\varphi_1(a_1)\cdots\varphi_k(a_k)\).
  2. Each leg \(a\mapsto1\otimes\cdots\otimes a\otimes\cdots\otimes1\) is an injective unital \(*\)-homomorphism \(A_j\to P\), and the legs commute.
  3. (Regrouping.) For a partition of \(\{1,\dots,k\}\) into consecutive blocks, \(P\) is canonically isomorphic to the maximal tensor product of the maximal tensor products of the blocks.
  4. (Functoriality.) Unital \(*\)-homomorphisms \(\psi_j\colon A_j\to B_j\) give a unital \(*\)-homomorphism \(\psi_1\otimes\cdots\otimes\psi_k\) from \(P\) to \(B_1\otimes_{\max}\cdots\otimes_{\max}B_k\).
  5. The identity of the algebraic tensor product extends to a surjective unital \(*\)-homomorphism \(P\to A_1\otimes_{\min}\cdots\otimes_{\min}A_k\).
  6. Every irreducible representation \(\pi\) of \(P\) is unital, and its restrictions \(\pi_j\) to the legs are unital representations with pairwise commuting ranges.
  7. For \(y\in A_1\otimes_{\max}A_2\) and \(c\in A_3\), \(\|y\otimes c\|\le\|y\|\|c\|\) in \(A_1\otimes_{\max}A_2\otimes_{\max}A_3\).

Proof. (1) Represent \(E\) faithfully on a Hilbert space (Gelfand–Naimark, Building representations from positive functionals, Theorem 7.2). The formula defines a \(*\)-representation of the algebraic tensor product, whose norm is at most \(\|\cdot\|_{\max}\) by definition; it extends to \(P\). Uniqueness holds because elementary tensors span a dense subalgebra. (2) The legs are \(*\)-homomorphisms; they are injective because the minimal norm, which is at most the maximal one, restricts to the norm of \(A_j\) on each leg (Tensor products of C*-algebras and the minimal norm, Lemma 3.1). (3) Both algebras have the universal property (1) for commuting unital homomorphisms of \(A_1,\dots,A_k\), the second by applying (1) twice; the maps given by (1) in both directions are mutually inverse on elementary tensors. (4) Apply (1) to the legs of the target composed with the \(\psi_j\). (5) The minimal norm is at most the maximal norm on the algebraic tensor product. (6) \(\pi(1)\) is a projection commuting with \(\pi(P)\), hence a scalar by Schur's lemma (Building representations from positive functionals, Corollary 2.3), and nonzero; so \(\pi(1)=1\). Compose with the legs of (2). (7) In every representation, \(\pi(y\otimes c)=\pi(y\otimes1)\pi(1\otimes1\otimes c)\), and both factors are images of \(*\)-homomorphisms of C*-algebras. \(\blacksquare\)

4. Dimension-drop algebras, separable reduction and commuting copies

Lemma 4.1. Every irreducible representation of \(I(2,3)\) is unitarily equivalent to the evaluation at a point \(t\in[0,1]\), followed by the identity representation of \(M_2\otimes1\cong M_2\) if \(t=0\), of \(1\otimes M_3\cong M_3\) if \(t=1\), and of \(M_6\) otherwise. So \(I(2,3)\) is subhomogeneous and has no characters, and the same holds for every quotient of \(I(2,3)\).

Proof. The scalar functions \(g(t)1\), \(g\in C([0,1])\), lie in \(I(2,3)\) and are central. Let \(\pi\) be irreducible. By Schur's lemma \(\pi(\iota1)=t_0\) for the function \(\iota(t)=t\) and a number \(t_0\in[0,1]\), and then \(\pi(g1)=g(t_0)\) by the functional calculus. If \(F(t_0)=0\), let \(g_n(t)=\min\{1,n|t-t_0|\}\); then \(\|F-Fg_n\|\to0\) and \(\pi(Fg_n)=\pi(F)g_n(t_0)=0\), so \(\pi(F)=0\). Hence \(\pi\) factors through the evaluation at \(t_0\), whose image is \(M_2\otimes1\), \(1\otimes M_3\) or \(M_2\otimes M_3=M_6\): functions \(g(t)X\) with \(g\) vanishing at the endpoints and \(g(t_0)=1\) give every \(X\) at an interior point. An irreducible representation \(\rho\) of a matrix algebra \(M_n\) on \(H\) is equivalent to the identity representation: for a unit vector \(\xi\) in \(\rho(e_{11})H\), the vectors \(\rho(e_{i1})\xi\) are orthonormal and span an invariant subspace, which is \(H\). So the irreducible representations have dimensions \(2\), \(3\) and \(6\), and none is a character. For a quotient map \(q\colon I(2,3)\to E\), an irreducible representation or a character of \(E\), composed with \(q\), is one of \(I(2,3)\). \(\blacksquare\)

Lemma 4.2 (separable reduction; [OpenAI-KRC, Lemma 6.1], after [KR, Lemma 3.5]). Every nonzero unital C*-algebra \(B\) without characters contains a separable C*-subalgebra without characters with the unit of \(B\).

Proof. Let \(J\) be the closed ideal generated by the commutators \([b,c]\) of self-adjoint elements. In \(B/J\) self-adjoint elements commute, so it is commutative; if \(J\ne B\), \(B/J\) is a nonzero unital commutative C*-algebra, which has a character (Gelfand), and composing with the quotient map gives a character of \(B\). So \(J=B\), and some element of the algebraic ideal generated by the commutators is within distance less than \(1\) of \(1\): \[ \Bigl\|1-\sum_{i=1}^rh_i[b_i,c_i]k_i\Bigr\|<1 . \] Let \(D\) be the C*-algebra generated by \(1\) and the elements \(h_i,k_i,b_i,c_i\); it is separable and unital. A character \(\chi\) of \(D\) kills the commutators, so \(\chi\) of the displayed element \(1-\sum_i\dots\) equals \(1\), while characters are contractive. \(\blacksquare\)

Lemma 4.3 (commuting copies; [OpenAI-KRC, Proposition 2.2(1)], after [KR, Corollary 2.3]). Let \(A\) be a nonzero unital separable C*-algebra, \(\omega\) a free ultrafilter, and \(D\subseteq F_\omega(A)\) a separable C*-subalgebra containing the unit. For every \(m\ge1\) there is a unital \(*\)-homomorphism \(D^{\otimes_{\max}m}\to F_\omega(A)\).

Proof. One new copy. Let \(E\subseteq F_\omega(A)\) be a separable C*-subalgebra. We construct a unital \(*\)-homomorphism \(\theta\colon D\to F_\omega(A)\cap E'\). Let \(D_0\) be a countable dense unital \(*\)-subalgebra of \(D\) over \(\mathbb Q+i\mathbb Q\). For \(x\in D_0\) choose a representing sequence \((a^x_l)_l\) with \(\sup_l\|a^x_l\|\le2\|x\|\): since \(\|x\|=\lim_{l\to\omega}\|a_l\|\) for any representative, replacing the terms of norm larger than \(2\|x\|\) by \(0\) changes the sequence only on a set outside \(\omega\) (if \(x\ne0\)); take the zero sequence for \(x=0\) and the constant unit for \(x=1\). Choose a countable dense subset \(\{a_k\}\) of \(A\), a countable dense subset \(\{e_j\}\) of \(E\), and bounded representing sequences \((b^j_s)_s\) of the \(e_j\).

The following countably many tests hold along \(\omega\): the defects \(a^{x+y}_l-a^x_l-a^y_l\), \(a^{\lambda x}_l-\lambda a^x_l\), \(a^{xy}_l-a^x_la^y_l\), \(a^{x^*}_l-(a^x_l)^*\) and \([a^x_l,a_k]\) tend to \(0\) in norm as \(l\to\omega\), for \(x,y\in D_0\), \(\lambda\in\mathbb Q+i\mathbb Q\) and all \(k\). Enumerate the tests and the pairs \((x,j)\). For each \(s\), choose an index \(l(s)\) at which the first \(s\) tests hold within \(1/s\) and \(\|[a^x_{l(s)},b^j_s]\|<1/s\) for the first \(s\) pairs \((x,j)\). Such an index exists: each of these finitely many requirements holds on a set of indices \(l\) belonging to \(\omega\), the last ones because \(b^j_s\) is a fixed element of \(A\) and \(x\) commutes with \(A\) in \(A_\omega\); a finite intersection of sets of \(\omega\) is in \(\omega\), hence nonempty.

The sequences \((a^x_{l(s)})_s\) satisfy every test in the ordinary limit \(s\to\infty\), hence along \(\omega\); being bounded and asymptotically central for a dense subset of \(A\), their classes \(\theta(x)\) lie in \(F_\omega(A)\), and \(\|[\theta(x),e_j]\|=\lim_\omega\|[a^x_{l(s)},b^j_s]\|=0\), so \(\theta(x)\in E'\). On \(D_0\), \(\theta\) is a unital \(*\)-homomorphism over \(\mathbb Q+i\mathbb Q\), and \(\|\theta(x)-\theta(y)\|=\|\theta(x-y)\|\le2\|x-y\|\). It extends by continuity to a unital \(*\)-homomorphism on \(D\), complex linear by continuity of scalar multiplication.

Commuting copies. Let \(\theta_1\) be the inclusion of \(D\), and for \(i=2,\dots,m\) let \(\theta_i\) be the map just constructed for the C*-algebra \(E\) generated by \(\theta_1(D),\dots,\theta_{i-1}(D)\). The ranges commute pairwise, and Lemma 3.1(1) gives the homomorphism. \(\blacksquare\)

5. Proofs of the main theorem and the corollary

Proof of Theorem 2.1. Suppose \(F_\omega(A)\) has no characters. It is nonzero and unital, so Lemma 4.2 gives a separable C*-subalgebra \(D\subseteq F_\omega(A)\) without characters containing the unit. Theorem 2.2 and Lemma 4.3 give unital \(*\)-homomorphisms \[ I(2,3)\longrightarrow D^{\otimes_{\max}m}\longrightarrow F_\omega(A). \] The image \(E\) of the composite is separable, contains the unit of \(F_\omega(A)\), and is a quotient of \(I(2,3)\), hence subhomogeneous and without characters (Lemma 4.1). Hypothesis (KR) gives \(A\cong A\otimes_{\min}\mathcal Z\). Conversely, if \(A\cong A\otimes_{\min}\mathcal Z\), Hypothesis (KR) gives a subalgebra \(E\subseteq F_\omega(A)\) without characters containing the unit, and a character of \(F_\omega(A)\) would restrict to a character of \(E\). \(\blacksquare\)

Proof of Corollary 2.3. The maps \(x\mapsto x\otimes1_D\) are injective unital \(*\)-homomorphisms (Tensor products of C*-algebras and the minimal norm, Corollary 2.3 and Lemma 3.1), so \(B\) is a nonzero unital separable C*-algebra and the canonical maps \(\iota_m\colon D^{\otimes_{\min}m}\to B\) are unital. Compose the homomorphism \(\varphi\) of Theorem 2.2 with the quotient map \(q_m\) of Lemma 3.1(5): \[ I(2,3)\xrightarrow{\ \varphi\ }D^{\otimes_{\max}m}\xrightarrow{\ q_m\ }D^{\otimes_{\min}m}\xrightarrow{\ \iota_m\ }B . \] The image of the unital composite is separable, contains \(1_B\), and is a quotient of \(I(2,3)\), hence subhomogeneous and without characters (Lemma 4.1), even if the composite has a kernel. Hypothesis (DT) gives \(B\cong B\otimes_{\min}\mathcal Z\). Since \(B\ne0\), \(z\mapsto1_B\otimes z\) embeds \(\mathcal Z\) unitally into \(B\otimes_{\min}\mathcal Z\), and the inverse of an isomorphism \(B\cong B\otimes_{\min}\mathcal Z\), which is unital, carries it into \(B\). \(\blacksquare\)

6. Exercises

6.1. Show that a unital C*-algebra has no characters if \(M_2\) embeds into it unitally, and that \(M_2\otimes_{\max}D\) has no characters for every unital \(D\).

6.2. Show that \(I(2,2)\) has no characters but \(I(1,3)\) has one.

6.3. Show that \(F_\omega(A)\) is a C*-subalgebra of \(A_\omega\), that is, closed under the operations and in norm.

6.4. Why does the proof of Lemma 4.3 choose the indices \(l(s)\) depending on \(s\), rather than taking a fixed subsequence of the given representatives?

6.5. Show that a character \(\chi\) of a C*-algebra satisfies \(|\chi(x)|\le\|x\|\).

7. Solutions

6.1. A character \(\chi\) would restrict to a unital homomorphism \(M_2\to\mathbb C\). Since scalars commute, \(\chi(e_{11})=\chi(e_{12})\chi(e_{21})\), which equals \(\chi(e_{21})\chi(e_{12})=\chi(e_{22})\), and \(\chi(e_{11})\chi(e_{22})=\chi(e_{11}e_{22})=0\); so both vanish, contradicting \(\chi(e_{11})+\chi(e_{22})=\chi(1)=1\). The leg \(M_2\to M_2\otimes_{\max}D\) is unital (Lemma 3.1(2)).

6.2. For \(I(2,2)\), every irreducible representation has dimension \(2\) or \(4\), by the argument of Lemma 4.1. For \(I(1,3)\), evaluation at \(0\) has the image \(\mathbb C\otimes1_3\cong\mathbb C\) and is a character.

6.3. \(A_\omega\) is a C*-algebra (quotient by a closed ideal), and the commutant of a set is closed under sums, products, adjoints and norm limits.

6.4. The given representatives are asymptotically central only along \(\omega\), and the commutation with \(E\) must hold against the moving terms \(b^j_s\); a single subsequence chosen in advance cannot respond to the elements \(b^j_s\), which change with \(s\).

6.5. \(\chi\) is a \(*\)-homomorphism into the C*-algebra \(\mathbb C\), hence contractive.

References