An explicit obstruction to nuclear norm-ultrapower embeddings

1 The embedding problem

Throughout, C^*-algebras are complex, and an embedding is an injective *-homomorphism. We state unit preservation explicitly. For a nonzero unital C^*-algebra B and a free ultrafilter \omega on \N, its norm ultrapower is

Here \ell^\infty(\N,B) is the algebra of bounded sequences, and the quotient uses the operator norm. The Cuntz algebra \Otwo is the universal unital C^*-algebra generated by two isometries s_1,s_2 satisfying s_1s_1^*+s_2s_2^*=1; it was introduced by Cuntz [2].

The embedding theorem of Kirchberg–Phillips [8] states that every separable unital exact C^*-algebra embeds unitally into \Otwo. Here exactness means that minimal tensoring preserves short exact sequences. Passing from \Otwo to its norm ultrapower asks how far this universality extends beyond exact algebras. The techniques behind the embedding theorem led Kirchberg to consider this possibility, as recounted in [6, Section 1].

Kirchberg’s embedding problem asks whether every separable unital C^*-algebra admits a unital embedding into \Otwo^\omega; see Goldbring–Sinclair [6, Section 3.1]. We give an explicit counterexample. The unital formulation is equivalent to the formulation without unit preservation [6, Section 3.1]. This question concerns norm ultrapowers of a nuclear C^*-algebra. It is distinct from the tensor-norm formulation of Kirchberg’s conjecture and from Connes’ embedding problem, as emphasized in [6, Section 1].

The group is countable and independent of both B and \omega; no separability assumption on B is needed. The proof treats an arbitrary fixed nuclear B throughout. Specializing to the nuclear algebra \Otwo gives the negative answer to the embedding problem; for its nuclearity, see [11, Section 4, pp. 80–82]. Section 6 gives the consequences for existential closure, permanence of \Otwo^\omega-embeddability under crossed products, and good nuclear witnesses, with the relevant notions defined there.

1.1 Previous work and the new obstruction

Recent approaches study computability and tensor norms. Fox–Goldbring–Hart [5] derive computability-theoretic consequences of a positive answer, using the computable presentation of \Otwo. Goldbring–Sinclair give a conditional route to a negative answer through a weakly explicit minimal tensor norm: an effective choice of universal formulas approximating the minimal tensor-norm predicates to a fixed accuracy. They leave this property of \Otwo open [7, Definition 4.16, Theorem 4.18 and Question 4.20]. Our obstruction instead comes from the representation theory of an explicit discrete group.

The group configuration already appears in the nonfiniteness arguments of Sauers [13, Section 2.4] and Eckhardt [4, Theorem 3.1 and Example 3.2]. They show that the full algebra of the ascending HNN extension G is nonfinite: it contains a proper isometry v, so v^*v=1 but vv^*\ne1. Its property (T) base group \Gamma=\Z^3\rtimes\SL_3(\Z) and its finite-index scalar self-embeddings are recorded in de Cornulier [3, Proposition 1.1 and its proof]. Sauers and Eckhardt also construct non-MF groups: groups admitting no injective homomorphism into the unitary group of any norm matrix corona \prod_j M_{d_j}/\bigoplus_j M_{d_j}, with positive integer dimensions d_j. Here the product consists of bounded sequences and the direct sum is the ideal of sequences tending to zero in norm [4, Theorem 4.3][13, Sections 3–5]. Eckhardt’s example is an associated wreath product. The target class here is different: nonfiniteness of C^*(G) alone cannot exclude a unital embedding into \Otwo^\omega, which itself contains proper isometries.

The additional ingredient is a finiteness statement for approximate finite-dimensional representation types inside a nuclear algebra (Lemma 4.3). Its tensor-product argument follows Wassermann’s method [16], as explained by Manuilov–Thomsen [10, Section 2.1]. The statement applies to an arbitrary tuple of unitaries, even when the tuple satisfies no group relations. This flexibility is essential because coordinate lifts of an ultrapower representation are only approximately multiplicative. A dimension-independent estimate allows the representation type to vary with the coordinate. Finite-index induction then relates members of the resulting finite sets by composition with the self-embedding. These two passages turn the group configuration into the norm-ultrapower obstruction.

1.2 Proof outline

The subgroup \Gamma=\Z^3\rtimes\SL_3(\Z) of G has property (T). Conjugation by the generator t of the last \Z-factor sends \Gamma onto the index-eight subgroup \theta(\Gamma), where \theta(v,M)=(2v,M). There is a finite-dimensional irreducible representation \pi_* of \Gamma such that \pi_*\circ\theta^j is inequivalent to \pi_* for every j\ge1. Section 2 verifies these facts.

Fix a finite generating set S of \Gamma for which property (T) gives a spectral gap. Suppose a unital embedding A\to B^\omega exists, and choose unitary lifts x_n(g)\in B of its group unitaries. For each n, consider the nonzero finite-dimensional irreducible unitary representations \pi for which the tuple (x_n(s)\otimes\bar\pi(s))_{s\in S} has spectral minimum below a fixed small threshold for the sum of squared displacements from 1. Here \bar\pi is the conjugate representation. Sections 3 and 4 specify the threshold and show that the resulting set D_n of representation classes is finite.

Write \pi\prec\rho when \pi occurs as a unitary subrepresentation of \rho, and call \pi' a predecessor of \pi if \pi\prec\pi'\circ\theta. Faithfulness of the embedding places [\pi_*] in D_n on a set belonging to \omega. In Section 5, the spectral gap for every sequence of finite-dimensional representations supplies, on one set belonging to \omega, the simultaneous implication

Starting at \pi_* and repeatedly choosing a predecessor must give a repetition in the finite set D_n. Irreducibility then makes \pi_* periodic under composition with \theta, a contradiction.

2 The group and a finite-dimensional representation

We construct a finite-dimensional representation inequivalent to each of its positive pullbacks along a fixed group endomorphism. The same group configuration appears in the nonfiniteness arguments of Sauers [13, Section 2.4] and Eckhardt [4, Theorem 3.1 and Example 3.2]. De Cornulier [3, Proposition 1.1 and its proof] records the property (T) affine group and its finite-index scalar self-embeddings. We use the dyadic case.

We use column vectors. In the group G of Theorem 1.1, write elements as (v,M,k), so that

Set

We suppress the final coordinate when writing elements of \Gamma, and define

The group G is countable, so its full group C^*-algebra C^*(G) is separable and unital. Every unitary representation of G extends to C^*(G). We will use this universal property after inducing \pi_* from \Gamma to G.

3 A spectral test for representation matching

We now introduce the spectral quantity used to form the finite sets in the proof outline. The discussion applies to any finitely generated discrete group \Gamma with property (T). Choose a nonempty finite generating set S containing a Kazhdan set. By property (T), after decreasing the constant if necessary, there is 0<\kappa\le4|S| such that every unitary representation \sigma with no nonzero invariant vector satisfies

For a tuple \mathbf z=(z_s)_{s\in S} of unitaries in a nonzero unital C^*-algebra C, set

Thus 0\le L(\mathbf z)\le R1, without any group relations on \mathbf z.

Every finite-dimensional representation below acts on a nonzero Hilbert space. If \pi:\Gamma\to\U(V_\pi) is such a representation, its conjugate \bar\pi acts on the conjugate Hilbert space \bar V_\pi by \bar\pi(g)\bar v=\overline{\pi(g)v}. Define

where the spectrum is taken in C\otimes\mathcal B(\bar V_\pi). Matrix algebras have a unique C^*-tensor norm. On any faithful unital Hilbert-space realization C\subseteq \mathcal B(H), the number in (4) equals

In such a faithful realization, if z_s=\rho(s) comes from a unitary representation and \pi is irreducible, then d(\mathbf z,\pi)=0 exactly when \pi occurs in \rho, and otherwise d(\mathbf z,\pi)\ge\kappa. Indeed, invariant vectors of \rho\otimes\bar\pi correspond to intertwiners from V_\pi to H, and the spectral gap separates their orthogonal complement from zero.

The number d(\mathbf z,\pi) is unchanged by unitary equivalence of \pi. For a finite direct sum, the same formula gives

Fix once and for all

The next section proves that, for a tuple in a nuclear algebra, only finitely many irreducible \pi can satisfy d(\mathbf z,\pi)<a. No group relations on that tuple will be needed.

4 Finiteness of approximate representation matches

We prove that, for a fixed unitary tuple \mathbf u in a nuclear algebra, only finitely many finite-dimensional irreducible representation classes satisfy d(\mathbf u,\pi)<a. The tuple need not satisfy any group relations. This distinction will allow us to apply the result to coordinate lifts from a norm ultrapower.

We retain the notation and constants of Section 3. For C^*-algebras B and C, the minimal tensor product B\otimes_{\min}C is realized by faithful representations on separate Hilbert spaces; the maximal tensor product B\otimes_{\max}C is characterized by pairs of commuting representations on one Hilbert space. We use the tensor-norm characterization of nuclearity developed by Takesaki and Lance [14, 9]: B is nuclear if these two tensor norms agree for every C. We first obtain a lower bound when the second tensor factor has no finite-dimensional subrepresentations. The passage from a vector to its reduced density operator removes the arbitrary unitaries in the first factor.

Bekka [1] proved that if a unitary representation of a property (T) group on a Hilbert space K admits a state on \mathcal B(K) invariant under conjugation by the representation, then it has a nonzero finite-dimensional subrepresentation; see also [15, proof of Proposition 2]. The next lemma gives the explicit spectral bound needed here for an arbitrary unitary tuple in the first tensor factor.

The following quotient argument uses Wassermann’s property (T) method [16]; see the exposition in [10, Section 2.1]. The preceding uniform bound allows us to use an arbitrary unitary tuple in the first factor. Nuclearity will transfer the bound to the coordinate quotient and contradict the assumed small spectral minima.

The only use of nuclearity was to identify the minimal and maximal tensor products before applying \Psi. In particular, the proof does not assume an identification of the minimal tensor product with the coordinate quotient, and it imposes no bound on the dimensions of the \pi_j.

5 From an ultrapower embedding to a finite contradiction

To prove Theorem 1.1, fix a nonzero unital nuclear C^*-algebra B and a free ultrafilter \omega on \mathbb N. A set of indices is called \omega-large if it belongs to \omega. Suppose, towards a contradiction, that there is a unital injective *-homomorphism

We will associate a finite set of irreducible representations of \Gamma to each coordinate of \Phi. Conjugation by t will force these sets to contain predecessors under restriction by \theta. The resulting finite chains contradict the choice of \pi_*.

5.1 Unitary lifts and a gap for varying matrix dimensions

Write u_g for the canonical unitary of g\in G in C^*(G). Each unitary in B^\omega has a representative consisting of unitaries in B. Indeed, if (y_n) represents a unitary, then

On an \omega-large set both norms are less than 1/2, so y_n is invertible and y_n(y_n^*y_n)^{-1/2} is unitary. The difference between this unitary and y_n tends to zero along \omega, by continuous functional calculus. Set the representative equal to 1 outside this set.

Choose such representatives x_n(g)\in\mathcal U(B) of \Phi(u_g) for every g\in G, with x_n(e)=1. For each fixed pair g,h\in G,

These choices impose no group relations at any individual coordinate. All uses of (9) below involve finitely many fixed group elements; no intersection over all pairs is needed. Put \mathbf x_n=(x_n(s))_{s\in S}. Among the unitary equivalence classes of nonzero finite-dimensional irreducible representations of \Gamma, define

Here brackets denote unitary equivalence classes. Each D_n is finite by Lemma 4.3, which applies to arbitrary unitary S-tuples.

The next observation provides uniform control even when the dimensions of the representations increase with n.

5.2 The initial representation occurs

We next use injectivity of \Phi to ensure that [\pi_*]\in D_n for an \omega-large set of indices.

5.3 A predecessor for every representation at one coordinate

For representations \alpha and \beta of the same group, write \alpha\prec\beta when \alpha is unitarily equivalent to a subrepresentation of \beta. The required quantifier order is the following: on one \omega-large set, every member of D_n has a predecessor in that same finite set.

5.4 Finite chains force periodicity

The analytic argument is complete. We finish with the elementary consequence of finiteness that makes the predecessor property incompatible with \pi_*. Consider a chain of finite-dimensional irreducible unitary representations \tau_0=\pi,\tau_1,\ldots of \Gamma whose classes lie in a finite set D and satisfy \tau_i\prec\tau_{i+1}\circ\theta. Finiteness gives indices l\geq0 and p\geq1 with \tau_l\simeq\tau_{l+p}. Figure 1 separates the initial segment of this chain from the repeated segment. The key point is that periodicity at the repeated class forces every pullback of that representation to remain irreducible, so that periodicity reaches the starting class.

Choose one index n in the intersection of the \omega-large sets from Lemmas 5.2 and 5.3. Apply Lemma 5.4 to D_n and \pi_*. It gives \pi_*\simeq\pi_*\circ\theta^p for some p\geq1. But \pi_* acts nontrivially on the translations \mathbb Z^3, whereas \pi_*\circ\theta^p acts trivially on them: \theta^p multiplies these translations by 2^p, and \pi_* factors through reduction modulo 2. This is the required contradiction.

The argument used only that B is nonzero, unital and nuclear, and that \omega is free. The group G and the algebra C^*(G) are independent of these choices. Thus the same algebra has no unital embedding into B^\omega for any such B and any free \omega. Taking B=\mathcal O_2 proves the stated negative answer to Kirchberg’s embedding problem.

6 Consequences for existential closure and nuclear approximation

We spell out the model-theoretic consequences of the negative answer to Kirchberg’s embedding problem. In the unital C^*-algebra language, quantifier-free formulas are continuous combinations of norms of *-polynomials; variables range over prescribed norm balls. A unital algebra E is existentially closed if, for every unital inclusion E\subseteq F, every finite parameter tuple a from E, and every quantifier-free formula \varphi(a,y) in a finite tuple y,

where y ranges over the corresponding products of prescribed norm balls. Thus an extension cannot improve the infimum of a finite quantifier-free test over E [6, Section 2.1].

A parameter-free finite condition consists of finitely many strict inequalities \varphi_j(x)<r_j with quantifier-free \varphi_j. It is satisfiable when some tuple in a unital C^*-algebra satisfies them. The last assertion below makes the failure of good nuclear witnesses explicit as a positive lower bound on the approximation error of every finite-dimensional completely positive contractive factorization of every realizing tuple.

The KEP equivalences of Goldbring–Sinclair [6, Proposition 3.2 and Theorems 3.3 and 3.7] give these direct consequences of Theorem 1.1.