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Model theory of operator algebras I: Stability
Abstract
Several authors have considered whether the ultrapower and the relative
commutant of a C*-algebra or II_1 factor depend on the choice of the
ultrafilter. We settle each of these questions, extending results of Ge-Hadwin
and the first author.
... Farah, Sherman, and the second named author took up this line of inquiry motivated by a question of Popa about isomorphisms of matrix ultraproducts (see Section 3 below). In a series of three papers [14,15,16], they introduced a natural language for which the class of tracial von Neumann algebras (as well as the subclasses of tracial factors and II 1 factors) become axiomatizable, used model-theoretic ideas to settle a variety of questions about isomorphism of tracial ultraproducts, and initiated the study of elementary equivalence of II 1 factors (see Section 4 for a summary of the results proven there). ...
... Farah, Sherman, and the second named author took up this line of inquiry motivated by a question of Popa about isomorphisms of matrix ultraproducts (see Section 3 below). In a series of three papers [14,15,16], they introduced a natural language for which the class of tracial von Neumann algebras (as well as the subclasses of tracial factors and II 1 factors) become axiomatizable, used model-theoretic ideas to settle a variety of questions about isomorphism of tracial ultraproducts, and initiated the study of elementary equivalence of II 1 factors (see Section 4 for a summary of the results proven there). Goldbring After these papers, the model theory of operator algebras became a vibrant area of research. ...
... Moreover, for each i " 1, 2, we can write M i " Mz i and τ i " 1 τpz i q τ, where z 1 , z 2 are central projections in M such that z 1`z2 " 1. It is straightforward to check that, for any ultrafilter U, one has pM, τq U " pM 1 , τ 1 q U ' pM 2 , τ 2 q U (see [14,Lemma 4.1]). ...
We survey the developments in the model theory of tracial von Neumann algebras that have taken place in the last fifteen years. We discuss the appropriate first-order language for axiomatizing this class as well as the subclass of II$_1$ factors. We discuss how model-theoretic ideas were used to settle a variety of questions around isomorphism of ultrapowers of tracial von Neumann algebras with respect to different ultrafilters before moving on to more model-theoretic concerns, such as theories of II$_1$ factors and existentially closed II$_1$ factors. We conclude with two recent applications of model-theoretic ideas to questions around relative commutants.
... This example shows that there is no analogue of Krivine-Maurey's famous theorem in the case of multiplicatively stable Banach algebras, at least not without additional assumptions. Note that a model-theoretical version of the stability of Banach algebras has been studied by Farah, Hart, and Sherman in [6,7]. Stability, in this sense, implies both additive and multiplicative stability of Banach algebras. ...
... An important example of a multiplicatively unstable Banach algebra is any separable unital infinitedimensional C * -algebra. It has been proved in [6,Lemma 5.3] that such a C * -algebra is not stable in the model-theoretical sense. In the proof, the authors used the formula φ(x, y) = xy − y to witness instability through violation of the double limit criterion. ...
In this paper, we introduce and study the concept of hyper-instability as a strong version of multiplicative instability. This concept provides a powerful tool to study the multiplicative instability of Banach algebras. It replaces the condition of the iterated limits in the definition of multiplicative instability with conditions that are easier to examine. In particular, special conditions are suggested for Banach algebras that admit bounded approximate identities. Moreover, these conditions are preserved under isomorphisms. This enlarges the class of studied Banach algebras. We prove that many interesting Banach algebras are hyper-unstable, such as $ C^* $-algebras, Fourier algebras, and the algebra of compact operators on Banach spaces, each under certain conditions.
... The study of W * -algebras or von Neumann algebras is a deep and challenging subject with many connections to fields as diverse as ergodic theory, geometric group theory, random matrix theory, quantum information, and model theory. Our present goal is to bring together two of these facets-the model theory of tracial W * -algebras studied in the work of Farah, Hart, and Sherman [7,8,9] and Voiculescu's free entropy theory which, roughly speaking, quantifies the amount of matrix approximations for the generators of W * -algebra (see eg Voiculescu [32,33], Ge [12], Jung [25] and ...
... This section sketches the setup of continuous model theory, or model theory for metric structures of Ben Yaacov, Berenstein, Henson, and Usvyatsov [3,4] and its application to operator algebras by Farah, Hart, and Sherman [7,8,9]. We strive to present a self-contained exposition for two reasons: First, some readers may not be familiar with the model-theoretic terminology. ...
We study embeddings of tracial $\mathrm{W}^*$-algebras into a ultraproduct of matrix algebras through an amalgamation of free probabilistic and model-theoretic techniques. Jung implicitly and Hayes explicitly defined \emph{$1$-bounded entropy} through the asymptotic covering numbers of Voiculescu's microstate spaces, that is, spaces of matrix tuples $(X_1^{(N)},X_2^{(N)},\dots)$ having approximately the same $*$-moments as the generators $(X_1,X_2,\dots)$ of a given tracial $\mathrm{W}^*$-algebra. We study the analogous covering entropy for microstate spaces defined through formulas that use suprema and infima, not only $*$-algebra operations and the trace | formulas such as arise in the model theory of tracial $\mathrm{W}^*$-algebras initiated by Farah, Hart, and Sherman. By relating the new theory with the original $1$-bounded entropy, we show that if $\mathcal{M}$ is a separable tracial $\mathrm{W}^*$-algebra with $h(\cN:\cM) \geq 0$, then there exists an embedding of $\cM$ into a matrix ultraproduct $\cQ = \prod_{n \to \cU} M_n(\C)$ such that $h(\cN:\cQ)$ is arbitrarily close to $h(\cN:\cM)$. We deduce that if all embeddings of $\cM$ into $\cQ$ are automorphically equivalent, then $\cM$ is strongly $1$-bounded and in fact has $h(\cM) \leq 0$.
... We define, The quotient of ∞ (A) by c U (A) is a C*-algebra and it is called the ultrapower of A w.r.t. U, see [13], [11] and [15]. We denote this C*-algebra by A U . ...
... If A is separable and the Continuum Hypothesis is assumed then the isomorphism class of A ∩ A U is independent of the choice of U, see [13]. In the absence of the Continuum Hypothesis (CH), A ∩ A U depends on the choice of U for every infinite-dimensional separable C*-algebra A, see [10] and [11]. ...
Let B(H) be the algebra of bounded linear operators on a separable infinite-dimensional Hilbert space H. In 2004 Kirchberg asked whether the relative commutant of B(H) in its ultrapower is trivial. In [13] the authors have shown that under the Continuum Hypothesis the commutant of B(H) in its ultrapower depends on the choice of the ultrafilter. We here give a combinatorial characterization of the class of non-principal ultrafilters for which this commutant is non-trivial, answering Question 5.2 of [13]. This reduces Kirchberg’s question to a purely set-theoretic question: Can the existence of non-flat ultrafilters be proven in ZFC? In addition, we introduce the notion of quasi P-points and show that for such ultrafilters, and for ultrafilters satisfying the three functions property, the relative commutant of B(H) in its ultrapower is trivial.
... In [39, §6.1] the present author suggested investigating non-commutative optimal transport theory in the framework of continuous model theory. Continuous model theory [7,5] which made its first formal contact with tracial von Neumann algebras in [29,20,21,22] and has since had many applications [25,26,24,1]; for a survey, see [28]. The non-commutative law of a tuple (x 1 , . . . ...
We investigate the connections between continuous model theory, free probability, and optimal transport/convex analysis in the context of tracial von Neumann algebras. In particular, we give an analog of Monge-Kantorovich duality for optimal couplings where the role of probability distributions on $\mathbb{C}^n$ is played by model-theoretic types, the role of real-valued continuous functions is played by definable predicates, and the role of continuous function $\mathbb{C}^n \to \mathbb{C}^n$ is played by definable functions. In the process, we also advance the understanding of definable predicates and definable functions by showing that all definable predicates can be approximated by "$C^1$ definable predicates" whose gradients are definable functions. As a consequence, we show that every element in the definable closure of $\mathrm{W}^*(x_1,\dots,x_n)$ can be expressed as a definable function of $(x_1,\dots,x_n)$. We give several classes of examples showing that definable closure can be much larger than $\mathrm{W}^*(x_1,\dots,x_n)$ in general.
... Some model-theoretic notions. We present a few necessary definitions, but omit much of the formal model theory; we refer the reader to [FHS13], [FHS14a], [FHS14b] for details on the model theory of tracial von Neumann algebras. We will also need the notion of an elementary embedding: For 1 < t < ∞, the notion of compression is extended to the notion of amplifications of M by taking tensors with matrix algebras. ...
We investigate the problem of elementary equivalence of the free group factors, that is, do all free group factors $L(\mathbb{F}_n)$ share a common first-order theory? We establish a trichotomy of possibilities for their common first-order fundamental group, as well as several possible avenues for establishing a dichotomy in direct analog to the free group factor alternative of Dykema and Radulescu. We also show that the $\forall \exists$-theories of the interpolated free group factors are increasing, and use this to establish that the dichotomy holds on the level of $\forall \exists$-theories. We conclude with some observations on related problems.
... In the standard modeltheoretic terminology, the theory of M is ℵ 0 -categorical (some authors write ω-categorical, as the ordinal ω is routinely identified with the cardinal ℵ 0 ). First we use the fact that the center Z(M) of a tracial von Neumann algebra M is definable (this is essentially [12,Lemma 4.1]). The proof shows that the lattice of projections in the center is also definable (this is not an immediate consequence of the fact that the set of projections is also definable, since by an unpublished result of Henson in continuous logic the intersection of definable sets is not necessarily definable). ...
Tensoring with type I algebras preserves elementary equivalence in the category of tracial von Neumann algebras. The proof involves a novel and general Feferman--Vaught-type theorem for direct integrals of metric structures.
... This is similar to the proof of the analogous part of Theorem C. Suppose that U is not a P-point and fix X n 2 U for n 2 N such that for every X 2 U the set X n X n is infinite for some n 2 N. In B, identified with N N M 2 .C/, we can choose unitaries u j and v j for all j such that lim j !1 kOEa; u j k D 0 for all a 2 B but kOEu j ; v n k D 2 if n j (the construction is similar to that in the proof of [18,Lemma 3.2], where the analogous statement for the tracial norm was proven). Define c 2 B U by its representing sequence c j D u n if j 2 X n n X nC1 (with X 1 D N). ...
We adapt a continuous logic axiomatization of tracial von Neumann algebras due to Farah, Hart and Sherman in order to prove a metatheorem for this class of structures in the style of proof mining, a research programme that aims to obtain the hidden computational content of ordinary mathematical proofs using tools from proof theory.
We show that no faithful functor from the category of Hilbert spaces with injective linear contractions into the category of sets preserves directed colimits. Thus Hilbert spaces cannot form an abstract elementary class, even up to change of language. We deduce an analogous result for the category of commutative unital C⁎-algebras with ⁎-homomorphisms. This implies, in particular, that this category is not axiomatizable by a first-order theory, a strengthening of a conjecture of Bankston.
Introduction
A metric structure is a many-sorted structure in which each sort is a complete metric space of finite diameter. Additionally, the structure consists of some distinguished elements as well as some functions (of several variables) (a) between sorts and (b) from sorts to bounded subsets of ℝ, and these functions are all required to be uniformly continuous. Examples arise throughout mathematics, especially in analysis and geometry. They include metric spaces themselves, measure algebras, asymptotic cones of finitely generated groups, and structures based on Banach spaces (where one takes the sorts to be balls), including Banach lattices, C*-algebras, etc.
The usual first-order logic does not work very well for such structures, and several good alternatives have been developed. One alternative is the logic of positive bounded formulas with an approximate semantics (see [23, 25, 24]). This was developed for structures from functional analysis that are based on Banach spaces; it is easily adapted to the more general metric structure setting that is considered here. Another successful alternative is the setting of compact abstract theories (cats; see [1, 3, 4]). A recent development is the realization that for metric structures the frameworks of positive bounded formulas and of cats are equivalent. (The full cat framework is more general.) Further, out of this discovery has come a new continuous version of first-order logic that is suitable for metric structures; it is equivalent to both the positive bounded and cat approaches, but has many advantages over them.
We initiate a general study of ultraproducts of C*-algebras, including topics on representations, homomorphisms, isomorphisms, positive linear maps and their ultraproducts. We partially settle a question of D. McDuff by proving, for a finite von Neumann algebra with a separable predual, that the continuum hypothesis implies the isomorphism of all of its tracial ultrapowers with respect to different free ultrafilters on the natural numbers. The analog for C*-ultrapowers of separable C*-algebras is equivalent to the continuum hypothesis. We also prove a finite local reflexivity theorem for operator spaces that implies that the second dual of a C*-algebra can be embedded in some ultra-power of the algebra using a completely positive completely isometric map.
We prove that a separable C*-algebra A has continuous trace if and only if each central sequence in A is trivial. This is used to show that the condition, that every derivation of A is determined by a multiplier of A, is equivalent to the condition that every summable central sequence in A is trivial, and again equivalent to a representation $A = A_1 \bigoplus A_2$, where A1 has continuous trace and A2 is the restricted direct sum of simple C*-algebras.
1. Homomorphisms from algebras of continuous functions 2. Partial orders, Boolean algebras, and ultraproducts 3. Woodin's condition 4. Independence in set theory 5. Martin's Axiom 6. Gaps in ordered sets 7. Forcing 8. Iterated Forcing.