Model theory of operator algebras III: elementary equivalence and II<sub>1</sub>factors

Farah, Ilijas; Hart, Bradd; Sherman, David K.

doi:10.1112/blms/bdu012

Cited by 95 publications

(74 citation statements)

References 49 publications

(84 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This situation is partially explained by the fact that elementary equivalence of II 1 factors is a much coarser notion of equivalence than isomorphism. An illuminating explanation of this fact is provided by a result in [FHS11] which states that any II 1 factor is elementarily equivalent to uncountably many non-isomorphic II 1 factors. R. B. was supported in part by NSF Grant DMS #1161047, NSF Career Grant DMS #1253402 and ANR Grant NEUMANN.…”

Section: Introductionmentioning

confidence: 99%

“…This problem has received a lot of attention (see e.g. [FHS11,GS14] and the survey [Fa14]). The connection between the two problems stems from the continuous version of the Keisler-Shelah theorem [Ke61,Sh71].…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, it was noticed in [FGL06,FHS11] that, for separable II 1 factors, property Gamma and the property of being McDuff are elementary properties (i.e. they are remembered by ultrapowers).…”

Section: Introductionmentioning

confidence: 99%

“…Recently, a closely related problem has been considered in the emerging field of continuous model theory of operator algebras [FHS09,FHS10,FHS11]: how many elementary equivalence classes of II 1 factors exist? This problem has received a lot of attention (see e.g.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

II1 factors with nonisomorphic ultrapowers

Boutonnet¹,

Chifan²,

Ioana³

2017

Duke Math. J.

View full text Add to dashboard Cite

Abstract. We prove that there exist uncountably many separable II1 factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic. In fact, we prove that the families of non-isomorphic II1 factors originally introduced by McDuff [MD69a,MD69b] are such examples. This entails the existence of a continuum of non-elementarily equivalent II1 factors, thus settling a well-known open problem in the continuous model theory of operator algebras.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

II1 factors with nonisomorphic ultrapowers

Boutonnet¹,

Chifan²,

Ioana³

2017

Duke Math. J.

View full text Add to dashboard Cite

show abstract

“…Fortunately, a continuous first-order logic and model theory for metric structures was introduced in [BYBHU08], and specialized to C * -algebras in [FHS14a] and [FHS14b]. Another great reference is [FHL + 16].…”

Section: Continuous Logicmentioning

confidence: 99%

Pseudocompact C*-Algebras

Hardy¹

View full text Add to dashboard Cite

Abstract. We study the class of pseudocompact C * -algebras, which are the logical limits of finite-dimensional C * -algebras. The pseudocompact C * -algebras are unital, stably finite, real rank zero, stable rank one, and tracial. We show that the pseudocompact C * -algebras have trivial K1 groups and the Dixmier property. The class is stable under direct sums, tensoring by finite-dimensional C * -algebras, taking corners, and taking centers. We give an explicit axiomatization of the commutative pseudocompact C * -algebras. We also study the subclass of pseudomatricial C * -algebras, which have unique tracial states, strict comparison of projections, and trivial centers. We give some information about the K0 groups of the pseudomatricial C * -algebras.

show abstract

Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture

Capraro

Lupini

2015

Lecture Notes in Mathematics

View full text Add to dashboard Cite

PrefaceAnalogy is one of the most effective techniques of human reasoning: When we face new problems we compare them with simpler and already known ones, in the attempt to use what we know about the latter ones to solve the former ones. This strategy is particularly common in Mathematics, which offers several examples of abstract and seemingly intractable objects: Subsets of the plane can be enormously complicated but, as soon as they can be approximated by rectangles, then they can be measured; Uniformly finite metric spaces can be difficult to describe and understand but, as soon as they can be approximated by Hilbert spaces, then they can be proved to satisfy the coarse Novikov's and Baum-Connes's conjectures.These notes deal with two particular instances of such a strategy: Sofic and hyperlinear groups are in fact the countable discrete groups that can be approximated in a suitable sense by finite symmetric groups and groups of unitary matrices. These notions, introduced by Gromov and Rȃdulescu, respectively, at the end of the 1990s, turned out to be very deep and fruitful, and stimulated in the last 15 years an impressive amount of research touching several seemingly distant areas of mathematics including geometric group theory, operator algebras, dynamical systems, graph theory, and more recently even quantum information theory. Several long-standing conjectures that are still open for arbitrary groups were settled in the case of sofic or hyperlinear groups. These achievements aroused the interest of an increasing number of researchers into some fundamental questions about the nature of these approximation properties. Many of such problems are to this day still open such as, outstandingly: Is there any countable discrete group that is not sofic or hyperlinear? A similar pattern can be found in the study of II 1 factors. In this case the famous conjecture due to Connes (commonly known as Connes' embedding conjecture) that any II 1 factor can be approximated in a suitable sense by matrix algebras inspired several breakthroughs in the understanding of II 1 factors, and stands out today as one of the major open problems in the field.The aim of this monograph is to present in a uniform and accessible way some cornerstone results in the study of sofic and hyperlinear groups and Connes' embedding conjecture. These notions, as well as the proofs of many results, are here presented in the framework of model theory for metric structures. We believe that this point of view, even though rarely explicitly adopted in the literature, can contribute to a better understanding iii iv of the ideas therein, as well as provide additional tools to attack many remaining open problems. The presentation is nonetheless self-contained and accessible to any student or researcher with a graduate-level mathematical background. In particular no specific knowledge of logic or model theory is required.Chapter 1 presents the conjectures and open problems that will serve as common thread and motivation for the rest of the survey: Connes' em...

show abstract

Model theory of operator algebras III: elementary equivalence and II₁factors

Cited by 95 publications

References 49 publications

II1 factors with nonisomorphic ultrapowers

II1 factors with nonisomorphic ultrapowers

Pseudocompact C*-Algebras

Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture

Contact Info

Product

Resources

About

Model theory of operator algebras III: elementary equivalence and II1factors

Cited by 95 publications

References 49 publications

II1 factors with nonisomorphic ultrapowers

II1 factors with nonisomorphic ultrapowers

Pseudocompact C*-Algebras

Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture

Contact Info

Product

Resources

About

Model theory of operator algebras III: elementary equivalence and II₁factors