scholarly journals Hyperlinear and Sofic Groups: A Brief Guide

2008 ◽  
Vol 14 (4) ◽  
pp. 449-480 ◽  
Author(s):  
Vladimir G. Pestov
Keyword(s):  
Symbolic Dynamics ◽  
Operator Algebras ◽  
Symmetric Groups ◽  
Unitary Groups ◽  
Embedding Problem ◽  
Open Questions ◽  
Sofic Groups

AbstractThis is an introductory survey of the emerging theory of two new classes of (discrete, countable) groups, called hyperlinear and sofic groups. They can be characterized as subgroups of metric ultraproducts of families of, respectively, unitary groups U(n) and symmetric groups Sn, n ∈ ℕ. Hyperlinear groups come from theory of operator algebras (Connes' Embedding Problem), while sofic groups, introduced by Gromov, are motivated by a problem of symbolic dynamics (Gottschalk's Surjunctivity Conjecture). Open questions are numerous, in particular it is still unknown if every group is hyperlinear and/or sofic.

scholarly journals STABILITY, COHOMOLOGY VANISHING, AND NONAPPROXIMABLE GROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
MARCUS DE CHIFFRE ◽  
LEV GLEBSKY ◽  
ALEXANDER LUBOTZKY ◽  
ANDREAS THOM
Keyword(s):  
Symmetric Groups ◽  
Unitary Groups ◽  
Frobenius Norm ◽  
Central Extensions ◽  
Finitely Presented Groups ◽  
Finite Dimensional ◽  
Open Questions ◽  
Higher Dimensional ◽  
First Time ◽  
Finitely Presented

Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $\text{Sym}(n)$ (in the sofic case) or the finite-dimensional unitary groups $\text{U}(n)$ (in the hyperlinear case)? In the case of $\text{U}(n)$ , the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist finitely presented groups which are not approximated by $\text{U}(n)$ with respect to the Frobenius norm $\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$ . Our strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain nonresidually finite central extensions of lattices in some simple $p$ -adic Lie groups. These groups act on high-rank Bruhat–Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

On the Structure of Certain Nest Algebra Modules

1987 ◽  
Vol 39 (6) ◽  
pp. 1405-1412
Author(s):  
G. J. Knowles
Keyword(s):  
Operator Algebras ◽  
Adjoint Operator ◽  
Systematic Investigation ◽  
Unique Determination ◽  
Nest Algebra ◽  
Minimal Elements ◽  
Open Questions ◽  
Weakly Closed ◽  
Order Homomorphisms

Let be a nest algebra of operators on some Hilbert space H. Weakly closed -modules were first studied by J. Erdos and S. Power in [4]. It became apparent that many interesting classes of non self-adjoint operator algebras arise as just such a module. This paper undertakes a systematic investigation of the correspondence which arises between such modules and order homomorphisms from Lat into itself. This perspective provides a basis to answer some open questions arising from [4]. In particular, the questions concerning unique “determination” and characterization of maximal and minimal elements under this correspondence, are resolved. This is then used to establish when the determining homomorphism is unique.

scholarly journals On Ultraproduct Embeddings and Amenability for Tracial von Neumann Algebras

2020 ◽  
Author(s):  
Scott Atkinson ◽  
Srivatsav Kunnawalkam Elayavalli
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Analogous Result ◽  
Equivalence Classes ◽  
Embedding Problem ◽  
Unitary Equivalence ◽  
Von Neumann ◽  
Sofic Groups

Abstract We define the notion of self-tracial stability for tracial von Neumann algebras and show that a tracial von Neumann algebra satisfying the Connes embedding problem (CEP) is self-tracially stable if and only if it is amenable. We then generalize a result of Jung by showing that a separable tracial von Neumann algebra that satisfies the CEP is amenable if and only if any two embeddings into $R^{\mathcal{U}}$ are ucp-conjugate. Moreover, we show that for a II$_1$ factor $N$ satisfying CEP, the space $\mathbb{H}$om$(N, \prod _{k\to \mathcal{U}}M_k)$ of unitary equivalence classes of embeddings is separable if and only $N$ is hyperfinite. This resolves a question of Popa for Connes embeddable factors. These results hold when we further ask that the pairs of embeddings commute, admitting a nontrivial action of $\textrm{Out}(N\otimes N)$ on ${\mathbb{H}}\textrm{om}(N\otimes N, \prod _{k\to \mathcal{U}}M_k)$ whenever $N$ is non-amenable. We also obtain an analogous result for commuting sofic representations of countable sofic groups.

scholarly journals Representations of the symmetric groups and the Clebsch—Gordan coefficients of the unitary groups

1968 ◽  
Vol 8 (3) ◽  
pp. 597-609
Author(s):  
A. Jucys
Keyword(s):  
Symmetric Groups ◽  
Unitary Groups

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: А. А. Юцис. Представления симметрических групп и коэффициенты Клебша—Гордана унитарных групп A. Jucys. Simetrinių grupių atvaizdavimai ir unitarinių grupių Klebšo—Gordano koeficientai

Type classification of extreme quantized characters

2019 ◽  
Vol 41 (2) ◽  
pp. 593-605
Author(s):  
RYOSUKE SATO
Keyword(s):  
Dynamical Systems ◽  
Quantum Groups ◽  
Asymptotic Representation ◽  
Operator Algebras ◽  
Complete Solution ◽  
Unitary Groups ◽  
Inductive System ◽  
Von Neumann ◽  
Type Classification

The notion of quantized characters was introduced in our previous paper as a natural quantization of characters in the context of asymptotic representation theory for quantum groups. As in the case of ordinary groups, the representation associated with any extreme quantized character generates a von Neumann factor. From the viewpoint of operator algebras (and measurable dynamical systems), it is natural to ask what is the Murray–von Neumann–Connes type of the resulting factor. In this paper, we give a complete solution to this question when the inductive system is of quantum unitary groups $U_{q}(N)$.

Unitary-group canonical states and matrix elements

1989 ◽  
Vol 67 (8) ◽  
pp. 774-780
Author(s):  
M. F. Soto Jr. ◽  
R. Mirman
Keyword(s):  
Symmetric Group ◽  
Unitary Group ◽  
Symmetric Groups ◽  
Unitary Groups ◽  
Matrix Elements ◽  
Group Representations ◽  
Group Basis

States of unitary groups are realized as multinomials in boson operators, symmetrized to give symmetric-group basis states. From these, matrix elements of the group generators are calculated using the procedures discussed here. A table of basis states and matrix elements of unitary-group representations, and the values of the invariants so generated, is given for SU(1) through SU(4), for all symmetric groups from S(1) through S(3).

scholarly journals ENFORCEABLE OPERATOR ALGEBRAS

2019 ◽  
pp. 1-33 ◽  
Author(s):  
Isaac Goldbring
Keyword(s):  
Positive Solution ◽  
Model Theory ◽  
Operator Algebras ◽  
Continuous Model ◽  
Continuous Functions ◽  
Embedding Problem ◽  
Prime Model ◽  
Canonical Operator ◽  
Classical Notion ◽  
Building Models

We adapt the classical notion of building models by games to the setting of continuous model theory. As an application, we study to what extent canonical operator algebras are enforceable models. For example, we show that the hyperfinite II$_{1}$factor is an enforceable II$_{1}$factor if and only if the Connes Embedding Problem has a positive solution. We also show that the set of continuous functions on the pseudoarc is an enforceable model of the theory of unital, projectionless, abelian$\text{C}^{\ast }$-algebras and use this to show that it is the prime model of its theory.

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