Model theory of operator algebras III: elementary equivalence and II1 factors
Reads0
Chats0
TLDR
In this paper, the authors use continuous model theory to obtain several results concerning isomorphisms and embeddings between II-1 factors and their ultrapowers, including a poor man's resolution of the Connes embedding problem.Abstract:
We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.read more
Citations
More filters
Journal ArticleDOI
On Kirchberg's embedding problem
Isaac Goldbring,Thomas Sinclair +1 more
TL;DR: In this article, the existence of good nuclear witnesses has been shown to be equivalent to a local approximate nuclearity condition that is equivalent to the local lifting property of Kirchberg.
Posted Content
Logic and operator algebras
TL;DR: The most recent wave of applications of logic to operator algebras is a young and rapidly developing field as mentioned in this paper, which is a snapshot of the current state of the art.
Book
Tensor Products of C*-Algebras and Operator Spaces: The Connes-Kirchberg Problem
TL;DR: In this article, completely bounded and completely positive maps have been studied in the context of the Connes embedding problem with respect to finite representability conjecture and Tsirelson's problem.
Posted Content
The Model Theory of Nuclear $\mathrm{C}^*$-algebras
Ilijas Farah,Bradd Hart,Martino Lupini,L. Robert,Aaron Tikuisis,Alessandro Vignati,W. Winter +6 more
TL;DR: In this paper, a model theoretic study of nuclear algebraic structures using the tools of continuous logic is presented. But this study is restricted to the case of nuclear √ c-algebras.
Journal ArticleDOI
II_1 factors with non-isomorphic ultrapowers
TL;DR: In this paper, it was shown that there exist uncountably many separable II$_1$ factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic.
References
More filters
Journal ArticleDOI
The analogues of entropy and of Fisher's information measure in free probability theory, II.
TL;DR: Monotonicity properties and an analogue of the Cramer-Rao inequality are proved and analogues of the entropy and Fisher information measure for random variables in the context of free probability theory are introduced.