Model theory of operator algebras II: model theory
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TLDR
In this article, a model-theoretic result was proved that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with non-principal ultrafilters on ℕ are isomorphic even when the Continuum Hypothesis fails.Abstract:
We introduce a version of logic for metric structures suitable for applications to C*-algebras and tracial von Neumann algebras. We also prove a purely model-theoretic result to the effect that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with nonprincipal ultrafilters on ℕ are isomorphic even when the Continuum Hypothesis fails.read more
Citations
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Journal ArticleDOI
Ultraproducts of von Neumann algebras
Hiroshi Ando,Uffe Haagerup +1 more
TL;DR: In this article, it was shown that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the Ultraproduct ∏ ω M introduced by Groh and Raynaud.
Journal ArticleDOI
Model theory of operator algebras III: elementary equivalence and II1 factors
TL;DR: 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.
Journal ArticleDOI
Model theory of operator algebras I: Stability
TL;DR: In this article, the ultrapower and the relative commutant of a C*-algebra or II_1 factor depend on the choice of the ultrafilter, and they extend the results of Ge-Hadwin and the first author.
Journal ArticleDOI
Model theory of operator algebras I: stability
TL;DR: In this article, it was shown that the negative answer to each of these questions is equivalent to the Continuum Hypothesis, extending results of Ge{Hadwin and the rst author.
Posted Content
Countable Saturation of Corona Algebras
TL;DR: In this paper, the authors present unied proofs of several properties of -unital C*-algebras such as AA-CRISP, SAW*, being sub-Stonean in the sense of Kirchberg, and the conclusion of Kasparov's Tech-nical Theorem.
References
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Classification of injective factors
TL;DR: Nann et al. as mentioned in this paper presented a survey of the main lines of the investigation in the classification of factors, culminating in the Connes-Takesakl structure theory of type III factors.
Book
Operator Algebras: Theory of C*-Algebras and von Neumann Algebras
TL;DR: In this article, the authors present a model for operators on Hilbert Space, including C*-Algebras, Von Neumann Algebra, and K-Theory and Finiteness.
Book ChapterDOI
Model theory for metric structures
TL;DR: A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter as mentioned in this paper, and there are functions (of several variables) between sorts, assumed to be uniformly continuous.