##### Department of Mathematics,

University of California San Diego

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### Model Theory and Operator Algebras Seminar

## Isaac Goldbring

#### University of Illinois, Chicago

## Omitting types in C* algebras

##### Abstract:

In typical applications of model theory to various areas of mathematics, deciding which properties are axiomatizable in the sense of first-order logic is usually key. Thus, when it was first realized that many important properties of C* algebras were not axiomatizable (e.g. nuclearity), it appeared that all hope waslost for an interesting development of the model theory of C* algebras. That pessimism was soon reversed when it was realized that many of these aforementioned properties were so-called â€œomitting typesâ€ properties, which is a specific type of infinitary axiomatizability. Using the Omitting Types Theorem, this has allowed for some interesting results to be proven about C* algebras. I will give a survey of some of the applications of omitting types in C* algebras, including the Kirchberg Embedding Problem, the failure of a finitary version of the Arveson Extension Theorem, and a question of Bankston on the pseudo-arc. I will also mention some open questions where the Omitting Types machinery might prove useful, e.g. a problem of Ozawa asking for an example of a non-nuclear C* algebra with both WEP and LLP. I will assume no knowledge of C* algebras or model theory (although some prior knowledge will help in some parts of the talk). Some of the talk will represent joint work with Thomas Sinclair, Martino Lupini, Alessandro Vignati, and Christopher Eagle.

Sam Buss and Adrian Ioana

### October 22, 2015

### 3:00 PM

### AP&M 5829

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