Model Theory and Operator Algebras (18w5155)


(University of California at Irvine)

(York University)

Dimitri Shlyakhtenko (UCLA)

(University of Muenster)


The Banff International Research Station will host the "Model Theory and Operator Algebras" workshop from November 25th to November 30th, 2018.

Operator algebra has developed into a field of its own since the time of von Neumann. C*-algebras and von Neumann algebras can be represented as algebras of bounded operators on a Hilbert space and although the relevant topology is different in the study of these algebras (the operator norm for C*-algebras and the weak-$\ast$ topology for von Neumann algebras), both classes have interesting ultraproduct constructions. The class of C*-algebras is closed under the usual norm ultraproduct while the class of II$_1$ factors is closed under the tracial ultraproduct. Although it took 40 years to notice, the existence of these ultraproduct constructions highlights that model theory has a role to play in the subject. There are many interesting current directions to pursue but we wish to concentrate on three general themes:

  • Interaction with the Elliott classification programme

  • Relationship with free probability

  • Model theoretic considerations

There are several things that model theory brings to the table in this endeavour. First of all, the theory of an algebra is an invariant which is complementary to many of the operator algebraic invariants on offer. The utility and consequences of recognizing when two algebras do not have the same theory will be highlighted below. Second, model theory provides methods of constructing examples which are different from those in operator algebra. The primary example is model theoretic forcing which plays a prominent role in [thebook]. Although to date the examples constructed have been modest, refocusing attention on the construction of specific examples with this technique in mind could pay dividends. Third, the notion of an elementary class in continuous logic is a common generalization of both the class of C*-algebras and II$_1$ factors.

Model theory can be used for both clarifying concepts and identifying good questions to ask. The key is to identify at an abstract level what role the language is playing and what model theoretic properties are present.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).