Frobenius C*-algebras and local adjunctions of C*-correspondences

Abstract: Many interesting and important C*-algebras do not have multiplicative identities, and C*-algebraists have long known how to deal with this fact by using approximate identities, multiplier algebras, etc. A similar situation arises when one attempts to use methods of category theory to study modules over C*-algebras: objects like "the category of compact operators on Hilbert spaces" don't fit neatly into the standard theory of categories, because they lack identity morphisms; but they do fit nicely into a theory of non-unital C*-categories and their multiplier categories, as developed by Kandelaki, Mitchener, Vasselli, Antoun-Voigt, and others. This talk concerns an adaptation of the important categorical notion of adjoint functors to this non-unital-category point of view. I will present a definition (taken from joint work with Pierre Clare and Nigel Higson) of adjoint functors between categories of compact operators on Hilbert C*-modules, and I will explain how this definition corresponds to a natural notion of Frobenius C*-algebra, mirroring a correspondence between two-sided adjunctions and Frobenius algebras in classical category theory.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | slides | video )

Noncommutative Geometry in NYC

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