Past Junior Functional Analysis Seminar

We consider the question of whether almost-homomorphisms are close to honest homomorphisms. I’ll survey a few historical results, with different source/target collections of algebras, and also consider what to take as the definition of “almost-homomorphisms”. If we end up having time, I will sketch an elementary proof that almost-characters from commutative C*-algebras are close to honest characters.

Descriptive set theory provides a useful framework for studying the complexity of classification problems in operator algebras. In this talk I will discuss how C*-algebras can be encoded as points in a Borel space, and introduce several equivalent parametrizations, including a new one in terms of ideals of a universal C*-algebra. I will then discuss examples of natural classes of C*-algebras that form Borel sets, as well as a parametrization of *-homomorphisms and recent results on the Borelness of certain functors. Time permitting, I will introduce KK-theory and the Kasparov product, and explain a new result showing that the Kasparov product is Borel in a certain appropriate parametrized setting.