C*-Algebras, Dynamics, and Classification

Abstract

Classification is a central theme in mathematics, and a particularly rich one in the theory of operator algebras. Indeed, one of the first major results in the theory is Murray and von Neumann’s type classification of factors (weakly closed self-adjoint algebras of operators on Hilbert space with trivial center), and one of its modern touchstones is the mid-1970s Connes-Haagerup classification of amenable factors with separable predual. Several significant themes in the classification theory of norm-separable C‡-algebras have emerged since the work of Connes-Haagerup, and these were the focus of our workshop. They include Elliott’s program to classify separable nuclear C‡-algebras via K-theoretic invariants, the role of C‡-algebras in the classification of orbit equivalence relations of discrete countable group actions, and the more recent contact between descriptive set theorists and operator algebraists which seeks to quantify the Borel complexity of the isomorphism relation for various natural classes of algebras.