The Classification Problem for Automorphisms of C*-Algebras

Abstract

We present an overview of the recent developments in the study of the classification problem for automorphisms of C*-algebras from the perspective of Borel complexity theory. §1. Introduction. Borel complexity theory is an area of logic that studies, generally speaking, the relative complexity of classification problems in mathematics. The main ideas and methods for such a study come from descriptive set theory, which can be described as the analysis of definable sets in Polish spaces and their properties. In the framework of Borel complexity theory, a classification problem is regarded as an equivalence relation on a Polish space. Perhaps after a suitable parametrization, this virtually covers almost all concrete classification problems in mathematics.Under the assumption that the classes of objects under consideration are naturally parameterized by the points of a Polish space, it is to be expected-and demanded-that a satisfactory classification satisfy some constructibility assumption. A sensible notion of constructibility is to be Borel measurable with respect to the given parameterizations. This leads to the following definition, first introduced and studied in [33]. Suppose that E, E are equivalence relations on standard Borel spaces X, X . A Borel reduction from E to E is a Borel function f : X → X such thatfor every x 1 , x 2 ∈ X . The function f can be seen as a constructive way to assign to the objects of X complete invariants up to E that are E -classes. Equivalently a Borel reduction can be seen as an embedding of the quotient space X/E into the quotient space X /E that is "definable", in the sense that it admits a Borel lifting from X to X .