C*-algebras and set theory

这是一个关于C*-代数和集合论之间的接口的跨学科建议。

This is an interdisciplinary proposal on the interface between C*-algebras and set theory. These two subjects were once considered to be only superficially related but exciting connections and deep theorems have been discovered during the last few years. Some well-known and long-standing open problems about C*-algebras were recently resolved using set theory. Moreover, it was proved that these problems have an inherent foundational aspect and that the use of set theory in their solution was necessary. I will describe two of several problems on C*-algebras that I plan to solve and whose solutions appear to require set theory. The first one is Naimark's problem. It asks whether every C*-algebra with a unique (up to conjugacy) irreducible representation is isomorphic to the algebra of compact operators on some Hilbert space. A negative answer to this problem was shown to be relatively consistent with the standard axioms of set theory by C. Akemann and N. Weaver in 2002. I conjecture that the positive answer to Naimark's problem is also relatively consistent with the standard axioms of set theory and that it would lead to an interesting dichotomy in representation theory for arbitrary C*-algebras. The second problem, or rather circle of problems, is of a very different nature since it does not involve independence results. Elliott's program to classify nuclear C*-algebras by K-theoretic invariants is a major theme in the modern theory of operator algebras. In recent years it underwent a transformation after the discovery of counterexamples by M. Rordam and A. Toms and the introduction of new invariants such as the Cuntz semigroup. I propose to analyze the classification problem for separable C*-algebras using the abstract classification theory developed by Hjorth, Kechris, and others. This theory provides the means for comparing the complexity of classification problems and it can isolate concrete obstructions to classification by simple invariants such as countable groups.