This project involves the interaction between set theory and C*-algebras. The primary goal is to settle old outstanding open problems using set theoretic tools, which could involve proving independence results. Specific problems that could be addressed include the following: Naimark's problem about the existence of nontrivial C*-algebras with only one irreducible representation; Anderson's conjecture about pure states restricting to pure states on diagonal masas; the Kadison-Singer problem that has many equivalent formulations and is significant to several areas of mathematics; the Stone-Weierstrass problem about the existence of proper subalgebras that separate pure states; the problem of Brown, Douglas, and Fillmore about the existence of an automorphism of the Calkin algebra taking the image of the unilateral shift to its adjoint; and Arveson's problem about the existence of sufficiently many boundary representations for nonseparable operator systems.
C*-algebras, a class of mathematical structures that were originally studied primarily for applications in physics, have over time found substantial applications in a broad variety of areas of both mathematics and physics. Much current work in this field focuses on developing further connections to these other areas, but fundamental questions of basic theory remain open. In the past decade a series of long-standing open problems in C*-algebras have been settled using, in an essential way, techniques from the seemingly unrelated area of set theory. This was a surprising connection that has enriched both fields. Further work in this direction is expected to lead to a better understanding of C*-algebras at a fundamental level.
