Interactions between C*-algebra and set theory

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