Grant: Discovery Grants Program - Individual

This is an interdisciplinary proposal at the interface between C*-algebras and logic. A C*-algebra is an algebra of bounded linear operators on a complex Hilbert space closed under the formation of adjoints and the norm topology. Hilbert space is the infinite-dimensional modification of our standard three-dimensional space. The study of C*-algebras began in the 1940s, and has since expanded to touch much of modern mathematics, including number theory, geometry, ergodic theory, mathematical physics, and topology.***Two areas of mathematical logic with connections to C*-algebras are set theory and model theory. Some well-known and long-standing open problems about C*-algebras were recently resolved using set theory and model theory. Moreover, in some cases it was proved that these problems have an inherent foundational aspect and that the use of logic in their solution was necessary. ***Most important questions that I will work on are the existence of a K-theory reversing automorphism of the Calkin algebra and Naimark's problem. The first question was asked by Brown, Douglas and Fillmore in 1977. I have proved that a negative answer is relatively consistent with the standard axioms of set theory, ZFC and (together with Phillips and Weaver) that a closely related problem of the existence of outer automorphisms of the Calkin algebra cannot be decided in ZFC. Naimark's problem 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. A confirmation of this conjecture would be a first step in extending Glimm's dichotomy to the realm of nonseparable C*-algebras.***Model theory studies the definable sets in mathematical structures and their first-order properties. The methods of continuous model theory were adapted to operator algebras less than a decade ago and much progress was made in understanding structure theory of massive algebras, such as ultrapowers and relative commutants. The latter algebras are more important and less understood, and I propose to investigate the exact formal properties which make them such an important tool in the study of operator algebras. ***A resolution of, or even a substantial progress on, any of these problems would provide new insight into the structure of C*-algebras. Mathematical logic (and set theory in particular) was developed in the `commutative' context and noncommutative problems pose new challenges. Further progress in applications of logic to operator algebras will require refinement of the existing techniques and development of new ones.