Operator Algebras Associated to Groups and Noncommutative Convexity

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Beginning with the work of von Neumann on the mathematical foundations of quantum physics, mathematicians have found it profitable to view various structures arising in the theory of operator algebras as noncommutative counterparts of classical mathematical objects. This philosophy underlies my work, which spans a wide range of topics within the field of operator algebras. My most significant recent work is concentrated in two areas: the structure of operator algebras associated to groups and the structure of noncommutative convex sets. The first major component of my proposal concerns the structure of operator algebras associated to groups and dynamical systems. On the one hand, group-theoretic constructions of operator algebras provide a valuable source of examples and questions. On the other hand, it is becoming increasingly clear that many problems about the analytic structure of groups and dynamical systems are most naturally studied within an operator-algebraic framework. The fundamental problem considered in my research is how to relate properties of the group or dynamical system to properties of the corresponding operator algebra. The second major component of my proposal concerns the theory of noncommutative convexity. In 1969, motivated by groundbreaking work on the structure of convex sets in infinite dimensional spaces, Arveson proposed a theory of noncommutative convexity as a framework for the study of objects arising from noncommutative mathematics. Despite the enormous potential of these ideas, they went undeveloped for many years until recent breakthroughs, the first by Arveson himself and the second by current author in joint work with K.R. Davidson. I plan to continue developing these ideas, which have already been applied with great success to problems in fields like quantum information theory, group theory and semidefinite optimization.