Groups and Operator Algebras Seminar
Speaker: Se-Jin Kim (KU Leuven)
Title: NC convex sets and operator systems
Abstract: This talk concerns operator systems which may not admit a unit and their dual geometric structure. An operator system in this new sense, sometimes called a self-adjoint operator space, is a Banach subspace that is closed under involution. These objects naturally arise in the study of operator algebras in the study of Furstenberg boundaries of groups and groupoids, in quantum information theory via quantum graphs and quantum channels, and in non-commutative metric geometry as what are now called spectral truncations.
While the theory of C*-algebras admit many non-commutative but not precise analogs of some topological structure, a result due to Webster--Winkler, Davidson--Kennedy, and Kennedy, Manor, and myself demonstrate that there is a robust non-commutative convex structure associated to each operator system coming from the spaces of completely positive and completely contractive maps. In this talk we give some examples of these nc convex sets, some basic properties of these sets, some successes in this manner of thinking, as well as current questions in this field.