PhD Thesis Defence

Sam Kim, Department of Pure Mathematics, University of Waterloo

"Operator Systems, Crossed Products, and Correlation Sets"

This talk is split into two parts. In the first part, we will talk about crossed products of operator systems and crossed products of operator algebras and their relation to a property known as Hyperrigidity. Crossed products of operator algebras are introduced by Katsoulis and Ramsey with a view towards solving the Hao-Ng isomorphism problem, which asks whether the crossed product construction commutes with the Cuntz-Pimsner algebra construction for C*-correspondences. Their techniques show that the Hao-Ng problem has a path towards a solution if their crossed product construction commutes with the C*-envelope of an operator algebra. In joint work with Sam Harris, we show that this is the case for groups for which C*(G) admits a trace if and only if the group G is amenable. Beyond this, we will describe a complete characterization of hyperridigity for the canonical subspaces generating Cuntz-Pimsner algebras. 

In the second part, we look at synchronous correlation sets. Synchronous correlation sets are introduced by Paulsen as a special instance of Tsirelsen's correlation sets so that they admit an operator algebraic interpretation. We describe what is currently known about these sets and its connection to an open problem in operator algebras called Connes' embedding problem. 

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