Rigidity properties for operator systems

Specialeforsvar ved Julius Beck Schwartz

Titel: Rigidity properties for operator systems

Abstract: In this thesis, we are concerned with intrinsic C*-algebraic properties for unital operator spaces, which are modeled in the so-called C*-envelope. We prove the existence of the C*-envelope in the framework of injective envelopes, which follows an approach of Hamana and Sinclair that uses tools from the theory of topological semigroups. Then, having established this, we turn our attention to representations with the unique extension property. We show that every unital completely positive map on an operator system admits a dilation to one with the unique extension property. This consequently gives a new proof of the existence of the C*-envelope of an operator system.

A central class of maps that have this unique extension property are the boundary representations, which form a noncommutative analogue of the Choquet boundary. Inspired by results in approximation theory, we relate these boundary representations to Arveson’s notion of a hyperrigid generating set. Hyperrigidity provides a strong link between operator systems and their C*-envelope, and Arveson conjectured that hyperrigidity can be encapsulated in terms of boundary representations. While we find may circumstances where Arveson's conjecture is true, we also present a recent counterexample to the conjecture.

Vejleder:  Ian Thompson