Non-commutative boundary theory and Arveson’s hyperrigidity conjecture

In his foundational work, Arveson extended the classical boundary theory for function algebras to the non-commutative world. His theory provided us with analogues of Shilov and Choquet boundaries which, classically, are certain “smallest” subsets for which the maximum modulus principle holds. Arveson’s theory has had a profound influence in the fields of operator theory and operator algebras, leading up to the resolution of several problems in dilation theory, approximation theory, C*-simplicity, and the structure theory of operator algebras.

In this talk I will survey Arveson’s boundary theory, leading up to his last major open conjecture known as Arveson’s hyperrigidity conjecture. This conjecture roughly states that if the Choquet boundary coincides with the whole spectrum of the generated C-algebra, then nets of unital completely positive maps which converge to the identity on generators must converge to the identity on the whole generated C-algebra. We will showcase a counterexample with a separable type I C*-algebra.

All necessary motivation and background will be provided throughout the talk, and the construction of the counterexample will be clear (at least) to undergraduate students who have taken an advanced course in functional analysis.

Based on joint work with Boris Bilich