Speaker: Adam Dor-On, Haifa University

In his foundational work, Arveson extended the classical boundary theory for subalgebras of functions to the non-commutative world. His theory provided us with non-commutative analogues of Shilov and Choquet boundaries which, classically, are smallest subsets of the space 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, non-commutative convexity, structure theory of C*-algebras and classification theory of operator algebras.

In this talk I will survey Arveson's non-commutative boundary theory, leading up to his last major open conjecture known as Arveson's hyperrigidity conjecture. This conjecture roughly states that if the non-commutative 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 background will be provided throughout the talk, and the construction of the counterexample will be clear (at least) to third-year undergraduate students.

*Based on joint work with Boris Bilich