Hyperrigidity conjecture and recent developments

Abstract

This work presents the recent developments in hyperrigidity conjecture in the theory of non commutative Choquet boundary. The pioneer work in the theory was done by Arveson who proposed the conjecture in 1969 among others. Arveson generalised the idea of boundary and Korovkin set in commutative C*-algebra through unique extension property (UEP) of representations in a non commutative C*-algebra. The counterpart of Korovkin set in commutative theory is called a hyperrigid set. The conjecture when proposed by Arveson claimed that a set is hyperrigid if and only if all the irreducible representations of the C*-algebra generated has UEP relative to the set. The forward implication was solved by Arveson himself in 2011 and the other implication is still open. For the sake of completion, this thesis surveys all the necessary results in the theory of non commutative Choquet boundary, most of them being very recent.