Abstract

We generalize the Gelfand-Naimark theorem for non-commutative C*-algebras in the context of CP-convexity theory. We prove that any C*-algebra A is *-isomorphic to the set of all B(H)-valued uniformly continuous quivariant functions on the irreducible representations Irr(A : H) of A on H vanishing at the limit 0 where H is a Hilbert space with a suficiently large dimension. As applications, we consider the abstract Dirichlet problem for the CP-extreme boundary, and generalize the notion of semi-perfectness to non-separable C*-algebras and prove its Stone-Weierstrass property. We shall also discuss a generalized spectral theory for non-normal operators.

Authors