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 sufficiently 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.
