A Gelfand-Naimark theorem for \(C^*\)-algebras
It is known, that the classical Gelfand-Naimark theorem states that every commutative \(C^*\)-algebra \(A\) is *-isomorphic to the algebra \(C_0(\Omega)\) of all continuous functions on the characters \(\Omega\) of \(A\) which vanish at infinity. In the non-commutative case there are various attempts to obtain such a representation theorem. One of the ways to approach is to use convexity theory, which was initiated by R. V. Kadison, representing the self-adjoint part of a unital \(C^*\)-algebra on the weak*-closure of the pure states as an order isomorphism in the context of Kadison's function representation theorem. Another is the approach from the method of sectional representation in \(C^*\)-bundles over the primitive ideal space, which was initiated by J. M. G. Fell, and proved that every \(C^*\)-algebra is *-isomorphic to the \(C^*\)-algebra of all continuous sections of a \(C^*\)-bundle over the spectrum of the center of its multiplier algebra. However, in this approach, the affine geometric aspect of the duality could not be restored. In the present paper the author combines the ideas of the above two different approaches in duality for \(C^*\)-algebras together with the CP-convexity theory, the last one being introduced and developed in the previous works of the author. The main result of this paper is the non-commutative Gelfand-Naimark theorem, which states that any \(C^*\)-algebra \(A\) is *-isomorphic to the set of all \(B(H)\)-valued uniformly continuous equivariant functions on the irreducible representations of \(A\) on the Hilbert space (with a sufficiently large dimension) \(H\) vanishing at the limit 0. Moreover, for the CP-extreme boundary the abstract Dirichlet problem is solved, and generalized the notion of semi-perfectness of non-separable \(C^*\)-algebras and proved its Stone-Weierstrass property.
Kadison's function representation theorem
sectional representation in \(C^*\)-bundles over the primitive ideal space
scientific article; zbMATH DE number 1186198