A Gelfand-Naimark theorem for \(C^*\)-algebras
type: Item
instance of: scholarly article
title: A Gelfand-Naimark theorem for \(C^*\)-algebras
zbMATH Open document ID: 0917.46053
author: Ichiro Fujimoto
published in: Pacific Journal of Mathematics
publication date: 1999-07-19T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L35
zbMATH DE Number: 1186198
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: Gelfand-Naimark theorem
zbMATH Keywords: abstract Dirichlet problem
zbMATH Keywords: convexity theory
zbMATH Keywords: Kadison's function representation theorem
zbMATH Keywords: sectional representation in \(C^*\)-bundles over the primitive ideal space
zbMATH Keywords: duality for \(C^*\)-algebras
zbMATH Keywords: CP-convexity
zbMATH Keywords: non-commutative Gelfand-Naimark theorem
zbMATH Keywords: CP-extreme boundary
zbMATH Keywords: semi-perfectness
zbMATH Keywords: Stone-Weierstrass property
reviewed by: A. A. Rakhimov
MaRDI profile type: Publication
full work available at URL: pjm.1998.184.95
OpenAlex ID: W4230848743
DOI: 10.2140/PJM.1998.184.95
description: scientific article; zbMATH DE number 1186198