A Gelfand-Naimark Theorem for C*-Algebras

Since the memorial work by I.M. Gelfand and M.A. Naimark [22], there have been various attempts to generalize the beautiful representation theorem for non-commutative C*-algebras (see e.g. [12] and its bibliography). Among them, one notable direction was the approach from the convexity theory, initiated by R.V. Kadison finding a functional representation of a C*-algebra on the w*- closure of the pure states as an order isomorphism in the context of Kadison’s function representation theorem (cf. [23]). This motivated the abstract Dirichlet problem for the extreme boundary of compact convex sets (cf. [2, 3]), and the duality arguments on the pure states for C*-algebras by F.W. Shultz [28] in the context of Alfsen-Shultz theory, which was further developed in C.A. Akemann and F.W. Shultz [1] based on Takesaki’s duality theorem (cf. [8], [30]). Another important approach was initiated by J.M.G. Fell [14] using the fiber bundle theory, and culminated in the Dauns-Hofmann theorem [11] (cf. also [13]), where they showed 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.