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A Gelfand-Naimark theorem for \(C^*\)-algebras. (English) Zbl 0826.46044

Curto, Raúl E. (ed.) et al., Algebraic methods in operator theory. Boston, MA: Birkhäuser. 124-133 (1994).
Since the memorable work by I. M. Gelfand and M. A. Naimark, there have been various attempts to generalize the beautiful representation theorem for non-commutative \(C^*\)-algebras. Among them, one notable direction was the approach from 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. This motivated the abstract Dirichlet problem for the extreme boundary of compact convex sets, and the duality arguments on the pure states for \(C^*\)-algebras by F. W. Shultz in the context of Alfsen-Shultz theory, which was further developed in C. A. Akemann and F. W. Shultz using Takasaki’s duality theorem. Another important approach was initiated by J. M. G. Fell using fiber bundle theory and culminated in the Dauns- Hofmann theorem showing 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.
Our purpose in this paper is to present a generalized Gelfand-Naimark theorem for non-commutative \(C^*\)-algebras on the extreme elements of the CP-state space in the sense of CP-convexity, or the irreducible representations of the algebra, which interpolates the above-mentioned approaches and recovers both the algebraic and affine geometric aspects of the duality. As applications, we discuss the abstract Dirichlet problem for the CP-extreme boundary, and propose a generalized spectral theory for non-normal operators.
For the entire collection see [Zbl 0790.00002].

MSC:

46L05 General theory of \(C^*\)-algebras
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