On a geometric realization of C^∗-algebras

Abstract

Further to the functional representations of C^∗-algebras proposed by R. Cirelli and A. Manià, we consider the uniform Kähler bundle (UKB) description of some C^∗-algebraic subjects. In particular, we obtain a one-toone correspondence between closed ideals of a C^∗-algebra Aand full uniform Kähler subbundles over open subsets of the base space of the UKB associated with A . In addition, we present a geometric description of the pure state space of hereditary C^∗-subalgebras and show that if B is a hereditary C^∗-subalgebra of A , the UKB of B is a kind of Kähler subbundle of the UKB of A . As a simple example, we consider hereditary C^∗-subalgebras of the C^∗-algebra of compact operators on a Hilbert space. Finally, we remark that each hereditary C^∗- subalgebra of A also can be naturally characterized as a uniform holomorphic Hilbert bundle.

Key words： C^∗-algebra; uniform Kähler bundle (UKB); uniform Kähler isomorphism; uniform holomorphic Hilbert bundle

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