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.

Keywords

C^∗-algebra / uniform Kähler bundle (UKB) / uniform Kähler isomorphism / uniform holomorphic Hilbert bundle

References

[1] CirelliR, ManiàA, PizzoccheroL. A functional representation of noncommutative C^∗-algebras. Rev Math Phys, 1994, 6(5): 675-697

[2] DixmierJ. C^∗-Algebras. Revised ed. Amsterdam: North-Holland Publishing Company, 1982

[3] ElliottG A, KawamuraK A. Hilbert bundle characterization of Hilbert C^∗-modules. Trans Amer Math Soc, 2008, 360(9): 4841-4862

[4] HusemollerD. Fibre Bundles, New York: Springer-Verlag, 1993

[5] KawamuraK. Serre-Swan theorem for non-commutative C^∗-algebras. J Geom Phys, 2003, 48: 275-296

[6] MurphyJ G. C^∗-Algebras and Operator Theory. San Diego: Academic Press, 1990

[7] UpmeierH. Symmetric Banach Manifolds and Jordan C^∗-algebras. Amsterdam: North-Holland Publishing Company, 1985

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Higher Education Press and Springer-Verlag Berlin Heidelberg