On a geometric realization of C${}^{\ast }$-algebras

Abstract

Further to the functional representations of C${}^{\ast }$-algebras proposed by R. Cirelli, A. Mania and L. Pizzocchero, we consider in this article the uniform Kähler bundle (in short, UKB) description of some C${}^{\ast }$-algebraic subjects. In particular, we obtain an one-to-one correspondence between closed ideals of a C${}^{\ast }$-algebra $\mathcal{A}$ and full uniform Kähler sub bundles over open subsets of the base space of the UKB associated with $\mathcal{A}$. In addition, we will present a geometric description of the pure state space of hereditary C${}^{\ast }$-subalgebras and show that that if $\mathcal{B}$ is a hereditary C${}^{\ast }$-subalgebra of $\mathcal{A}$, the UKB of $\mathcal{B}$ is a kind of Kähler subbundle of the UKB of $\mathcal{A}$. As a simple example, we consider hereditary C${}^{\ast }$-subalgebras of the C${}^{\ast }$-algebra of compact operators on a Hilbert space. Finally, we remark that hereditary C${}^{\ast }$-subalgebras also naturally can be characterized as uniform holomorphic Hilbert subbundles.

9 pages

References in corpus (1)

• Serre-Swan theorem for non-commutative C${}^{\ast }$-algebras