On a geometric realization of C$^*$-algebras

TL;DR

This paper explores a geometric framework for C$^*$-algebras using uniform K"ahler bundles, establishing correspondences with ideals and subalgebras, and providing geometric insights into their state spaces.

Contribution

It introduces a novel geometric description of C$^*$-algebras via UKB, linking ideals and hereditary subalgebras to K"ahler subbundles and holomorphic structures.

Findings

One-to-one correspondence between closed ideals and K"ahler subbundles.

Geometric description of pure state spaces of hereditary subalgebras.

Hereditary C$^*$-subalgebras characterized as holomorphic Hilbert subbundles.

Abstract

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