Serre-Swan theorem for non-commutative C$^{*}$-algebras

TL;DR

This paper extends the classical Serre-Swan theorem to non-commutative C*-algebras by associating Hilbert modules with hermitian vector bundles and establishing an isomorphism between modules and sections of these bundles.

Contribution

It introduces a novel framework linking Hilbert C*-modules over non-commutative algebras to hermitian vector bundles, generalizing the Serre-Swan theorem.

Findings

Established a correspondence between Hilbert modules and hermitian vector bundles.

Constructed a flat connection on the associated bundle.

Proved the isomorphism between the module and sections of the bundle.

Abstract

We generalize the Serre-Swan theorem to non-commutative C$^{*}$-algebras. For a Hilbert C$^{*}$-module $X$ over a C$^{*}$-algebra ${\cal A}$, we introduce a hermitian vector bundle $\exx$ associated to $X$. We show that there is a linear subspace $\Gamma_{X}$ of the space of all holomorphic sections of ${\cal E}_{X}$ and a flat connection $D$ on ${\cal E}_{X}$ with the following properties: (i) $\Gamma_{X}$ is a Hilbert ${\cal A}$-module with the action of ${\cal A}$ defined by $D$, (ii) the C$^{*}$-inner product of $\Gamma_{X}$ is induced by the hermitian metric of ${\cal E}_{X}$, (iii) ${\cal E}_{X}$ is isomorphic to an associated bundle of an infinite dimensional Hopf bundle, (iv) $\Gamma_{X}$ is isomorphic to $X$.