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

Abstract

We generalize the Serre-Swan theorem to non-commutative C${}^{\ast }$-algebras. For a Hilbert C${}^{\ast }$-module $X$ over a C${}^{\ast }$-algebra $\mathcal{A}$, 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 ${\mathcal{E}}_X$ and a flat connection $D$ on ${\mathcal{E}}_X$ with the following properties: (i) ${\Gamma }_X$ is a Hilbert $\mathcal{A}$-module with the action of $\mathcal{A}$ defined by $D$, (ii) the C${}^{\ast }$-inner product of ${\Gamma }_X$ is induced by the hermitian metric of ${\mathcal{E}}_X$, (iii) ${\mathcal{E}}_X$ is isomorphic to an associated bundle of an infinite dimensional Hopf bundle, (iv) ${\Gamma }_X$ is isomorphic to $X$.

16 pages, LaTeX

Cited by in corpus (2)

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