Serre-Swan theorem for non-commutative C*-algebras

[摘要] For a Hilbert C*-module X over a C*-algebra A, we introduce a vector bundle epsilon(X) associated to X. We prove that epsilon(X) has an hermitian metric and a flat connection. We introduce a vector space Gamma(X) of holomorphic sections of epsilon(X) with the following properties: (i) Gamma(X) is a Hilbert A-module, (ii) the action of A on Gamma(X) is defined by means of the connection of A, (iii) the C*-inner product of Gamma(X) is induced by the hermitian metric of epsilon(X). We prove that the Hilbert C*-module Gamma(X) is isomorphic to X. This sectional representation is a generalization of the Serre-Swan theorem to non-commutative C*-algebras. We show that epsilon(X) is isomorphic to an associated bundle of an infinite dimensional Hopf bundle with the structure group U(1). (C) 2003 Elsevier Science B.V. All rights reserved.

[发布日期] 2003-11-01

[关键词] non-commutative geometry;Serre-Swan theorem;Hilbert C*-module