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A characterization of Hilbert C * -modules over finite dimensional C * -algebras. (English) Zbl 1188.46036

Hilbert C * -modules are generalizations of Hilbert spaces, but the theory of Hilbert C * -modules is different from the theory of Hilbert spaces; for example, not all closed submodules of a given Hilbert C * -modules are orthogonally complemented. It is well-known that the closed unit ball of a Hilbert space H is weakly sequentially compact. This result is not true for Hilbert C * -modules.

A sequence {ξ n } n in a Hilbert C * -module E over a C * -algebra A is weakly convergent to an element ξE if the sequence {ξ n ,η} n converges to ξ,η with respect to the C * -norm on A, for each ηE. The authors prove that the closed unit ball of a full Hilbert C * -module E over a C * -algebra A is weakly sequentially compact if and only if the C * -algebra A is finite-dimensional.


MSC:
46L08C * -modules
46L05General theory of C * -algebras
46L10General theory of von Neumann algebras