A characterization of Hilbert $C^{\ast }$-modules over finite dimensional $C^{\ast }$-algebras

Abstract

We show that the unit ball of a full Hilbert $C^{\ast }$-module is sequentially compact in a certain weak topology if and only if the underlying $C^{\ast }$-algebra is finite dimensional. This provides an answer to the question posed in J. Chmieliński et al [Perturbation of the Wigner equation in inner product $C^{\ast }$-modules, J. Math. Phys. 49 (2008), no. 3, 033519; arXiv:0801.2726].

7 Pages; to appear in Operators and Matrices (OAM)

References in corpus (1)

• Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules