A characterization of Hilbert $C^*$-modules over finite dimensional $C^*$-algebras

TL;DR

This paper characterizes when the unit ball of a full Hilbert $C^*$-module is sequentially compact in a weak topology, showing it occurs if and only if the underlying $C^*$-algebra is finite dimensional.

Contribution

It provides a precise characterization linking the sequential compactness of the unit ball to the finite dimensionality of the underlying $C^*$-algebra.

Findings

Unit ball is sequentially compact iff the algebra is finite dimensional

Answers a question posed in prior research

Connects topological properties with algebraic dimensionality

Abstract

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