Hilbert modules revisited: orthonormal bases and Hilbert-Schmidt operators

Abstract

The concept of a Hilbert module (over an H * -algebra) arises as a generalization of that of a complex Hilbert space when the complex field is replaced by an (associative) H * -algebra with zero annihilator. P. P. Saworotnow [13] introduced Hilbert modules and extended to its context some classical theorems from the theory of Hilbert spaces, J. F. Smith [17] gave a complete structure theory for Hilbert modules, and G. R. Giellis [9] obtained a nice characteristization of Hilbert modules.