A certain class of pro $C^*$-algebras and bounded elements of a Hilbert module

Abstract

In this paper, introducing the notion of topologically simple pro $C^*$-algebras, we show that direct product of $C^*$-algebras $K(H_i)$, as the set of all compact operators on a Hilbert space $H_i$, is a topologically simple pro $C^*$-algebra. Applying this fact, we prove that the set of all bounded elements of a certain class of Hilbert modules are dense in the same module.