Convergence of bounded sequences in Hilbert C*-modules

Abstract

In this paper, we provide the necessary conditions for a monotone bounded nets(with respect to diamond order) to be convergent. Also, we show that each bounded sequence in a Hilbert A-module has a convergent subsequence. Moreovere, we prove that a module map *-preserves the orthonormality of a basis whenever it is a surjective '-morphism. Finaly, we present the sufficient conditions for a Hilbert A-module to be finite dimensional.