Discrete spectra of 𝐶*-algebras and orthogonally closed submodules in Hilbert 𝐶*-modules

Abstract

Let A A and B B be C ∗ C^{*} -algebras and let X X be an A A - B B -imprimitivity bimodule. Then it is shown that if the spectrum A ^ \widehat A of A A (resp. B ^ \widehat B of B B ) is discrete, then every closed A A - B B -submodule of X X is orthogonally closed in X X , and conversely that if A ^ \widehat A (resp. B ^ \widehat B ) is a T 1 T_{1} -space and if every closed A A - B B -submodule of X X is orthogonally closed in X X , then A ^ \widehat A (resp. B ^ \widehat B ) is discrete.