Original scientific paper

https://doi.org/10.3336/gm.56.2.08

Hilbert \(C^{*}\)-modules in which all relatively strictly closed submodules are complemented

Boris Guljaš ; Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia

Download JATS file

Abstract

We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given.

As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in \(\ell_\infty\) and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

Keywords

Hilbert modules, orthogonal complementing, reletive strict topology for modules, hereditary subalgebras and hereditary modules, direct sums of algebras and modules

Hrčak ID:

267557

URI

https://hrcak.srce.hr/267557

Publication date:

23.12.2021.

Visits: 611 *