Original scientific paper

On a class of module maps of Hilbert C*-modules

D. Bakić
B. Guljaš

Abstract

The paper describes some basic properties of a class of module maps of Hilbert C*-modules.
In Section 1 ideal submodules are considered and the canonical Hilbert C*-module structure on the quotient of a Hilbert C*-module over an ideal submodule is described.
Given a Hilbert C*-module V, an ideal submodule $V_{\inss}$, and the quotient $V/V_{\inss}$, canonical morphisms of the corresponding C*-algebras of adjointable operators are discussed.
In the second part of the paper a class of module maps of Hilbert C*-modules is introduced.Given Hilbert C*-modules V and W and a morphism $\varphi : \ass \rightarrow \bss$ of the underlying \cez-algebras, a map $\Phi : V \rightarrow W$ belongs to the class under consideration if it preserves inner products modulo $\varphi$: $\langle \Phi(x), \Phi(y) \rangle = \varphi(\langle x,y \rangle)$ for all $x,y \in V$.
It is shown that each morphism Φ of this kind is necessarily a contraction such that the kernel of Φ is an ideal submodule of V. A related class of morphisms of the corresponding linking algebras is also discussed.

Keywords

C*-algebra; Hilbert C*-module; adjointable operator

Hrčak ID:

775

URI

https://hrcak.srce.hr/775

Publication date:

17.12.2002.

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