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

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