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

By D. Bakić, B. Guljaš. 2002

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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 submod- ule VI, and the quotient V/V I, 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 ϕ : A→B of the underlying C ∗ -algebras, a map Φ: V → W belongs to the class under consideration if it preserves inner products modulo ϕ: � Φ(x), Φ(y)� = ϕ(� x, y� ) for all x, y ∈ 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.

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