Morphisms of extensions of Hilbert C*-modules

Abstract

Abstract. We consider the condition for a morphism of (between) extensions of Hilbert C * -modules to exist and give the description of a morphism out of an extension of a Hilbert C * -module in a general case. Preliminary definitionsThe definition of extensions of Hilbert C * -modules is relatively recent. It was announced in [2, Example 2.10] and developed in a series of papers ([3, 4, 5]). For the sake of completeness we recall here some basic definitions.Let V be a (right) Hilbert C * -module over a C * -algebra A (a Hilbert Amodule) and let W be a Hilbert B-module over some C * -algebra B. A map Φ :Each morphism of Hilbert C * -modules is necessarily linear, contractive and a module map (in a sense that Φ(va) = Φ(v)ϕ(a) is valid for all v ∈ V, a ∈ A).The ideal submodule V I of V associated to an ideal I ⊆ A iswe denote canonical quotient maps. A quotient V | VI has a natural Hilbert A| I -module structure via operations:A Hilbert A-module V is said to be full if (V | V ) (the (closed) ideal in A generated by elements (v 1 | v 2 ), v 1 , v 2 ∈ V ) is all of A. We may freely suppose that V is full because we can always consider V as a full Hilbert (V | V )-module.