Morphisms out of a split extension of a Hilbert C*-module

Abstract

Abstract. Let us have a split extension W of a Hilbert C * -module V by a Hilbert C * -module Z. Like in the case of C * -algebras (well known theorem of T. A. Loring), every morphism out of W , more precisely from W to an arbitrary Hilbert C * -module U , can be described as a pair of morphisms from V and Z, respectively, into U that satisfies certain conditions. It turns out that besides the generalization of the Loring's condition, an additional condition has to be posed. PreliminariesA Hilbert C * -module V over a C * -algebra A (a Hilbert A-module) is a generalization of a Hilbert space in the sense that the "inner product" (· | ·) : V × V → A defined on it takes values in a C * -algebra A instead of the field of complex numbers C (see [3,6]). V is said to be full if the (closed) ideal in A generated by elements (When making quotients of a Hilbert C * -module by its submodule, only the quotient of a Hilbert C * -module by its ideal submodule is again a Hilbert C * -module. What is an ideal submodule? The ideal submodule V I of V associated to an ideal I ⊆ A is V I = {vb : v ∈ V, b ∈ I} = {v ∈ V : (v | x) ∈ I, ∀x ∈ V }. Denote by Π : V → V | VI , π : A → A| I canonical quotient maps. V | VI has a natural Hilbert A| I -module structure with the operation of right multiplication and the inner product given by:A sum of submodules

Morphisms of extensions of Hilbert C*-modules

“…It is done in [9] for the case of split extensions. In order to do it in a general case, one has to define an idealizer of a Hilbert C * -module.…”