Extensions of Hilbert C*-modules II

Abstract

Abstract. We describe the pullback construction in the category of Hilbert C * -modules (with a suitable class of morphisms) in terms of pullbacks of underlying C * -algebras. In the second section the Busby invariant for extensions of Hilbert C * -modules is introduced and it is proved that each extension is uniquely determined, up to isomorphism, by the corresponding Busby map. The induced extensions of the underlying C * -algebras as well as of the corresponding linking algebras are also discussed. The paper ends with a Hilbert C * -module version of a familiar result which states that a C * -algebra is projective if and only if it is corona projective.

A note on extensions of Hilbert C*-modules and their morphisms

“…The objective of this section is to provide a connection between Blecher's Hilbert C * -extensions of operator spaces and the C * -extensions of Hilbert C * -modules as introduced by D. Bakić and B. Guljaš ( [2]). In order to obtain such a connection, it is necessary to generalize Blecher's definition in a way which will ensure that taking an operator space which is also o Hilbert C * -module one generally gets more than one extension.…”

“…A strict completion of a (full) Hilbert A-module V is a Hilbert B-module W which is V -strictly complete and such that A is an essential ideal in B. It is proven in [2] that the strict completion of a Hilbert A-module V is (up to isomorphism) the Hilbert…”

“…It is well known that the multiplier algebra M(A) of a C * -algebra A can be realized as the strict completion of A. D. Bakić and B. Guljaš have generalized this concept for Hilbert C * -modules (for details, see [2]), by introducing the strict topology on a Hilbert C * -module induced by an ideal submodule V . A strict completion of a (full) Hilbert A-module V is a Hilbert B-module W which is V -strictly complete and such that A is an essential ideal in B.…”

“…If Im(ϕ) is an essential ideal in B, we speak of essential extensions. The above mentioned strict completion is an C * -extension (using natural maps γ : A → M(A) and Γ : V → M(V )) and it is shown in [2] that it is the maximal C * -extension of V . The second part of this paper is an attempt of providing a connection between this concept of C * -extensions of a Hilbert C * -module and of Hilbert C * -extensions Hilbert C * -extensions of operator spaces defined in [5].…”

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

“…A definition of an extension (see [1,2]) of a Hilbert C * -module is given inside of the category which objects are Hilbert C * -modules and morphisms are given as follows: for a Hilbert A-module V and a Hilbert B-module W a map Φ : V → W is a morphism (or a ϕ-morphism) of Hilbert C * -modules if there is a morphism (a * -homomorphism) ϕ :…”