Injective envelopes of Banach modules

Injective Envelopes of Banach Spaces

Injective Envelopes of C∗-algebras as Operator Modules

In this paper we give some characterizations of M. Hamana’s injective envelope I(A) of a C∗-algebra A in the setting of operator spaces and completely bounded maps. These characterizations lead to simplifications and generalizations of some known results concerning completely bounded projections onto C∗-algebras. We prove that I(A) is rigid for completely bounded A-module maps. This rigidity yi...

Kantorovich’s Principle in Action: Aw ∗-modules and Injective Banach Lattices

The aim of this note is to demonstrate that Kaplansky–Hilbert lattices and injective Banach lattices may be produced from each other by means of the well known convexification procedure. This is done via the Boolean valued analysis approach. The subject gives a good opportunity to discuss also the relationship between the Kantorovich’s heuristic principle and the Boolean value transfer principl...

Generalizations of principally quasi-injective modules and quasiprincipally injective modules

LetR be a ring andM a rightR-module with S= End(MR). The moduleM is called almost principally quasi-injective (or APQ-injective for short) if, for any m∈M, there exists an S-submodule Xm of M such that lMrR(m) = Sm ⊕ Xm. The module M is called almost quasiprincipally injective (or AQP-injective for short) if, for any s∈ S, there exists a left ideal Xs of S such that lS(ker(s)) = Ss ⊕ Xs. In thi...

Separably Injective Banach Spaces

It is no exaggeration to say that the theory of separably injective spaces is quite different from that of injective spaces. In this chapter we will explain why. Indeed, we will enter now in the main topic of the monograph, namely, separably injective spaces and their “universal” version. After giving the main definitions and taking a look at the first natural examples one encounters, we presen...