Extreme flatness and Hahn-Banach type theorems for normed modules over c_0

TL;DR

This paper characterizes extremely flat homogeneous modules over c_0 and derives Hahn-Banach type theorems for c_0-modules, revealing new insights into module extensions and tensor product injectivity.

Contribution

It provides a complete description of essential homogeneous modules over c_0 that are extremely flat, and establishes new Hahn-Banach type theorems for c_0-modules.

Findings

All l_p-sums of normed spaces of integrable functions are extremely flat.

Topologically injective morphisms tensor with any module remain injective.

Non-essential homogeneous c_0-modules are not extremely flat.

Abstract

Let A be a commutative normed algebra, K a class of normed A-modules. A normed A-module Z is called extremely flat with respect to K, if, for every isometric morphism of normed A-modules, belonging to K, the non-completed projective A-module tensor product of this morphism and the identity morphism on Z, is also isometric. In the present paper we take, in the capacity of A, the algebra c_0 of vanishing sequences and consider the class of the so-called homogeneous modules, over the latter algebra, denoted by H. The main theorem gives a full description of essential homogeneous modules over the mentioned algebra that are extremely flat with respect to H. (In particular, all l_p-sums; p<infinity of normed spaces of integrable functions on different measure spaces have the indicated property). As a corollary, some theorems of Hahn-Banach type, concerning extensions of c_0-module morphisms with preservation of their norms, are obtained. (In particular, l_p-sums of normed spaces of essentially bounded functions play, in the relevant context, the same role as the scalar field in the classical Hahn-Banach Theorem). Besides, the following related result is established. If I is a topologically injective morphism of normed c_0-modules, then for every normed c_0-module Z, the non-completed projective tensor product of I and the identity morphism on Z is also injective (despite it is, generally speaking, not topologically injective). If I is a (just) injective bounded morphism, then such a tensor product is not bound to be injective. Finally, it is shown that we can not omit the word "essential" in the formulation of the main theorem. Namely, the non-essential homogeneous c_0-module of all bounded sequences is not extremely flat with respect to H.