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Extreme flatness and Hahn-Banach type theorems for normed modules over c_0 | | A. Ya. Helemskii
; | | Date: |
4 May 2010 | | 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. | | Source: | arXiv, 1005.0489 | | Services: | Forum | Review | PDF | Favorites | |
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