Extreme flatness of normed modules and Arveson-Wittstock type theorems

Details Extreme flatness of normed modules and Arveson-Wittstock type theorems Extreme flatness of normed modules and Arveson-Wittstock type theorems

2008-04-09

arxiv.org/abs/0804.1434v1

Mathematics

Functional Analysis

19 pages

Scientific paper

We show in this paper that a certain class of normed modules over the algebra of all bounded operators on a Hilbert space possesses a homological property which is a kind of a functional-analytic version of the standard algebraic property of flatness. We mean the preservation, under projective tensor multiplication of modules, of the property of a given morphism to be isometric. As an application, we obtain several extension theorems for different types of modules, called Arveson-Wittstock type theorems. These, in their turn, have, as a straight corollary, the `genuine' Arveson-Wittstock Theorem in its non-matricial presentation. We recall that the latter theorem plays the role of a `quantum' version of the classical Hahn-Banach theorem on the extension of bounded linear functionals. It was originally proved by Wittstock (1981), and a crucial preparatory step was done by Arveson (1969).

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