Extreme flatness of normed modules and Arveson-Wittstock type theorems

TL;DR

This paper demonstrates a homological property of certain normed modules over bounded operators, leading to new extension theorems that generalize the classical Arveson-Wittstock theorem in a quantum context.

Contribution

It introduces a homological notion of flatness for normed modules over operator algebras and derives new extension theorems generalizing Arveson-Wittstock results.

Findings

Identification of a flatness-like property in normed modules

Derivation of several extension theorems for modules

Establishment of a non-matricial Arveson-Wittstock theorem

Abstract

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).