On matricially normed spaces

Abstract

Arveson and Wittstock have proved a non-commutati ve HahnBanach Theorem for completely hounded operator-valued maps on of operators. In this paper it is shown that if T is a linear map from the dual of an operator space into a C*-algebra, then the usual operator norm of T coincides with the completely bounded norm. This is used to prove that the Arveson-Wittstock theorem does not generalize to normed An elementary proof of the Arveson-Wittstock result is presented. Finally a simple bimodule interpretation is given for the Haagerup and matricial tensor products of matricially normed spaces. 1. Introduction. A. function space V on a set X is a linear subspace of the bounded complex functions on X. With the uniform norm, this is a normed vector space. Conversely, any (complex) normed vector space V may be realized as a function space on the closed unit ball X of the dual space V*. Thus one may regard a normed vector space as simply an abstract function space. An operator space V on a Hubert space H is a linear subspace of the bounded operators on H. For each «GN, the operator norm associated with Hn determines a distinguished norm on the n x n matrices over V. The second author recently gave an abstract characterization for the operator by taking into consideration these systems of matrix norms. The operator V are characterized among the normed spaces (see §2), by the L°°-property: given matrices v = [v/7], w = [wkl] with vzy, wkι e V, ||vθH>||=max{||v||,|M|}.