Journal of Operator Theory

Volume 64, Issue 1, Summer 2010 pp. 171-188.

Extreme flatness of normed modules and Arveson-Wittstock type theorems

Authors: A.Ya Helemskii
Author institution: Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119992, Russia

Summary: Let

$L$ be a separable infinite-dimensional Hilbert space

$\bb:=\bb(L)$ . A contractive right

$\bb$ -module

$X$ is called {\it semi-Ruan module}, if for every

$u,v\in X$ and mutually orthogonal projections

$P,Q\in\bb$ we have

$\|u\cd P+v\cd Q\|\leqslant (\|u\cd P\|^2+\|v\cd Q\|^2)^{1/2}.$ For an arbitrary Hilbert space

$H$ we consider the Hilbert tensor product

$L\otimes H$ as a left

$\bb$ -module with the outer multiplication

$a\cd\zeta:=(a\otimes\id_H)\zeta; a\in\bb, \zeta\in L\otimes H.$ We prove that \textit{for every isometric morphism

$\alpha : Y \to Z$ of right semi-Ruan modules the operator

$\alpha\mmbd \id_{L\otimes H}$ is also isometric}. As corollaries, we obtain several theorems on extensions of morphisms with the preservation of their norms and a new proof of the Arveson--Wittstock Extension Theorem in operator theory.

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