Journal of Operator Theory

Volume 47, Issue 1, Winter 2002 pp. 145-167.

Perturbation of $l^1$-copies and measure convergence

Authors: Hermann Pfitzner
Author institution: Universite d'Orleans, BP 6759, F-45067 Orleans Cedex 2, France

Summary: Let $L^1$ be the predual of a von Neumann algebra with a finite faithful normal trace. We show that a bounded sequence in $L^1$ converges to $0$ in measure if and only if each of its subsequences admits another subsequence which converges to $0$ in norm or spans $l^1$ almost isometrically. Furthermore we give a quantitative version of an essentially known result concerning the perturbation of a sequence spanning $l^1$ isomorphically in the dual of a $C^*$-algebra.

Keywords: asymptotically isometric copies of $l_1$, James' distortion, L-summands, L-embedded, measure topology

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