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Journal of Operator Theory

Volume 75, Issue 2, Spring 2016  pp. 259-288.

Haagerup approximation property and positive cones associated with a von Neumann algebra

Authors:  Rui Okayasu (1) and Reiji Tomatsu (2)
Author institution:(1) Department of Mathematics Education, Osaka Kyoiku, University, Osaka 582-8582, Japan
(2) Department of Mathematics, Hokkaido University, Hokkaido 060-0810, Japan

Summary: We introduce the notion of the $\alpha$-Haagerup approximation property ($\alpha$-HAP) for $\alpha\in[0,1/2]$ using a one-parameter family of positive cones studied by Araki and show that the $\alpha$-HAP actually does not depend on the choice of $\alpha$. This enables us to prove the fact that the Haagerup approximation properties introduced in two ways are actually equivalent, one in terms of the standard form and the other in terms of completely positive maps. We also discuss the $L^p$-Haagerup approximation property ($L^p$-HAP) for a non-commutative $L^p$-space associated with a von Neumann algebra for $p\in(1,\infty)$ and show the independence of the $L^p$-HAP on the choice of $p$.

Keywords: von Neumann algebra, Haagerup approximation property, non-commutative $L^p$-space

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