# 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 $p$.

**DOI: **http://dx.doi.org/10.7900/jot.2015feb24.2058

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

Contents
Full-Text PDF