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

We introduce the notion of the a-Haagerup approximation property (α-HAP) for α ∈ [0, 1/2] using a one-parameter family of positive cones studied by Araki and show that the a-HAP actually does not depend on the choice of α. 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 noncommutative L^p-space associated with a von Neumann algebra for p ∈ (1,∞) and show the independence of the L^p-HAP on the choice of p.