SOME REMARKS ON HAAGERUP'S APPROXIMATION PROPERTY

A finite von Neumann algebra M with a faithful normal trace tau has Haagerup's approximation property if there exists a pointwise deformation of the identity in 2-norm by subtracial compact completely positive maps. In this paper we prove that the subtraciality condition can be removed. This enables us to provide a description of Haagerup's approximation property in terms of correspondences. We also show that if N subset of M is an amenable inclusion of finite von Neumann algebras and N has Haagerup's approximation property, then M also has Haagerup's approximation property.