Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras

Abstract

The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalised to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in term of the standard form and the other in term of the approximating maps with respect to a fixed faithful normal semifinite weight. Several stability properties, in particular regarding the crossed product construction are established and certain examples are introduced.