Generalisations of the Haagerup approximation property to arbitrary von   Neumann algebras
  The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalized to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in terms of the standard form and the other in terms 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. 
Martijn Caspers, Rui Okayasu, Adam Skalski, Reiji Tomatsu
math.OA math.FA