## Mathematical Sciences: Operator Algebras

##### Abstract

DMS-9622457 Kraus SUNY @ Buffalo For discrete groups G, there is a close connection between approximation properties of the group von Neumann algebra VN(G) and properties of G. In particular, VN(G) is injective if and only if G is amenable. The proposer and Uffe Haagerup have defined a property for locally compact groups (called the approximation property, or AP) which is weaker than amenability, and have shown that a discrete group G has the AP if and only if VN(G) has the weak* operator approximation property (the weak* OAP). The weak* OAP is a normal matricial version (for dual operator spaces) of the approximation property for Banach spaces. A dual operator space has the weak* OAP if and only if it satisfies the slice map property for arbitrary weak* closed subspaces of B(H), and so von Neumann algebras with the weak* OAP behave well with respect to taking tensor products with such subspaces. Notions of amenabilty have been defined for homogeneous spaces, correspondences and Kac algebras. The proposer has defined appropriate versions of the AP in each of these cases and during this project will investigate a number of problems concerning the weak* OAP and approximation properties for groups, homogeneous spaces, correspondences and Kac algebras. Von Neumann algebras are an important tool in the study of quantum mechanics and quantum field theory, and also play an important role in the study of group representations (which themselves are used in quantum physics). The von Neumann algebras that arise in these applications are usually infinite dimensional spaces. One way of obtaining information about the structure and properties of such infinite dimensional objects is to approximate them in an appropriate way by finite dimensional objects. The approximation properties of von Neumann algebras that will be investigated in this project are of this type. The weakest (least restrictive) such property is weak enough that it is satisfied by a large class of von Neumann algebras, but strong enough that knowing it is valid gives us important information about the von Neumann algebra. In the case of discrete groups, much is known about the connection between properties of the groups and approximation properties of the associated von Neumann algebras. The theory of Kac algebras can be viewed as a generalization of the theory of group representations. One of the goals of this project is to investigate to what extent the results concerning approximation properties for groups can be extended to the more general setting of Kac algebras.