## Mathematical Sciences: Operator Algebras

##### Abstract

9401397 Kraus The investigator will continue the investigation of tensor product properties (including the OAP and the w*-OAP) for sigma- weakly closed subspaces of the full algebra of operators on a Hilbert space and operator spaces. The operator approximation property (OAP) is a matricial version (for operator spaces) of the approximation property for Banach spaces. The investigator has shown that a sigma-weakly closed subspace has the general slice map property for sigma-weakly closed subspaces if and only if it has the w*-OAP. The investigator and Uffe Haagerup have defined a property for locally compact groups (called the AP) which is weaker than amenability, and have shown that a discrete group has the AP if and only if its group von Neumann algebra has the w*-OAP (if and only if its reduced group C*-algebra has the OAP). Specific problems that will be attacked under this project include:(1) Does SL(3,Z) have the AP?(2)If A is a C*-algebra does A have the OAP if and only if A** has the w*-OAP?(3)Does every exact C*-algebra have the OAP? Hilbert space operators can be considered as an infinite dimensional analog of matrices. There is a natural addition and multiplication defined on operators so that some collections of operators can be considered as operator algebras. Usually operator algebras come equipped with a topological structure. Operator algebras themselves can be used to form product algebras and investigations in this project center around a certain product structure on operator algebras called the tensor product. Operator approximation problems are related in a natural way to questions about the tensor product of operator algebras. The focus here is on these connections between approximation and tensor products.