Weak Exactness and Amalgamated Free Product of von Neumann Algebras

Abstract

Weakly exact von Neumann algebras were introduced by Kirchberg in 1995 in parallel with exact C$^{}$-algebras and have since become an effective tool for allowing the use of C$^{}$-algebraic techniques in the theory von Neumann algebras. It is known that for von Neumann algebras with separable predual, weak exactness is preserved under taking subalgebras with conditional expectations, tensor products, crossed products with exact groups, and increasing unions. We show that the amalgamated free product of weakly exact von Neumann algebras is weakly exact. This is done by using a universal property of Toeplitz-Pimsner algebras and a locally convex topology on bimodules of von Neumann algebras, which is used to characterize weakly exact von Neumann algebras. As a corollary, we deduce that weak exactness is preserved under HNN extensions and graph products. We also introduce a generalized notion of exactness on C$^{}$-bimodules which extends both exactness of C$^{}$-algebras and weak exactness of von Neumann algebras.