The Amalgamated Free Product of Hyperfinite von Neumann Algebras

Abstract

We examine the amalgamated free product of hyperfinite von Neumann algebras. First we describe the amalgamated free product of hyperfinite von Neumann algebras over finite dimensional subalgebras. In this case the result is always the direct sum of a hyperfinite von Neumann algebra and a finite number of interpolated free group factors. We then show that this class is closed under this type of amalgamated free product. After that we allow amalgamation over possibly infinite dimensional multimatrix subalgebras. In this case the product of two hyperfinite von Neumann algebras is the direct sum of a hyperfinite von Neumann algebra and a countable direct sum of interpolated free group factors. As before, we show that this class is closed under amalgamated free products over multimatrix algebras.