Free products of hyperfinite von Neumann algebras and free dimension

Abstract

The free product of an arbitrary pair of finite hyperfinite von Neumann algebras is examined, and the result is determined to be the direct sum of a finite dimensional algebra and an interpolated free group factor $L(\freeF_r)$. The finite dimensional part depends on the minimal projections of the original algebras and the "dimension", r, of the free group factor part is found using the notion of free dimension. For discrete amenable groups $G$ and $H$ this implies that the group von Neumann algebra $L(G*H)$ is an interpolated free group factor and depends only on the orders of $G$ and $H$.