A description of amalgamated free products of finite von Neumann algebras over finite-dimensional subalgebras

Abstract

We show that a free product of a II 1 -factor and a finite von Neumann algebra with amalgamation over a finite-dimensional subalgebra is always a II 1 -factor, and provide an algorithm for describing it in terms of free products (with amalgamation over the scalars) and compression/dilation. As an application, we show that the class of direct sums of finitely many von Neumann algebras that are interpolated free group factors, hyperfinite II 1 -factors, type I n algebras for n finite and finite-dimensional algebras, is closed under taking free products with amalgamation over finite-dimensional subalgebras.