Compressions of free products of von Neumann algebras

Abstract

A reduction formula for compressions of von Neumann algebra II $_1$ –factors arising as free products is proved. This shows that the fundamental group is ${\bf R}^*_+$ for some such algebras. Additionally, by taking a sort of free product with an unbounded semicircular element, continuous one parameter groups of trace scaling automorphisms on II $_\infty$ –factors are constructed; this produces type III $_1$ factors with core $\mathcal{M}\otimes B(\mathcal{H})$ , where $\mathcal{M}$ can be a full II $_1$ –factor without the Haagerup approximation property.