Rescalings of free products of II₁–factors

Abstract

We introduce the notation Q ( 1 ) ∗ ⋯ ∗ Q ( n ) ∗ L ( F r ) \mathcal {Q}(1)*\cdots *\mathcal {Q}(n)*L(\mathbf F_r) for von Neumann algebra II 1 _1 –factors where r r is allowed to be negative. This notation is defined by rescalings of free products of II 1 _1 –factors, and is proved to be consistent with known results and natural operations. We also give two statements which we prove are equivalent to isomorphism of free group factors.