The definition of amalgamated free product for general von Neumann algebras

Question

When discussing the amalgamated free product von Neumann algebras, people often assume that the algebras are $\sigma$-finite. I am wandering if there is any literature on the amalgamated free product for general (non $\sigma$-finite) von Neumann algebras? Thank you in advance.

Answer 1

Let's just consider the case of amalgamation over $\mathbb C$ as the general case is similar. Almost all definitions of free product of von Neumann algebras call for faithful normal states. This condition is equivalent to the algebras being $\sigma$-finite by Takesaki (volume I, II.3.19).

However, one can still do the construction using non-faithful normal states but with the realization that your original von Neumann algebras may fail to be embedded in the free product. The only instance of this I could find was in a few of Dykema's papers where he looks at von Neumann algebras with normal states whose GNS representations are faithful, a condition that is weaker than having a faithful normal state.