On Topics Related to Tensor Products with Abelian von Neumann Algebras

Abstract

In this note, we developed several results concerning abelian von Neumann algebras, their spectrums, and their tensor products with other von Neumann algebras. In particular, we developed a theory connecting elements of the spectrum of $L^\infty(\Omega, \mu)$ with ultrafilters on a Boolean algebra associated with $(\Omega, \mu)$, generalizing the case where $\Omega$ is a discrete space and its spectrum is identified with the space of ultrafilters on $\Omega$. These ultrafilters were also used to define a new kind of tracial ultrapowers. We also developed a theory representing elements of $L^\infty(\Omega, \mu) \bar{\otimes} M$ as functions that are continuous from the spectrum to $M$ equipped with the weak* topology, analogous to the classical direct integral theory, but which can apply to non-separable von Neumann algebras. We then applied this theory to prove several structural results of isomorphisms $L^\infty(\Omega, \mu) \bar{\otimes} M \rightarrow L^\infty(\Omega, \mu) \bar{\otimes} N$ where $M$ and $N$ are $\mathrm{II}_1$ factors and placed several constraints on $M$ and $N$. The ultrafilter representation of the spectrum was then used to perform certain calculations to provide specific examples regarding the spectrum representations of tensor products as well as the structures of isomorphisms $L^\infty(\Omega, \mu) \bar{\otimes} M \rightarrow L^\infty(\Omega, \mu) \bar{\otimes} N$.