I came across the following sentence which I do not know how to prove when reading a paper,
"Suppose $(X_0,\mu_0)$ is a non-trivial atomic probability space, then we can identify $L^{\infty}(\mathbb{T},\lambda)$ wih ${\overline{\otimes}_{n\geq 0}}L^{\infty}(X_n,\mu_n)$, where $(X_n,\mu_n)=(X_0,\mu_0), \forall n\geq 0$."
Toward this, I guess we need to understand (in the context of von neumann algebras):
1, Show that ${\overline{\otimes}_{n\geq 0}}L^{\infty}(X_n,\mu_n)=L^{\infty}(\prod_{n\geq 0}X_n, \prod_{n\geq 0}\mu_n)$ (the RHS should not be $L^{\infty}(\oplus_{n\geq 0}X_n, ?))$, since $\oplus X_n$ may be countable)
2, Understand when we would have $L^{\infty}(X,\mu)\cong L^{\infty}(Y,\nu)$.
Thanks in advance!
For your first question, what do you mean by $\bigoplus X_n$?
As far as two goes, the category of abelian von Neumann algebras is dual to the category of standard measure spaces. So $L^\infty(X)\simeq L^\infty(Y)\Leftrightarrow X\simeq Y$ as measure spaces. As far as the last part all diffuse standard measure spaces are isomorphic. So inparticular $\mathbb{T}$ is isomorphic to $\Pi X_n$ for any $X_n$.