MASA in separable hyperfinite type II_1 factor

Question

Does the separable hyperfine type $II_1$ factor contain a copy of $L_\infty([0,1])$?

Answer 1

Lots and lots of copies (as von Neumann algebras). In fact, any masa in any II$_1$-factor is isomorphic to $L^\infty[0,1]$.

This way more general result holds:

Theorem. Let $A$ be a diffuse separable abelian von Neumann algebra with faithful normal trace $τ$. Then there exists a $∗$-isomorphism $θ:A\to L^∞[0, 1]$ such that $$ τ(x) =\int_0^1θ(x)(t) dt,\qquad\qquad x ∈ A. $$

This can be found for instance as Theorem 3.5.2 in Finite von Neumann Algebras and Masas by Sinclair & Smith.