the isomorphism of $L^\infty$ spaces

Question

If we have an unitary operator from $L^2(\mathbb{T})$ to $L^2(X,d\mu)$ ,$\mathbb{T}$ is boundary of the unit ball ,$d\mu$is the Borel probability measure.is there an isomorphism between $L^\infty(\mathbb{T})$ to $L^\infty(X,d\mu)$? Many thanks.

Answer 1

The answer is yes, because vaguely speaking localizable measure spaces are nothing more than a commutative von Neumann algebras. For more details see this answer on mathoverflow.com.