A question about von Neumann algebras

Question

I am trying to solve the following exercise:

Prove that the von Neumann algebras $\ell^{\infty}(\mathbb{Z})$ and $L^{\infty}(\mathbb{T})$ are not isomorphic.

Moreover, prove that $B(\ell^2(\mathbb{Z}))$ contains at least two maximal abelian von Neumann algebras that are not isomorphic.

Any hints are welcome! Thank you in advance

Answer 1

For the isomorphism, consider the existence of minimal projections.

The second part of the question consists of representing $\ell^\infty(\mathbb Z)$ and $L^\infty(\mathbb Z)$ as masas in $B(\ell^2(\mathbb Z)$. The former is trivial; for the latter, note that $\ell^2(\mathbb Z)\simeq L^2(\mathbb Z)$.