Maximal abelian von Neumann subalgebra

A unital abelian von Neumann subalgebra $A\subseteq M$ is maximal abelian if no strictly larger abelian von Neumann subalgebra of $M$ contains it. Equivalently,

A'\cap M=A.

It is often called a MASA.

Example and use

Diagonal matrices form a MASA in $M_n(\mathbb C)$ . For an essentially free action of a countable group on a standard probability space, the function algebra $L^\infty(X)$ is a MASA in the group–measure-space crossed product. This forces every central element to belong to the function algebra.

References

Sorin Popa, Ergodic theory of group actions. Theorem 4.1.1 and §4.3.