A maximal abelian subalgebra of a von Neumann algebra 
is a unital abelian star-subalgebra 
not properly contained in any other abelian star-subalgebra
of 
.
It is abbreviated MASA and satisfies 
. For example, the diagonal matrices form
a maximal abelian subalgebra of 
. Such subalgebras need not be the range of a normal conditional expectation. Paving
over a maximal abelian subalgebra uses finite partitions
of the identity by its projections.
Maximal Abelian Subalgebra
See also
Commutant, Operator Algebra Conditional Expectation, Operator Paving, von Neumann AlgebraExplore with Wolfram|Alpha
References
Anantharaman, C. and Popa, S. An Introduction to
Factors. https://www.math.ucla.edu/~popa/Books/IIun.pdf.
Cite this as:
Weisstein, Eric W. "Maximal Abelian Subalgebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MaximalAbelianSubalgebra.html