Central support of a projection

Question

With support of a measurable function in measure space can we make the analogy with the Central support of a projection in von Neumann algebra using functional calculus?? The question I asked because central support of normal functional in von Neumann algebra is as same as support of a measure in measure space.

Answer 1

I don't think you'll have a good analogy with the abelian case. In the abelian case, the centre is everything so the notion of central projection is irrelevant (the central support of any projection is the projection itself).

The notion of central support, basically, tells you in which "blocks" the projection lies. For instance, if $$ M=M_2(\mathbb C)\oplus M_3(\mathbb C)\oplus M_4(\mathbb C) $$ and $$ p=E_{11}\oplus 0\oplus E_{22}, $$ then the central support of $p$ is $$ I_2\oplus 0\oplus I_4. $$