real rank for abelian von Neumann algebra

Question

By Gelfand duality, any abelian von Neumann algebra is correspond to stonean space, which has zero Lebesgue covering dimension. The real rank of abelian C* algebra is the dimension of the its spectrum. So is it true that all abelian von Neumann algebra has real rank zero?

Answer 1

Every von Neumann algebra, abelian or not, has real rank zero. This follows from the Spectral Theorem and the Double Commutant Theorem (or alternatively, the Borel functional calculus), which allow you to approximate any selfadjoint element, in norm, with linear combinations of projections.

In other words, any von Neumann algebra is the norm-closed linear span of its projections.