High Quality Content by WIKIPEDIA articles! In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute. Though there is a theory for von Neumann algebras on non-separable Hilbert spaces (and indeed much of the general theory still holds in that case) the theory is considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces. Note that if the measure spaces (X, ) is a standard measure space (that is X – N is a standard Borel space for some null set N and is a -finite measure) then L2 (X) is separable.

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