Hilbert C*-module
Hilbert C*-modules are mathematical objects which generalise the notion of a Hilbert space (which itself is a generalisation of Euclidean space), in that they endow a linear space with an "inner product" which takes values in a C*-algebra. Hilbert C*-modules were first introduced in the work of Irving Kaplansky in 1953, which developed the theory for commutative, unital algebras (though Kaplansky observed that the assumption of a unit element was not "vital"). In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke and Marc Rieffel, the latter in a paper which used Hilbert C*-modules to construct a theory of induced representations of C*-algebras. Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory, and provide the right framework to extend the notion of Morita equivalence to C*-algebras. They can be viewed as the generalization of vector bundles to noncommutative C*-algebras and as such play an important role in noncommutative geometry, notably in C*-algebraic quantum group theory, and groupoid C*-algebras.
Abelian von Neumann algebra
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", μ) is a standard measure space (that is "X" − "N" is a standard Borel space for some null set "N" and μ is a σ-finite measure) then "L"("X", μ) is separable.