Von Neumann Algebra - Examples

Examples

• The essentially bounded functions on a σ-finite measure space form a commutative (type I₁) von Neumann algebra acting on the L2 functions. For certain non-σ-finite measure spaces, usually considered pathological, L∞(X) is not a von Neumann algebra; for example, the σ-algebra of measurable sets might be the countable-cocountable algebra on an uncountable set.

• The bounded operators on any Hilbert space form a von Neumann algebra, indeed a factor, of type I.

• If we have any unitary representation of a group G on a Hilbert space H then the bounded operators commuting with G form a von Neumann algebra G′, whose projections correspond exactly to the closed subspaces of H invariant under G. Equivalent subrepresentations correspond to equivalent projections in G′. The double commutant G′′ of G is also a von Neumann algebra.

• The von Neumann group algebra of a discrete group G is the algebra of all bounded operators on H = l2(G) commuting with the action of G on H through right multiplication. One can show that this is the von Neumann algebra generated by the operators corresponding to multiplication from the left with an element g ∈ G. It is a factor (of type II₁) if every non-trivial conjugacy class of G is infinite (for example, a non-abelian free group), and is the hyperfinite factor of type II₁ if in addition G is a union of finite subgroups (for example, the group of all permutations of the integers fixing all but a finite number of elements).

• The tensor product of two von Neumann algebras, or of a countable number with states, is a von Neumann algebra as described in the section above.

• The crossed product of a von Neumann algebra by a discrete (or more generally locally compact) group can be defined, and is a von Neumann algebra. Special cases are the group-measure space construction of Murray and von Neumann and Krieger factors.

• The von Neumann algebras of a measurable equivalence relation and a measurable groupoid can be defined. These examples generalise von Neumann group algebras and the group-measure space construction.