Spatial representation of groups of automorphisms of von Neumann algebras with properly infinite commutant

Abstract

Theorem. Let a topological groupG be represented (a→φ _a ) by *-automorphisms of a von Neumann algebraR acting on a separable Hilbert spaceH. Suppose that

• (a) G is locally compact and separable,

• (b) R′ is properly infinite,

• (c) for anyT∈R,x,y∈H the function

$$a \to \left\langle {\phi _a (T)x,y} \right\rangle _H $$

is measurable onG. Then there exists a strongly continuous unitary representation ofG onH,a→U _a , such that forT∈R,a∈G,

$$\phi _\alpha (T) = U_a TU_a *.$$

.