Crossed Product - Duality

Duality

If is a von Neumann algebra on which a locally compact Abelian acts, then, the dual group of characters of, acts by unitaries on :

These unitaries normalise the crossed product, defining the dual action of . Together with the crossed product, they generate, which can be identified with the iterated crossed product by the dual action . Under this identification, the double dual action of (the dual group of ) corresponds to the tensor product of the original action on and conjugation by the following unitaries on :

The crossed product may be identified with the fixed point algebra of the double dual action. More generally is the fixed point algebra of in the crossed product.

Similar statements hold when is replaced by a non-Abelian locally compact group or more generally a locally compact quantum group, a class of Hopf algebra related to von Neumann algebras. An analogous theory has also been developed for actions on C* algebras and their crossed products.

Duality first appeared for actions of the reals in the work of Connes and Takesaki on the classification of Type III factors. According to Tomita–Takesaki theory, every vector which is cyclic for the factor and its commutant gives rise to a 1-parameter modular automorphism group. The corresponding crossed product is a Type von Neumann algebra and the corresponding dual action restricts to an ergodic action of the reals on its centre, an Abelian von Neumann algebra. This ergodic flow is called the flow of weights; it is independent of the choice of cyclic vector. The Connes spectrum, a closed subgroup of the positive reals, is obtained by applying the exponential to the kernel of this flow.

• When the kernel is the whole of, the factor is type .
• When the kernel is for in (0,1), the factor is type .
• When the kernel is trivial, the factor is type .

Connes and Haagerup proved that the Connes spectrum and the flow of weights are complete invariants of hyperfinite Type III factors. From this classification and results in ergodic theory, it is known that every infinite-dimensional hyperfinite factor has the form for some free ergodic action of .