Operator Algebra Crossed Product

An operator algebra crossed product adjoins operators implementing a group action to an operator algebra. For an action of a discrete group on a unital C^*-algebra , its full crossed product is generated by and implementing unitaries satisfying

It is universal for covariant representations, while the reduced crossed product is obtained from the regular representation. For amenable groups these constructions agree. An action on a von Neumann algebra also has a von Neumann crossed product, formed using the regular representation and weak closure. For a homeomorphism of a compact Hausdorff space , uses . For the trivial action of on , the crossed product is . Modular actions of give continuous cores of von Neumann algebras.