冯．诺依曼代数交叉积的一点注记

A Note on the Crossed Product of von Neumann Algebras

设α是可数离散群G和H的半直积G■_σH在冯·诺依曼代数M上的作用,则β_h=α_((e,h))AdU_h定义了群H在冯·诺依曼代数交叉积M■_αG上的作用β.本文证明了交叉积冯·诺依曼代数M■_α(G■_σH)与(M■_αG)■_βH是*-同构的,因此在一定条件下,冯·诺依曼代数的交叉积满足结合律.

Let α be an action of the semi-direct product G ×σ H of countably discrete groups G and H on yon Neumann algebra M. Then βh = α（e,h）× AdUh is an action of H on the yon Neumann algebra crossed product M×αG. We show that M×α（G×σH） is ＊-isomorphic to （ M×α G） ×β H, therefore the crossed product of yon Neumann algebras has the associative law.