On the extensions of finite factors, I
Masahiro Nakamura, Zirô Takeda · Proceedings of the Japan Academy Series A Mathematical Sciences · 1959
In the previous paper 3, we have introduced the concept of crossed product of von Neumann algebras and using it we have generalized the construction of finite factors given by F. J. Murray and J. von Neumann 2. However the definition given there is rather restricted comparing with the usual concept given for Notherian rings since it corresponds only for cases with special factor sets.In this note we modify the definition of crossed product of von Neumann algebras to fill up this gap and show some properties of this generalized product.We shall show in the below that the crossed product of a finite factor with respect to a normalized factor set of unitary operators can be defined as usual and that the product is related deeply with the extension of a group described in 1.Also we shall show that the product satisfies the usual elementary properties of the crossed product and that our new product gives a way of construction of the crossed product basing on an intermediate subfactor.