On Infinite Tensor Products of Von Neumann Algebras

Abstract

Q-?) 0ij. This problem has been completely solved in a series of papers [1, 3, 9, 10, 15] when each j if of type I, and partial results have been obtained in the general case [4, 13]. In order that 2 be of type I it is necessary that each Nj is of type I [11], hence that case is completely taken care of in the quoted papers. Similarly, if 2 is finite each 2?j is finite, hence a result of Bures [4, Thm. 4.3] gives necessary and sufficient conditions in order that 2 be finite. If one of the gj is of type III then 2 is of type III [11]. Therefore, what is left is to give necessary and sufficient conditions in order that 2 be semi-finite under the assumption that each ?j is semi-finite. This will be done in the present paper: the result is then quite analogous to that when each 2?j is a type I factor. Our proof differs from that in the type I situation in that we avoid most of the measure theoretic arguments of Moore [9] and some of the manipulations with infinite series by him and Takenouchi [15]. Instead we use some powerful results of Tomita [17] and Takesaki [16] on the modular automorphisms of von Neumann algebras induced by faithful normal states. For our purposes the simplest defintion of modular automorphisms is given by Takesaki [16, Thms. 13. 1 and 13. 2]. Let 2 be a von Neumann algebra and p a faithful normal state of S. p is said to satisfy the KMS boundary condition with respect to a strongly continuous one parameter *-automorphism group at, -oo 0 such that for each pair of operators A, B in 2 there is a function F(z) holomorphic in the strip 0 < Im z < 8 with boundary values, F(t)-1p (gt (A)B) and F(t + iP) p (Bgt (A) ). Let now p be a faithful