A family of type III factors without propertyL

Abstract

IntroduetlonRecently Connes [5] made a significant progress in the classification of factors of type III. In particular, he proved that there exists a one-parameter family of non-isomorphic factors of type III without non-trivial central sequences, namely, the non-hyperfinite factors P~ (M2 in [10]) of Pukanszky. In another important paper [6], Connes proved that if a factor M has no non-trivial central sequence then the group Int(M) of all inner automorphisms of M is closed in the group Aut(M) of all automorphisms of M equipped with an appropriate topology (this result was obtained independently by Sakai [11] for the IIl-case ). In [3] Ching constructed a lit-factor M2.M 3, the free product of a 2 x 2 matrix algebra by a 3 x 3 matrix algebra, without non-trivial central sequence, and at present, we are not able to find an outer automorphism for it. For this factor M 2 . M 3 there remains the possibility that Int(M2.M3)=Aut(M2*M3). Takesaki [12] has shown, however, that a factor of type III always has an outer automorphism. Hence, it is of interest to see a type III version of the factor M 2 . M 3. The purpose of this paper is to construct such factors of type III, and classify them accordingly.We first generalize the crossed product ofa von Neumann algebra by a discrete group of automorphisms to the crossed product of avon Neumann algebra by a yon Neumann algebra with an ortho-unitary basis. We then construct a oneparameter family {Ma} of factors of type III using crossed products of hyperfinite factors of type III by M2*M 3. {M~} is shown to be a family of factors of type III without non-trivial central sequences. Finally, we prove that M~ is not isomorphic to M~, for 2 . 2 ' . The proof rely on a formula in Connes [51 but it is different from Conne's proof of the pairwise non-isomorphism of {Pa} as crossed products (Theorem 3.6.5 [5]) in the following respects: The "ratio set" r(G) of a transformation group introduced by Krieger [8] (based on the asymptotic ratio set of Araki and Woods) is not used here, and hence our approach is more direct in certain sense.