Free products of von Neumann algebras

“…A countable set O = { Ug}g~G of unitary operators in M containing the identity operator I is called an ortho-unitary basis for M provided (i) (Ug{tU,¢)=0 ifg=l=h (ii) for all geG, U*=cgUl for some UleO and a complex number c o of modulus 1, and for every pair g, heG, UgUh=cg,hUk for some UkeO and a complex number co, h of modulus 1, (iii) the set of all linear combinations of elements of O is strongly dense in M ( [3], § 3), As remarked in [3], G is a group under the multiplication g . h = k if UgUh=CO,hUk.…”

Section: Crossed Product Of Von Neumann Algebrasmentioning

confidence: 99%

“…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 .…”

mentioning

confidence: 91%

A family of type III factors without propertyL

Ching

Tsau

1976

Math. Ann.

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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.

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“…Let H be a complex, separable Hilbert space, and let M ⊆ B(H) be a factor of type II 1 . If S is a non-empty set, we say that a family of norm-closed subspaces {H i } i∈S of H is transitive relative to M if for each i ∈ S, the projection P i of H onto H i lies in M and only the scalar operators leave all of the H i invariant.…”

Section: Introductionmentioning

confidence: 99%

“…To shed light on this question we consider free families of projections. A family {P i } n i=1 of projections in a factor of type II 1 is free if each P i has trace 1 2 and the P i are free with respect to the trace (in the sense of Voiculescu, see [7] and [1]). We shall exhibit a free transitive family of twelve projections.…”

Section: Introductionmentioning

confidence: 99%

Transitive families of projections in factors of type $II_{1}$

Bannon

2004

Proc. Amer. Math. Soc.

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Abstract. We introduce a notion of transitive family of subspaces relative to a type II 1 factor, and hence a notion of transitive family of projections in such a factor. We show that whenever M is a factor of type II 1 and M is generated by two self-adjoint elements, then M ⊗ M 2 (C) contains a transitive family of 5 projections. Finally, we exhibit a free transitive family of 12 projections that generate a factor of type II 1 .

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Free products of von Neumann algebras

Cited by 57 publications

References 16 publications

On spectra and Brown's spectral measures of elements in free products of matrix algebras

On spectra and Brown's spectral measures of elements in free products of matrix algebras

A family of type III factors without propertyL

Transitive families of projections in factors of type $II_{1}$

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