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Maximal abelian subalgebras of von Neumann algebras Nielsen, Ole A.

Abstract

We are concerned with constructing examples of maximal abelian von Neumann subalgebras (MA subalgebras) in hyperfinite factors of type III. Our results will show that certain phenomena known to hold for the hyperfinite factor of type 11₁ also hold for type III factors. Let M and N be subalgebras of the factor α . We call M and N equivalent if M is the image of N by some automorphism of α . Let N(M) denote the subalgebra of α generated by all those unitary operators in α which induce automorphisms of M, and let N²(M), N³(M),... be defined in the obvious inductive fashion. Following J. Dixmier and S. Anastasio, we call a MA subalgebra M of α singular if N(M) = M, regular if N(M) = α, semi-regular if N(M) is a factor distinct from α, and m-semi-regular (m ≥ 2) if N(M),. . .N(m-1)(M) are not factors but N(m)(M) is a factor. The MA subalgebras of the hyperfinite 11₁ factor β have received much attention in the literature, in the papers of J. Dixmier, L. Pukanszky, Sister R. J. Tauer, and S. Anastasio. It is known that β contains a MA subalgebra of each type. Further, β contains pairwise inequivalent sequences of singular, semi-regular, 2-semi-regular, and 3-semi-regular MA subalgebras. The only hitherto known example of a MA subalgebra in a type III factor is regular. In 1956 Pukanszky gave a general method for constructing MA subalgebras in a class of (probably non-hyperfinite) type III factors. Because of an error in a calculation, the types of these subalgebras is not known. The main result of this thesis is the construction, in each of the uncountably many mutually non-isomorphic hyperfinite type III factors of R. Powers, of: (i) a semi-regular MA subalgebra (ii) two sequences of mutually inequivalent 2-semi-regular MA subalgebras 1 (iii) two sequences of mutually inequivalent 3-semi-regular MA subalgebras. Let α denote one of these type III factors and let β denote the hyperfinite 11₁ factor. Roughly speaking, whenever a non-singular MA subalgebra of β is constructed by means of group operator algebras, our method will produce a MA subalgebra of α of the same type. H. Araki and J. Woods have shown that α ⊗ β ≅ α, and it is therefore only necessary to construct MA subalgebras of α ⊗ β of the desired type. We obtain MA subalgebras of α ⊗ β by tensoring a MA subalgebra in α with one in β. In order to determine the type of such a MA subalgebra, we realize β as a constructible algebra and then regard α ⊗ β as a constructible algebra; this allows us to consider operators in α ⊗ β as functions from a group into an abelian von Neumann algebra. As a corollary to our calculations, we are able to construct mutually inequivalent sequences of 2-semi-regular and 3-semi-regular MA subalgebras of the hyperfinite 11₁ factor which differ from those of Anastasio.

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