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Given a function (more generally, a measure) on a locally compact Abelian group, one can define the Toeplitz operators as certain integral transforms of functions on the dual group, where the kernel is the Fourier transform of the original function or measure. In the case of the unit circle, this corresponds to forming a matrix out of the Fourier coefficients in a particular way. We will study the asymptotic eigenvalue distributions of these Toeplitz operators.

Thesis (Ph.D.)--Kent State University, 2008. / Title from PDF t.p. (viewed Sept. 17, 2009). Advisor: Donald White. Keywords: Huppert's Conjecture; character degrees; nonabelian finite simple groups Includes bibliographical references (p. 103-105).

Thesis (Ph.D.)--Baylor University, 2007. / In abstract "s and z " are subscript. Includes bibliographical references (p. 54-55).

Thesis (Ph. D.)--West Virginia University, 1998. / Title from document title page. Document formatted into pages; contains vii, 97 p. : ill. Includes abstract. Includes bibliographical references (p. 93-96).

Thesis (Ph. D.)--University of Washington, 2005. / Vita. Includes bibliographical references (p. 84-85).

Thesis--Upsala.

The broad goal is to classify all fuzzy subgroups of a given type of finite group. P.S. Das introduced the ntion of level subgroups to characterize fuzzy subgroups of finite grouops. The notion of equivalence of fuzzy subgroups which is used in this thesis was first introduced by Murali and Makamba. We use this equivalence to charterise fuzzy subgroups of inite Abelian groups (p-groups in particular) for a specified prime p. We characterize some crisp subgroups of p-groups and investigate some cases on equi valent fuzzy subgroups.

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. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis carries out some of classical integration theory in the context of an operator algebra. The starting point is measure on the projections of an abelian von Neumann algebra. This yields an integral on the self-adjoint operators whose spectral projections lie in the algebra. For this integral a Radon-Nikodym theorem, as well as the usual convergence theorems is proved. The methods and results of this thesis generalize, to non-commutative von Neumann Algebras [2, 3, 5]. (1) J. Dixmier Les Algèbres d'Opérateurs dans l'Espace Hilbertien. Paris, 1957. (2) H.A. Dye The Radon-Nikodym theorem for finite rings of operators, Trans. Amer. Math. Soc, 72, 1952, 243-230. (3) F.J. Murray and J. von Neumann, On Rings of Operators, Ann. Math. 37, 1936, 116-229. (4) F. RIesz and B. v. Sz.-Nagy, Functional Analysis, New York, 1955. (5) I.E. Segal A non-commutative extension of abstract integration, Ann. of Math. (2) 57, 1953, 401-457. / Science, Faculty of / Mathematics, Department of / Graduate