The following thesis is divided into two main chapters. In Chapter 2 we study isometric representations of product systems of correspondences over the semigroup 𝐍ᵏ which are minimal dilations of finite dimensional, fully coisometric representations. We show the existence of a unique minimal cyclic coinvariant subspace for all such representations. The compression of the representation to this subspace is shown to be a complete unitary invariant. For a certain class of graph algebras the nonself-adjoint WOT-closed algebra generated by these representations is shown to contain the projection onto the minimal cyclic coinvariant subspace. This class includes free semigroup algebras. This result extends to a class of higher-rank graph algebras which includes higher-rank graphs with a single vertex. In chapter 3 we move onto semicrossed product algebras. Let 𝒮 be the semigroup 𝒮=Σ𝒮ᵢ, where 𝒮ᵢ is a countable subsemigroup of the additive semigroup 𝐑₊ containing 0. We consider representations of 𝒮 as contractions {Tᵣ }ᵣ on a Hilbert space with the Nica-covariance property: Tᵣ*Tᵤ=TᵤTᵣ* whenever t^s=0. We show that all such representations have a unique minimal isometric Nica-covariant dilation. This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant representations of the action of 𝒮 on an operator algebra 𝒜 by completely contractive endomorphisms. We conclude by calculating the C*-envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis).

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This dissertation is devoted to several questions that arise in operator algebra theory. In the first part of the work we study the dilations of homomorphisms of subalgebras to the algebras that contain them. We consider the question whether a contractive homomorphism of a logmodular algebra into B(H) is completely contractive, where B(H) denotes the algebra of all bounded operators on a Hilbert space H. We show that every logmodular subalgebra of Mn(C) is unitary equivalent to an algebra of block upper triangular matrices, which was conjectured by V. Paulsen and M. Raghupathi. In particular, this shows that every unital contractive representation of a logmodular subalgebra of Mn(C) is automatically completely contractive. In the second part of the dissertation we investigate certain matrices composed of mixed, second?order moments of unitaries. The unitaries are taken from C??algebras with moments taken with respect to traces, or, alternatively, from matrix algebras with the usual trace. These sets are of interest in light of a theorem of E. Kirchberg about Connes' embedding problem and provide a new approach to it. Finally, we give a modification of I. Klep and M. Schweighofer?s algebraic reformulation of Connes' embedding problem by considering the ?-algebra of the countably generated free group. This allows us to consider only quadratic polynomials in unitary generators instead of arbitrary polynomials in self-adjoint generators.

We construct a new operation among representations of the symmetric group that interpolates between the classical internal and external products, which are defined in terms of tensor product and induction of representations. Following Malvenuto and Reutenauer, we pass from symmetric functions to non-commutative symmetric functions and from there to the algebra of permutations in order to relate the internal and external products to the composition and convolution of linear endomorphisms of the tensor algebra. The new product we construct corresponds to the Heisenberg product of endomorphisms of the tensor algebra. For symmetric functions, the Heisenberg product is given by a construction which combines induction and restriction of representations. For non-commutative symmetric functions, the structure constants of the Heisenberg product are given by an explicit combinatorial rule which extends a well-known result of Garsia, Remmel, Reutenauer, and Solomon for the descent algebra. We describe the dual operation among quasi-symmetric functions in terms of alphabets.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2008. / Title from title screen (site viewed Sept. 18, 2008). PDF text: v, 109 p. : ill. ; 459 K. UMI publication number: AAT 3303652. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 80-85). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--University of Missouri-Columbia, 1998. / Typescript. Vita. Includes bibliographical references (leaves 244-246). Also available on the Internet.

Thesis (Ph. D.)--University of Oregon, 2001. / Typescript. Includes vita and abstract. Includes bibliographical references (leaf 49). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

Thesis (Ph. D.)--University of Missouri-Columbia, 1998. / Typescript. Vita. Includes bibliographical references (leaves 244-246). Also available on the Internet.