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).

The following thesis is divided into two main parts. In the first part we study the problem of characterizing algebras of functions living on analytic varieties. Specifically, we consider the restrictions M_V of the multiplier algebra M of Drury-Arveson space to a holomorphic subvariety V of the unit ball as well as the algebras A_V of continuous multipliers under the same restriction. We find that M_V is completely isometrically isomorphic to cM_W if and only if W is the image of V under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. Furthermore, when V and W are homogeneous varieties then A_V is isometrically isomorphic to A_W if and only if the defining polynomial relations are the same up to a change of variables. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. In the continuous homogeneous case, two algebras are isomorphic if and only if they are similar. However, in the multiplier algebra case the problem is much harder and several examples will be given where no such characterization is possible. In the second part we study the triangular subalgebras of UHF algebras which provide new examples of algebras with the Dirichlet property and the Ando property. This in turn allows us to describe the semicrossed product by an isometric automorphism. We also study the isometric automorphism group of these algebras and prove that it decomposes into the semidirect product of an abelian group by a torsion free group. Various other structure results are proven as well.

Research Doctorate - Doctor of Philosophy (PhD) / There is a strong connection between directed graphs and the shifts of finite type which are an important family of dynamical systems. Higher-rank graphs (or k-graphs) and their C*-algebras were introduced by Kumjian and Pask to generalise directed graphs and their C*-algebras. Kumjian and Pask showed how higher-dimensional shifts of finite type can be associated to k-graphs, but did not discuss how one might associate k-graphs to k-dimensional shifts of finite type. In this thesis we construct a family of 2-graphs A arising from a certain type of algebraic two-dimensional shift of finite type studied by Schmidt, and analyse the structure of their C*-algebras. Graph algebras and k-graph algebras provide a rich source of examples for the classication of simple, purely infinite, nuclear C*-algebras. We give criteria which ensure that the C*-algebra C*(A) is simple, purely infinite, nuclear, and satisfies the hypotheses of the Kirchberg-Phillips Classification Theorem. We perform K-theory calculations for a wide range of our 2-graphs A using the Magma computational algebra system. The results of our calculations lead us to conjecture that the K-groups of C*(A) are finite cyclic groups of the same order. We are able to prove under mild hypotheses that the K-groups have the same order, but we have only numerical evidence to suggest that they are cyclic. In particular, we find several examples for which K1(C*(A)) is nonzero and has torsion, hence these are examples of 2-graph C*-algebras which do not arise as the C*-algebras of directed graphs. Finally, we consider a subfamily of 2-graphs with interesting combinatorial connections. We identify the nonsimple C*-algebras of these 2-graphs and calculate their K-theory.

Research Doctorate - Doctor of Philosophy (PhD) / Directed graphs are combinatorial objects used to model networks like fluid-flow systems in which the direction of movement through the network is important. In 1980, Enomoto and Watatani used finite directed graphs to provide an intuitive framework for the Cuntz-Krieger algebras introduced by Cuntz and Krieger earlier in the same year. The theory of the C*-algebras of directed graphs has since been extended to include infinite graphs, and there is an elegant relationship between connectivity and loops in a graph and the structure theory of the associated C*-algebra. Higher-rank graphs are a higher-dimensional analogue of directed graphs introduced by Kumjian and Pask in 2000 as a model for the higher-rank Cuntz-Krieger algebras introduced by Robertson and Steger in 1999. The theory of the Cuntz-Krieger algebras of higher-rank graphs is relatively new, and a number of questions which have been answered for directed graphs remain open in the higher-rank setting. In particular, for a large class of higher-rank graphs, the gauge-invariant ideal structure of the associated C*-algebra has not yet been identified. This thesis addresses the question of the gauge-invariant ideal structure of the Cuntz-Krieger algebras of higher-rank graphs. To do so, we introduce and analyse the collections of relative Cuntz-Krieger algebras associated to higher-rank graphs. The first two main results of the thesis are versions of the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem which apply to relative Cuntz-Krieger algebras. Using these theorems, we are able to achieve our main goal, producing a classification of the gauge-invariant ideals in the Cuntz-Krieger algebra of a higher-rank graph analogous to that developed for directed graphs by Bates, Hong, Raeburn and Szymañski in 2002. We also demonstrate that relative Cuntz-Krieger algebras associated to higher-rank graphs are always nuclear, and produce conditions on a higher-rank graph under which the associated Cuntz-Krieger algebra is simple and purely infinite.

Thesis (Dr. rer. nat.)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2002. / Includes bibliographical references (p. 136-140) and index.

In the classical theory of operators on a Banach space a beautiful interplay exists between Riesz and Fredholm theory, and the theory of traces and de¬terminants for operator ideals. In this thesis we obtain a complete Riesz de¬composition theorem for Riesz elements in a semi prime Banach algebra and on the other hand extend the existing theory of traces and determinants to a more general setting of Banach algebras. In order to obtain some of these results we use the notion of finite multiplicity of spectral points to give a characterization of the essential spec¬trum for elements in a Banach algebra. As an immediate corollary we obtain the well-known characterization of Riesz elements namely that their non-zero spectral points are isolated and of finite multiplicities. In the final chapter of the thesis we use Plemelj's type formulas to define a determinant on the ideal of finite rank elements and show that it extends continuously to the ideal of nuclear elements. / Thesis (PhD (Mathematics))--University of Pretoria, 2006. / Mathematics and Applied Mathematics / unrestricted

The goal of this thesis is to examine the actions of finite symmetry groups on aperiodic tilings. To an aperiodic tiling with finite local complexity arising from a primitive substitution rule one can associate a metric space, transformation groupoids, and C*-algebras. Finite symmetry groups of the tiling act on each of these objects and we investigate appropriate constructions on each, namely the orbit space, semidirect product groupoids, and crossed product C*-algebras respectively. Of particular interest are the crossed product C*-algebras; we derive important structure results about them and compute their K-theory.

In this thesis, we report on two projects applying representation theoretic techniques to solve enumerative and geometric problems, which were carried out by the author during his pursuit of Ph.D. at Columbia. We first study the relative Gromov-Witten theory on T*P¹ x P¹ and show that certain equivariant limits give relative invariants on P¹ x P¹. By formulating the quantum multiplications on Hilb(T*P¹) computed by Davesh Maulik and Alexei Oblomkov as vertex operators and computing the product expansion, we demonstrate how to get the insertion operator computed by Yaim Cooper and Rahul Pandharipande in the equivariant limits. Brenti proves a non-recursive formula for the Kazhdan-Lusztig polynomials of Coxeter groups by combinatorial methods. In the case of the Weyl group of a split group over a finite field, a geometric interpretation is given by Sophie Morel via weight truncation of perverse sheaves. With suitable modifications of Morel's proof, we generalize the geometric interpretation to the case of finite and affine partial flag varieties. We demonstrate the result with essentially new examples using sl₃ and sl₄..

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