Let X be a locally compact Hausdorff space along with n proper continuous maps σ = (σ1 , · · · , σn ). Then the pair (X, σ) is called a dynamical system. To each system one can associate a universal operator algebra called the tensor algebra A(X, σ). The central question in this theory is whether these algebras characterize dynamical systems up to some form of natural conjugacy. In the n = 1 case, when there is only one self-map, we will show how this question has been completely determined. For n ≥ 2, isomorphism of two tensor algebras implies that the two dynamical systems are piecewise conjugate. The converse was only established for n = 2 and 3. We introduce a new construction of the unitary group U (n) that allows us to prove the algebraic characterization question in n = 2, 3 and 4 as well as translating this conjecture into a conjecture purely about the structure of the unitary group.

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.

Let X be a locally compact Hausdorff space along with n proper continuous maps σ = (σ1 , · · · , σn ). Then the pair (X, σ) is called a dynamical system. To each system one can associate a universal operator algebra called the tensor algebra A(X, σ). The central question in this theory is whether these algebras characterize dynamical systems up to some form of natural conjugacy. In the n = 1 case, when there is only one self-map, we will show how this question has been completely determined. For n ≥ 2, isomorphism of two tensor algebras implies that the two dynamical systems are piecewise conjugate. The converse was only established for n = 2 and 3. We introduce a new construction of the unitary group U (n) that allows us to prove the algebraic characterization question in n = 2, 3 and 4 as well as translating this conjecture into a conjecture purely about the structure of the unitary group.

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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Thesis (M.S.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains vi, 41 p. : ill. (some col.). Vita. Includes abstract. Includes bibliographical references (p. 40-41).

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A well-known theorem of Blackadar and Handelman states that every unital stably finite C*-algebra has a bounded quasitrace. Rather strong generalizations of stable finiteness to the non-unital case can be obtained by either requiring the multiplier algebra to be stably finite, or alternatively requiring it to be at least stably not properly infinite. My thesis deals with the question whether the Blackadar-Handelman result can be extended to the non-unital case with respect to these generalizations of stable finiteness. For suitably well-behaved C*-algebras there is a positive result, but none of the non-unital versions holds in full generality. Two examples of C*-algebras are constructed. The first one is a non-unital, stably commutative C*-algebra A that contradicts the weakest possible generalization of the Blackadar-Handelman theorem: The multiplier algebras of all matrix algebras over A are finite, while A has no bounded quasitrace. The second example is a non-unital, simple C*-algebra B that is stably non-stable, i.e. no matrix algebra over B is a stable C*-algebra. In fact, the multiplier algebras over all matrix algebras of this C*-algebra are not properly infinite. Moreover, the C*-algebra B has no bounded quasitrace and therefore gives a simple counterexample to a possible generalization of the Blackadar-Handelman theorem.

A well-known theorem of Blackadar and Handelman states that every unital stably finite C*-algebra has a bounded quasitrace. Rather strong generalizations of stable finiteness to the non-unital case can be obtained by either requiring the multiplier algebra to be stably finite, or alternatively requiring it to be at least stably not properly infinite. My thesis deals with the question whether the Blackadar-Handelman result can be extended to the non-unital case with respect to these generalizations of stable finiteness. For suitably well-behaved C*-algebras there is a positive result, but none of the non-unital versions holds in full generality. Two examples of C*-algebras are constructed. The first one is a non-unital, stably commutative C*-algebra A that contradicts the weakest possible generalization of the Blackadar-Handelman theorem: The multiplier algebras of all matrix algebras over A are finite, while A has no bounded quasitrace. The second example is a non-unital, simple C*-algebra B that is stably non-stable, i.e. no matrix algebra over B is a stable C*-algebra. In fact, the multiplier algebras over all matrix algebras of this C*-algebra are not properly infinite. Moreover, the C*-algebra B has no bounded quasitrace and therefore gives a simple counterexample to a possible generalization of the Blackadar-Handelman theorem.

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.