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This Dissertation shows how the theory of C*-algebra of graphs relates to the theory of C*-algebras of sofic shifts. C*-algebras of sofic shifts are generalizations of Cuntz-Krieger algebras [8]. It is shown that if X is a sofic shift, then the C*-algebra of the sofic shift, Oₓ, is isomorphic to the C*-algebra of a directed graph E, C *(E). The graph E is shown to be the well known past set presentation of X constructed in [13]. We focus on the consequences of this result: In particular uniqueness of the generators of Oₓ, pure infiniteness, and ideal structure of the algebra Oₓ. We show the existence of an ideal I ⊂ Oₓ such that when we form the quotient, Oₓ/I, it is isomorphic to C*( F), and F is the left Krieger cover graph of X—a well known, canonical graph one can associate with a sofic shift. The dual cover, the right Krieger cover, can also be related to the structure of Oₓ, and we illustrate this relationship. Chapter 6 shows what happens when we label a directed graph E in a left resolving way. When the graph E and the labeling satisfy certain technical conditions, we can generate a C*-algebra Lₓ ⊂ C*(E), with Lₓ ≅ Oₓ provided that X an irreducible sofic shift. / Graduate

In this thesis we present two important closely related examples of Bratteli diagrams: the Pascal triangle and the Eulerian Bratteli diagram. The former is well-known and related to binomial coefficients. The latter, which is the main object of the thesis, is related to the Eulerian numbers. Bratteli diagrams were introduced in 1972 by Ola Bratteli in his study of approximately finite dimensional (AF) C*-algebras. In 1976, George Arthur Elliott associated to an AF C*-algebra or to a corresponding Bratteli diagram an ordered group, he called dimension group. In the first part of the thesis we study the space of infinite paths of the Eulerian diagram, and we realize it as a projective limit of finite permutation groups. In the second part, we study the state space of the dimension group associated to the Eulerian Bratteli diagram. It is a compact convex set and we describe its extremal points. Finally, we use this description to give a necessary and sufficient condition for an element of this dimension group to be positive.

In this work we examine C*-algebras of Toeplitz operators over the unit ball in ℂⁿ and the unit polydisc in ℂ². Toeplitz operators are interesting examples of non-normal operators that generate non-commutative C*-algebras. Moreover, in the nice cases (depending on the geometry of the domain) of algebras of Toeplitz operators we can recover some analogues of the spectral theorem up to compact operators. In this setting, we can capture the index of a Fredholm operator which is a fundamental numerical invariant in Operator Theory. / Master of Science

In the first half of the thesis a necessary and sufficient condition is given for a separable C*-algebra to be *-isomorphic to a maximal full algebra of cross-sections over a base-space such that the fibre algebras are primitive throughout a dense subset. The condition is that the relation of inseparability for pairs of points in the primitive ideal space should be an open equivalence relation. In the second half of the thesis a characterisation is given of those C*- algebras A for which each self-adjoint inner derivation D(α, A) satisfies ∥D(α, A)∥ = 2 inf {∥α-z∥ : z ∈Z(A), the centre of A}. This time the characterisation is that A should be quasicentral and the relation of inseparability for pairs of points in the primitive ideal space should be an equivalence relation. Those C*-algebras for which every inner derivation satisfies the equation are characterised in a similar way.

The Cuntz semigroup is an isomorphism invariant for C*-algebras consisting of a semigroup with a compatible (though not algebraic) ordering. Its construction is similar to that of the Murray-von Neumann semigroup (from which the ordered K_0-group arises by the Grothendieck construction), but using positive elements in place of projections. Both rich in structure and sensitive to subtleties of the C*-algebra, the Cuntz semigroup promises to be a useful tool in the classification program for nuclear C*-algebras. It has already delivered on this promise, particularly in the study of regularity properties and the classification of nonsimple C*-algebras. The first part of this thesis introduces the Cuntz semigroup, highlights structural properties, and outlines some applications. The main result of this thesis, however, contributes to the understanding of what the Cuntz semigroup looks like for particular examples of (nonsimple) C*-algebras. We consider separable C*-algebras given as the tensor product of a commutative C*-algebra C_0(X) with a simple, approximately subhomogeneous algebra A, under the regularity hypothesis that A is Z-stable. (The Z-stability hypothesis is needed even to describe of the Cuntz semigroup of A.) For these algebras, the Cuntz semigroup is described in terms of the Cuntz semigroup of A and the Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. This result is a marginal improvement over one proven by the author in [Tikuisis, A. "The Cuntz semigroup of continuous functions into certain simple C*-algebras." Internat. J. Math., to appear] (there, A is assumed to be unital), although improvements have been made to the techniques used. The second part of this thesis provides the basic theory of approximately subhomogeneous algebras, including the important computational concept of recursive subhomogeneous algebras. Theory to handle nonunital approximately subhomogeneous algebras is novel here. In the third part of this thesis lies the main result. The Cuntz semigroup computation is achieved by defining a Cuntz-equivalence invariant I(.) on the positive elements of the C*-algebra, picking out certain data from a positive element which obviously contribute to determining its Cuntz class. The proof of the main result has two parts: showing that the invariant I(.) is (order-)complete, and describing its range.

The Cuntz semigroup is an isomorphism invariant for C*-algebras consisting of a semigroup with a compatible (though not algebraic) ordering. Its construction is similar to that of the Murray-von Neumann semigroup (from which the ordered K_0-group arises by the Grothendieck construction), but using positive elements in place of projections. Both rich in structure and sensitive to subtleties of the C*-algebra, the Cuntz semigroup promises to be a useful tool in the classification program for nuclear C*-algebras. It has already delivered on this promise, particularly in the study of regularity properties and the classification of nonsimple C*-algebras. The first part of this thesis introduces the Cuntz semigroup, highlights structural properties, and outlines some applications. The main result of this thesis, however, contributes to the understanding of what the Cuntz semigroup looks like for particular examples of (nonsimple) C*-algebras. We consider separable C*-algebras given as the tensor product of a commutative C*-algebra C_0(X) with a simple, approximately subhomogeneous algebra A, under the regularity hypothesis that A is Z-stable. (The Z-stability hypothesis is needed even to describe of the Cuntz semigroup of A.) For these algebras, the Cuntz semigroup is described in terms of the Cuntz semigroup of A and the Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. This result is a marginal improvement over one proven by the author in [Tikuisis, A. "The Cuntz semigroup of continuous functions into certain simple C*-algebras." Internat. J. Math., to appear] (there, A is assumed to be unital), although improvements have been made to the techniques used. The second part of this thesis provides the basic theory of approximately subhomogeneous algebras, including the important computational concept of recursive subhomogeneous algebras. Theory to handle nonunital approximately subhomogeneous algebras is novel here. In the third part of this thesis lies the main result. The Cuntz semigroup computation is achieved by defining a Cuntz-equivalence invariant I(.) on the positive elements of the C*-algebra, picking out certain data from a positive element which obviously contribute to determining its Cuntz class. The proof of the main result has two parts: showing that the invariant I(.) is (order-)complete, and describing its range.

Thesis (Ph. D.)--University of Adelaide, Dept. of Mathematical Physics,1986. / Erratum (14 leaves) in pocket. Includes bibliographical references (leaves 124-128).

Thesis (Ph. D.)--University of Oregon, 2008. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 105-107). Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.