A continuous time quantum walk consists of a particle moving among the vertices of a graph G. Its movement is governed by the structure of the graph. More formally, the adjacency matrix A is the Hamiltonian that determines the movement of our particle. Quantum walks have found a number of algorithmic applications, including unstructured search, element distinctness and Boolean formula evaluation. We will examine the properties of periodicity and state transfer. In particular, we will prove a result of the author along with Godsil, Kirkland and Severini, which states that pretty good state transfer occurs in a path of length n if and only if the n+1 is a power of two, a prime, or twice a prime. We will then examine the property of strong cospectrality, a necessary condition for pretty good state transfer from u to v. We will then consider quantum walks involving more than one particle. In addition to moving around the graph, these particles interact when they encounter one another. Varying the nature of the interaction term gives rise to a range of different behaviours. We will introduce two graph invariants, one using a continuous-time multi-particle quantum walk, and the other using a discrete-time quantum walk. Using cellular algebras, we will prove several results which characterize the strength of these two graph invariants. Let A be an association scheme of n × n matrices. Then, any element of A can act on the space of n × n matrices by left multiplication, right multiplication, and Schur multiplication. The set containing these three linear mappings for all elements of A generates an algebra. This is an example of a Jaeger algebra. Although these algebras were initially developed by Francois Jaeger in the context of spin models and knot invariants, they prove to be useful in describing multi-particle walks as well. We will focus on triply-regular association schemes, proving several new results regarding the representation of their Jaeger algebras. As an example, we present the simple modules of a Jaeger algebra for the 4-cube.

This thesis consists of studies in the separate fields of operator algebras and non-associative algebras. Two natural operator algebra structures, A ⊗_max B and A ⊗_min B, exist on the tensor product of two given unital operator algebras A and B. Because of the different properties enjoyed by the two tensor products in connection to dilation theory, it is of interest to know when they coincide (completely isometrically). Motivated by earlier work due to Paulsen and Power, we provide conditions relating an operator algebra B and another family {C_i}_i of operator algebras under which, for any operator algebra A, the equality A ⊗_max B = A ⊗_min B either implies, or is implied by, the equalities A ⊗_max C_i = A ⊗_min C_i for every i. These results can be applied to the setting of a discrete group G pre-ordered by a subsemigroup G⁺, where B ⊆ C*_r(G) is the subalgebra of the reduced group C*-algebra of G generated by G⁺, and C_i = A(Q_i) are digraph algebras defined by considering certain pre-ordered subsets Q_i of G. The 16-dimensional algebra A₄ of real sedenions is obtained by applying the Cayley-Dickson doubling process to the real division algebra of octonions. The classification of subalgebras of A₄ up to conjugacy (i.e. by the action of the automorphism group of A₄) was completed in a previous investigation, except for the collection of those subalgebras which are isomorphic to the quaternions. We present a classification of quaternion subalgebras up to conjugacy.

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Includes bibliographical references / 123, [10] leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1984

Bibliography: leaves 143-146 / vii, 147 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Mathematical Physics, 1986

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Research Doctorate - Doctor of Philosophy (PhD) / (**Note: this abstract is a plain text version of the author's abstract, the original of which contains characters and symbols which cannot be accurately represented in this format. The properly formatted abstract can be viewed in the Abstract and Thesis files above.**) Let (G,G+) be a quasi-lattice-ordered group with positive cone G+ Laca and Raeburn have shown that the universal C*-algebra C*(G,G+)introduced by Nica is a crossed product BG+ Xɑ G+ by a semigroup of endomorphisms. Subsequent research centered on totally ordered abelian groups. We generalize the results in [2], [3] and [5] to extend it to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H+ of G+ we introduce a closed ideal IH+ of the C*-algebra BG+. We construct an approximate identity for this ideal and show that IH+ is extendibly a-invariant. It follows that there is an isomorphism between C*-crossed products (BG+/IH+) XɑG+ and B(G/H)+ XβG+. This leads to one of our main results that B(G/H)+ XβG+ is realized as an induced C*-algebra IndG-H (B(G/H+ Xt(G/H)+). Then we use this result to show the existence of the following short exact sequence of C*-algebras 0-IH+ XɑG+ → BG+ XɑG+ → IndG-H (B(G/H+ Xt(G/H)+) → 0. This leads to show that the ideal IH+ XɑG+ is generated by {iBG+(1-1u):u∊H+} and therefore contained in the commutator ideal CG of the C*-algebra BG+ XɑG+. Moreover, we use our short exact sequence to study the primitive ideals of the C* algebra BG+ XɑG+ which is isomorphic to the Toeplitz albebra T(G) of G.

Research Doctorate - Doctor of Philosophy (PhD) / (**Note: this abstract is a plain text version of the author's abstract, the original of which contains characters and symbols which cannot be accurately represented in this format. The properly formatted abstract can be viewed in the Abstract and Thesis files above.**) Let (G,G+) be a quasi-lattice-ordered group with positive cone G+ Laca and Raeburn have shown that the universal C*-algebra C*(G,G+)introduced by Nica is a crossed product BG+ Xɑ G+ by a semigroup of endomorphisms. Subsequent research centered on totally ordered abelian groups. We generalize the results in [2], [3] and [5] to extend it to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H+ of G+ we introduce a closed ideal IH+ of the C*-algebra BG+. We construct an approximate identity for this ideal and show that IH+ is extendibly a-invariant. It follows that there is an isomorphism between C*-crossed products (BG+/IH+) XɑG+ and B(G/H)+ XβG+. This leads to one of our main results that B(G/H)+ XβG+ is realized as an induced C*-algebra IndG-H (B(G/H+ Xt(G/H)+). Then we use this result to show the existence of the following short exact sequence of C*-algebras 0-IH+ XɑG+ → BG+ XɑG+ → IndG-H (B(G/H+ Xt(G/H)+) → 0. This leads to show that the ideal IH+ XɑG+ is generated by {iBG+(1-1u):u∊H+} and therefore contained in the commutator ideal CG of the C*-algebra BG+ XɑG+. Moreover, we use our short exact sequence to study the primitive ideals of the C* algebra BG+ XɑG+ which is isomorphic to the Toeplitz albebra T(G) of G.