http://nova.newcastle.edu.au/vital/access/services/Feed ${session.getAttribute("locale")} 5 http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:9929 We consider Exel’s new construction of a crossed product of a C*-algebra A by an endomorphism α. We prove that this crossed product is universal for an appropriate family of covariant representations, and we show that it can be realised as a relative Cuntz–Pimsner algbera. We describe a necessary and sufficient condition for the canonical map from A into the crossed product to be injective, and present several examples to demonstrate the scope of this result. We also prove a gauge-invariant uniqueness theorem for the crossed product. 2012-02-08T22:20:28.398Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:9935 We show that important structural properties of C*-algebras and the multiplicity numbers of representations are preserved under Morita equivalence. 2012-02-08T22:20:03.968Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:9761 Imprimitivity theorems provide a fundamental tool for studying the representation theory and structure of crossed-product C*-algebras. In this work, we show that the Imprimitivity Theorem for induced algebras, Green's Imprimitivity Theorem for actions of groups, and Mansfield's Imprimitivity Theorem for coactions of groups can all be viewed as natural equivalences between various crossed-product functors among certain equivariant categories. The categories involved have C*-algebras with actions or coactions (or both) of a fixed locally compact group G as their objects, and equivariant equivalence classes of right-Hilbert bimodules as their morphisms. Composition is given by the balanced tensor product of bimodules. The functors involved arise from taking crossed products; restricting, inflating, and decomposing actions and coactions; inducing actions; and various combinations of these. Several applications of this categorical approach are also presented, including some intriguing relationships between the Green and Mansfield bimodules, and between restriction and induction of representations. 2012-01-10T02:20:05.510Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:4303 Research Doctorate - Doctor of Philosophy (PhD) 2011-12-19T05:20:02.177Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:8055 We study Hecke algebras of groups acting on trees with respect to geometrically defined subgroups. In particular, we consider Hecke algebras of groups of automorphisms of locally finite trees with respect to vertex and edge stabilizers and the stabilizer of an end relative to a vertex stabilizer, assuming that the actions are sufficiently transitive. We focus on identifying the structure of the resulting Hecke algebras, give explicit multiplication tables of the canonical generators and determine whether the Hecke algebra has a universal C*-completion. The paper unifies algebraic and analytic approaches by focusing on the common geometric thread. The results have implications for the general theory of totally disconnected locally compact groups. 2011-07-04T06:30:06.487Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:3803 Graph algebras are a family of operator algebras which are associated to directed graphs. These algebras have an attractive structure theory in which algebraic properties of the algebra are related to the behaviour of paths in the underlying graph. In the past few years there has been a great deal of activity in this area, and graph algebras have cropped up in a surprising variety of situations, including non-abelian duality, non-commutative geometry, and the classification of simple C-algebras. 2010-07-12T02:21:21.852Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:6439 We consider a class of proper actions of locally compact groups on imprimitivity bimodules over C*-algebras which behave like the proper actions on C*-algebras introduced by Rieffel in 1988. We prove that every such action gives rise to a Morita equivalence between a crossed product and a generalized fixed-point algebra, and in doing so make several innovations which improve the applicability of Rieffel’s theory. We then show how our construction can be used to obtain canonical tensor-product decompositions of important Morita equivalences. Our results show, for example, that the different proofs of the symmetric imprimitivity theorem for actions on graph algebras yield isomorphic equivalences, and this gives new information about the amenability of actions on graph algebras. 2010-06-10T03:30:03.276Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:1853 We classify the gauge-invariant ideals in the C*-algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant primitave ideals in terms of the structural properties of the graph, and describe the K-theory of the C*-algebras of arbitrary infinite graphs. 2010-04-27T06:35:34.175Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:1837 For an arbitrary countable directed graph E we show that the only possible values of the stable rank of the associated Cuntz-Krieger algebra C*(E) are 1, 2 or ∞. Explicit criteria for each of these three cases are given. We characterize graph algebras of type I, and graph algebras which are inductive limits of C*-algebras of type I. We also show that a gauge-invariant ideal of a graph algebra is itself isomorphic to a graph algebra. 2010-04-27T06:35:14.980Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:1144 We give necessary and sufficient conditions for simplicity of Cuntz-Krieger algebras corresponding to infinite 0–1 matrices and of C*-algebras corresponding to countable directed graphs. We show that simple algebras within these two classes are either purely infinite or AF. 2010-04-27T06:05:34.753Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:4492 For a finitely aligned k-graph Λ with X a set of vertices in Λ, we define a universal C*-algebra called C* (Λ, X) generated by partial isometries. We show that C* (Λ, X) is isomorphic to the corner PXC*(Λ) PX, where PX is the sum of vertex projections in X. We then prove a version of the Gauge Invariant Uniqueness theorem for C*(Λ, X) and then use the theorem to prove various results involving fullness, simplicity and Morita equivalence as well as results relating to application in symbolic dynamics. 2010-04-27T04:58:06.134Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:4294 Let T₁ and T₂ be homogeneous trees of even degree ≥ 4. A BM group Γ is a torsion-free discrete subgroup of Aut(T₁)×Aut(T₂) which acts freely and transitively on the vertex set of T₁×T₂. This article studies dynamical systems associated with BM groups. A higher rank Cuntz-Krieger algebra A(Γ) is associated both with a 2-dimensional tiling system and with a boundary action of a BM group Γ. An explicit expression is given for the K-theory of A(Γ). In particular K₀=K₁. A complete enumeration of possible BM groups Γ is given for a product homogeneous trees of degree 4, and the K-groups are computed. 2010-04-27T04:57:07.805Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:5590 Noncommutative analogues of n-dimensional balls are defined by repeated application of the quantum double suspension to the classical low-dimensional spaces. In the ‘even-dimensional’ case they correspond to the twisted canonical commutation relations of Pusz and Woronowicz. Then quantum spheres are constructed as double manifolds of noncommutative balls. Both C*-algebras and polynomial algebras of the objects in question are defined and analysed, and their relations with previously known examples are presented. Our construction generalizes that of Hajac, Matthes, and Szymaski for ‘dimension 2’, and leads to a new class of quantum spheres (already on the C*-algebra level) in all ‘even dimensions’. 2010-04-27T04:39:22.900Z ]]>