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Let G be a locally compact group, w be a continuous weight function on G, and L^1(G,w) be the corresponding Beurling algebra. In this thesis, we study weak amenability of L^1(G,w) and of its centre ZL^1(G,w) for non-commutative locally compact groups G. We first give examples to show that the condition that characterizes weak amenability of L^1(G,w) for commutative groups G is no longer sufficient for the non-commutative case. However, we prove that this condition remains necessary for all [IN] groups G. We also provide a necessary condition for weak amenability of L^1(G,w) of a different nature, which, among other things, allows us to obtain a number of significant results on weak amenability of l^1(F_2,w) and l^1((ax+b),w). We then study the relation between weak amenability of the algebra L^1(G,w) on a locally compact group G and the algebra L^1(G/H,^w) on the quotient group G/H of G over a closed normal subgroup H with an appropriate weight ^w induced from w. We give an example showing that L^1(G,w) may not be weakly amenable even if both L^1(G/H,^w) and L^1(H,w|_H) are weakly amenable. On the other hand, by means of constructing a generalized Bruhat function on G, we establish a sufficient condition under which weak amenability of L^1(G,w) implies that of L^1(G/H,^w). In particular, with this approach, we prove that weak amenability of the tensor product of L^1(G_1,w_1) and L^1(G_2,w_2) implies weak amenability of both Beurling algebras L^1(G_1,w_1) and L^1(G_2,w_2), provided the weights w_1, w_2 are bounded away from zero. However, given a general weight on the direct product G of G_1 and G_2, weak amenability of L^1(G,w) usually does not imply that of L^1(G_1,w|_{G_1}), even if both G_1, G_2 are commutative. We provide an example to illustrate this. While studying the centres ZL^1(G,w) of L^1(G,w), we characterize weak amenability of ZL^1(G,w) for connected [SIN] groups G, establish a necessary condition for weak amenability of ZL^1(G,w) in the case when G is an [FC] group, and give a sufficient condition for the case when G is an [FD] group. In particular, we obtain some positive results on weak amenability of ZL^1(G,w) for a compactly generated [FC] group G with a polynomial weight w. Finally, we briefly discuss the derivation problem for weighted group algebras and present a partial solution to it.

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

Thesis (Ph.D.)--York University, [2004]. Graduate Programme in [Mathematics]. / Typescript. Includes bibliographical references (leaves 163-166). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ99158

We show that any Borel action on a standard Borel space of a group which is topologically isomorphic to the sum of a countable abelian group with a countable sum of lines and circles induces an orbit equivalence relation which is hypersmooth. We also show that any Borel action of a second countable locally compact abelian group on a standard Borel space induces an orbit equivalence relation which is essentially hyperfinite, generalizing a result of Gao and Jackson for the countable abelian groups.

The main theme of this thesis is to study the isometric algebra isomorphisms and the bipositive algebra isomorphisms between various Banach algebras associated with locally compact groups. Let $LUC(G)$ denote the $C^*$-algebra of left uniformly continuous functions with the uniform norm and let $C_0(G)^{\perp}$ denote the annihilator of $C_0(G)$ in $LUC(G)^*$. In Chapter 2 of this thesis, among other results, we show that if $G$ is a locally compact group and $H$ is a discrete group then whenever there exists a weak-star continuous isometric isomorphism between $C_0(G)^{\perp}$ and $C_0(H)^{\perp}$, $G$ is isomorphic to $H$ as a topological group. In particular, when $H$ is discrete $C_0(H)^{\perp}$ determines $H$ within the class of locally compact topological groups. In Chapter 3 of this thesis, we show that if $M(G,\omega_1)$ (the weighted measure algebra on $G$) is isometrically algebra isomorphic to $M(H,\omega_2)$, then the underlying weighted groups are isomorphic, i.e. there exists an isomorphism of topological groups $\phi:G\to H$ such that $\small{\displaystyle{\frac{\omega_1}{\omega_2\circ\phi}}}$ is multiplicative. Similarly, we show that any weighted locally compact group $(G,\omega)$ is completely determined by its Beurling group algebra $L^1(G,\omega)$, $LUC(G,\omega^{-1})^*$ and $L^1(G,\omega)^{**}$, when the two last algebras are equipped with an Arens product. Here, $LUC(G,\omega^{-1})$ is the weighted analogue of $LUC(G)$, for weighted locally compact groups. In Chapter 4 of this thesis, we show that the order structure combined with the algebra structure of each of the Banach algebras $L^1(G,\omega)$, $M(G,\omega)$, $LUC(G,\omega^{-1})^*$ and $L^1(G,\omega)^{**}$ completely determines the underlying topological group structure together with a constraint on the weight. In particular, we obtain new proofs for a previously known result of Kawada and results of Farhadi as special cases of our results. Finally, we provide an example of a bipositive algebra isomorphism between Beurling measure algebras that is not an isometry. We conclude this thesis with a selective list of open problems. / February 2016

Indiana University-Purdue University Indianapolis (IUPUI) / In 1970, Serge Novikov made a statement which is now called, "The Novikov Conjecture" and is considered to be one of the major open problems in topology. This statement was motivated by the endeavor to understand manifolds of arbitrary dimensions by relating the surgery map with the homology of the fundamental group of the manifold, which becomes diffi cult for manifolds of dimension greater than two. The Novikov Conjecture is interesting because it comes up in problems in many different branches of mathematics like algebra, analysis, K-theory, differential geometry, operator algebras and representation theory. Yu later proved the Novikov Conjecture holds for all closed manifolds with discrete fundamental groups that are coarsely embeddable into a Hilbert space. The class of groups that are uniformly embeddable into Hilbert Spaces includes groups of Property A which were introduced by Yu. In fact, Property A is generally a property of metric spaces and is stable under quasi-isometry. In this thesis, a new version of Yu's Property A in the case of locally compact groups is introduced. This new notion of Property A coincides with Yu's Property A in the case of discrete groups, but is different in the case of general locally compact groups. In particular, Gromov's locally compact hyperbolic groups is of Property A.

In the first part of this thesis, suitable function spaces for analysis of partial differ- ential equations in unbounded domains are introduced and studied. The results are then applied in the second part on semilinear wave equation in Rd with non- linear source term and nonlinear damping. The source term is supposed to be bounded by a polynomial function with a subcritical growth. The damping term is strictly monotone and satisfying a polynomial-like growth condition. Global existence is proved using finite speed of propagation. Dissipativity in locally uni- form spaces and the existence of a locally compact attractor are then obtained after additional conditions imposed on the damping term.

For a locally compact group G, the papers [13] and [7] have many results about G-invariant subsets of G-modules, and the relationship between G-module maps, L1(G)-module maps and M(G)-module maps. In both papers, the results were given for one specific module action. In this thesis we extended many of their results to arbitrary Banach G-modules. In addition, we give detailed proofs of most of the results found in the first section of the paper [21].