Isomorphisms of Banach algebras associated with locally compact groups
| dc.contributor.author | Safoura, Zaffar Jafar Zadeh | |
| dc.contributor.examiningcommittee | Zhang,Yong (Mathematics) Wang, Xikui (Statistics) Forest, Brian (University of Waterloo) | en_US |
| dc.contributor.supervisor | Ghahramani, Fereidoun (Mathematics) Stokke, Ross (Mathematics) | en_US |
| dc.date.accessioned | 2015-11-16T18:10:56Z | |
| dc.date.available | 2015-11-16T18:10:56Z | |
| dc.date.issued | 2015 | |
| dc.degree.discipline | Mathematics | en_US |
| dc.degree.level | Doctor of Philosophy (Ph.D.) | en_US |
| dc.description.abstract | 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. | en_US |
| dc.description.note | February 2016 | en_US |
| dc.identifier.uri | http://hdl.handle.net/1993/30932 | |
| dc.language.iso | eng | en_US |
| dc.rights | open access | en_US |
| dc.subject | Locally compact group | en_US |
| dc.subject | LUC-compactification | en_US |
| dc.subject | Isometric isomorphism | en_US |
| dc.subject | Beurling algebras | en_US |
| dc.subject | Arens product | en_US |
| dc.subject | Bipositive isomorphism | en_US |
| dc.subject | Topological group isomorphism | en_US |
| dc.subject | Algebra isomorphism | en_US |
| dc.title | Isomorphisms of Banach algebras associated with locally compact groups | en_US |
| dc.type | doctoral thesis | en_US |