Thesis (Ph. D.)--University of Adelaide, Dept. of Pure Mathematics, 1984. / Includes bibliographical references.

Thesis (Ph. D.)--University of California, Riverside, 2009. / Includes abstract. Includes bibliographical references (leaves 67-68). Issued in print and online. Available via ProQuest Digital Dissertations.

This thesis develops the theory of operator algebras from the perspective of abstract harmonic analysis, and in particular, the theory of von Neumann algebras. Results from operator algebras are applied to the study of spaces of coefficient functions of unitary representations of locally compact groups, and in particular, the Fourier algebra of a locally compact group. The final result, which requires most of the material developed in earlier sections, is that the group von Neumann algebra of a locally compact group is in standard form.

This thesis develops the theory of operator algebras from the perspective of abstract harmonic analysis, and in particular, the theory of von Neumann algebras. Results from operator algebras are applied to the study of spaces of coefficient functions of unitary representations of locally compact groups, and in particular, the Fourier algebra of a locally compact group. The final result, which requires most of the material developed in earlier sections, is that the group von Neumann algebra of a locally compact group is in standard form.

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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

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Cette thèse présente quelques résultats de la théorie des probabilités non commutatives, et traite en particulier des inégalités de martingales dans des algèbres de von Neumann et de leurs espaces de Hardy associés. La première partie démontre un analogue non commutatif de la décomposition de Davis faisant intervenir la fonction carrée. Les arguments classiques de temps d'arrêt ne sont plus valides dans ce cadre, et la preuve se base sur une approche duale. Le deuxième résultat important de cette partie détermine ainsi le dual de l'espace de Hardy conditionnel h_1(M). Ces résultats sont ensuite étendus au cas 1<p<2. La deuxième partie transfère une décomposition atomique pour les espaces de Hardy h_1(M) et H_1(M) aux martingales non commutatives. Des résultats d'interpolation entre les espaces h_p(M) et bmo(M) sont également établis, relativement aux méthodes complexe et réelle d'interpolation. Les deux premières parties concernent des filtrations discrètes. Dans la troisième partie, on introduit des espaces de Hardy de martingales non commutatives relativement à une filtration continue. Les analogues des inégalités de Burkholder/Gundy et de Burkholder/Rosenthal sont obtenues dans ce cadre. La dualité de Fefferman-Stein ainsi que la décomposition de Davis sont également transférées avec succès à cette situation. Les preuves se basent sur des techniques d'ultraproduit et de L_p-modules. Une discussion sur une décomposition impliquant des atomes algébriques permet d'obtenir les résultats d'interpolation attendus / This thesis presents some results of the theory of noncommutative probability. It deals in particular with martingale inequalities in von Neumann algebras, and their associated Hardy spaces. The first part proves a noncommutative analogue of the Davis decomposition, involving the square function. The usual arguments using stopping times in the commutative case are no longer valid in this setting, and the proof is based on a dual approach. The second main result of this part determines the dual of the conditioned Hardy space h_1(M). These results are then extended to the case 1<p<2. The second part proves that an atomic decomposition for the Hardy spaces h_1(M) and H_1(M) is valid for noncommutative martingales. Interpolation results between the spaces h_p(M) and bmo(M) are also established, with respect to both complex and real interpolations. The two first parts concern discrete filtrations. In the third part, we introduce Hardy spaces of noncommutative martingales with respect to a continuous filtration. The analogues of the Burkholder/Gundy and Burkholder/Rosenthal inequalities are obtained in this setting. The Fefferman-Stein duality and the Davis decomposition are also successfully transferred to this situation. The proofs are based on ultraproduct techniques and L_p-modules. A discussion about a decomposition involving algebraic atoms gives the expected interpolation results

The validity of the von-Neumann inequality for commuting $n$ - tuples of $3\times 3$ matrices remains open for $n\geq 3$. We give a partial answer to this question, which is used to obtain a necessary condition for the Carathéodory-Fejérinterpolation problem on the polydisc$\D^n. $ in the special case of $n=2$ (which follows from Ando's theorem as well), this necessary condition is made explicit. We discuss an alternative approach to the Carathéodory-Fejérinterpolation problem, in the special case of $n=2$, adapting a theorem of Korányi and Pukánzsky. As a consequence, a class of polynomials are isolated for which a complete solution to the Carathéodory-Fejér interpolation problem is easily obtained. Many of our results remain valid for any $n\in \mathbb N$, however the computations are somewhat cumbersome. Recall the well known inequality due to Varopoulos, namely, $\lim{n\to \infty}C_2(n)\leq 2 K^\C_G$, where $K^\C_G$ is the complex Grothendieck constant and \[C_2(n)=sup\{\|p(\boldsymbolT)\|:\|p\|_{\D^n,\infty}\leq 1, \|\boldsymbol T\|_{\infty} \leq 1\}.\] Here the supremum is taken over all complex polynomials $p$ in $n$ variables of degree at most $2$ and commuting $n$ - tuples$\boldsymbolT:=(T_1,\ldots,T_n)$ of contractions. We show that \[\lim_{n\to \infty} C_2 (n)\leq \frac{3\sqrt{3}}{4} K^\C_G\] obtaining a slight improvement in the inequality of Varopoulos. We also discuss several finite and infinite dimensional operator space structures on $\ell^1(n) $, $n>1. $