In this thesis we consider continuity and positivity properties of pseudo-differential operators in Gelfand-Shilov and Pilipović spaces, and their distribution spaces. We also investigate composition property of pseudo-differential operators with symbols in quasi-Banach modulation spaces. We prove that positive elements with respect to the twisted convolutions, possesing Gevrey regularity of certain order at origin, belong to the Gelfand-Shilov space of the same order. We apply this result to positive semi-definite pseudo-differential operators, as well as show that the strongest Gevrey irregularity of kernels to positive semi-definite operators appear at the diagonals. We also prove that any linear operator with kernel in a Pilipović or Gelfand-Shilov space can be factorized by two operators in the same class. We give links on numerical approximations for such compositions and apply these composition rules to deduce estimates of singular values and establish Schatten-von Neumann properties for such operators.   Furthermore, we derive sufficient and necessary conditions for continuity of the Weyl product with symbols in quasi-Banach modulation spaces.

Ce travail est au confluent de la théorie des algèbres d'opérateurs<br />et des probabilités non-commutatives. Nous étudions les propriétés<br />des algèbres de von Neumann, $\Gamma_{q}(H_{\R})$, engendrées par<br />des variables Gaussiennes non-commutatives et $q$-déformées. Ces<br />variables $q$-Gaussiennes sont des opérateurs agissant sur l'espace<br />de Fock $q$-déformé, où sont réalisées les relations de<br />$q$-commutations canoniques.<br /><br />Dans la première partie de ce mémoire, nous établissons des<br />inégalités à coefficients opérateurs de type Khintchine-$L^{\infty}$<br />pour les produits de Wick des algèbres $q$-Gaussiennes. Ces<br />inégalités étendent d'un côté les inégalités scalaires dues à<br />Haagerup dans le cas libre et d'un autre côté les inégalités à<br />coefficients opérateurs, pour les $q$-Gaussiens, dues à Bo\.zejko et<br />Speicher. A l'aide de ces inégalités nous en déduisons que les<br />algèbres $\Gamma_q(H_{\R})$ sont non injectives dès que<br />$\dim_{\R}(H_{\R})\ge 2$.<br /><br />La deuxième partie est dédiée à la construction d'un modèle<br />asymptotique matriciel pour les variables $q$-Gaussiennes.<br />L'existence d'un tel modèle nous permet de prouver que les algèbres<br />$\Gamma_{q}(H_{\R})$ sont QWEP.<br /><br />Chemin faisant, nous traitons également le cas $C^*-$algébrique et<br />étudions diverses généralisations des résultats précédents pour les<br />déformations par opérateur de Yang-Baxter et pour les déformations<br />$q$-Gaussiennes de type $I\!I\!I$.

We examine the amalgamated free product of hyperfinite von Neumann algebras. First we describe the amalgamated free product of hyperfinite von Neumann algebras over finite dimensional subalgebras. In this case the result is always the direct sum of a hyperfinite von Neumann algebra and a finite number of interpolated free group factors. We then show that this class is closed under this type of amalgamated free product. After that we allow amalgamation over possibly infinite dimensional multimatrix subalgebras. In this case the product of two hyperfinite von Neumann algebras is the direct sum of a hyperfinite von Neumann algebra and a countable direct sum of interpolated free group factors. As before, we show that this class is closed under amalgamated free products over multimatrix algebras.

No description available.

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.

We give a systematic study of the Hardy spaces of functions with values in the non-commutative Lp-spaces associated with a semifinite von Neumann algebra M. This is motivated by matrix valued harmonic analysis (operator weighted norm inequalities, operator Hilbert transform), as well as by the recent development of non-commutative martingale inequalities. Our non-commutative Hardy spaces are defined by non-commutative Lusin integral functions. It is proved in this dissertation that they are equivalent to those defined by the non-commutative Littlewood-Paley G-functions. We also study the Lp boundedness of operator valued dyadic paraproducts and prove that their Lq boundedness implies their Lp boundedness for all 1 < q < p < ∞.

No description available.

In this thesis a new solution concept is developed for n-player, nonzero sum games. The solution concept is based in reinterpreting the n-player nonzero sum game into 2-player zero sum games. The n-player nonzero sum game is first rewritten as an n + 1 player coalition game. The definition of zero sum payment is that one player pays the other what he gets in a given outcome (coalition of the n + 1 player game). Who pays whom depends on the coalition. More than one 2-player zero sum interpretation game always results from the procedure, and criteria are established to select one of the zero sum interpretation games. The solution concept defines results identical to the minimax concept when applied directly to zero sum 2-player games. When applied to 2-player prisoner’s dilemma games, the solution procedure assigns mixed strategies to the prisoners, thereby “resolving” the dilemma. The mixed strategies vary with the payoffs (up to a linear transformation). For prisoner’s dilemma matrices which have been used in large numbers of gaming experiments, the solution concept predicts dynamically, i.e., by play number, the “fraction of cooperative choices” for (approximately) the first 30 plays. In addition, the mixed strategy appears in a game between each subject (prisoner) and the n + 1st player (district attorney), suggesting that the subjects have been playing against the experimenter. Empirical evidence for this conclusion is given. A theorem is proved for n-player prisoner’s dilemma games. Game theory is reviewed to show the roots of this solution concept in the heuristic use of zero sum n-player games in the von Neumann and Morgenstern theory, and in rational decision making models, e.g., “games against Nature.” The empirical and formal difficulties of the equilibrium point solution concept for nonzero sum games are discussed. Detailed connections between game theory and cybernetics are discussed.

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