Positive Definite Kernels on the Complex Hilbert Sphere.
The Q-matrix of a connected graph = (V,E) is , where ∂(x,y) is the graph distance. Let q() be the range of q ∈ (-1,1) for which the Q-matrix is strictly positive. We obtain a sufficient condition for the equality q(̃) = q() where ̃ is an extension of a finite graph by joining a square. Some concrete examples are discussed.
We show that each group in a class of finitely generated groups introduced in [2] and [3] has Kazhdan’s property (T), and calculate the exact Kazhdan constant of with respect to its natural set of generators. These are the first infinite groups shown to have property (T) without making essential use of the theory of representations of linear groups, and the first infinite groups with property (T) for which the exact Kazhdan constant has been calculated. These groups therefore provide answers...
It is proved that every real metrizable locally convex space which is not nuclear contains a closed additive subgroup K such that the quotient group G = (span K)/K admits a non-trivial continuous positive definite function, but no non-trivial continuous character. Consequently, G cannot satisfy any form of the Bochner theorem.
On considère l’espace où et sont deux fonctions définies-négatives, réelles et continues sur . On étudie la possibilité d’approcher, au sens de la norme de , tout élément de par des combinaisons linéaires d’éléments de qui sont transformés de Fourier de mesures positives de support inclus dans le spectre de . Des méthodes de théorie du potentiel permettent de donner une réponse positive (sous certaines hypothèses additionnelles). On obtient ainsi des généralisations, au cas de ,...
Some relationships between representations of a hypergroup X, its algebras, and positive definite functions on X are studied. Also, various types of convergence of positive definite functions on X are discussed.
Semiperfect semigroups are abelian involution semigroups on which every positive semidefinite function admits a disintegration as an integral of hermitian multiplicative functions. Famous early instances are the group on integers (Herglotz Theorem) and the semigroup of nonnegative integers (Hamburger's Theorem). In the present paper, semiperfect semigroups are characterized within a certain class of semigroups. The paper ends with a necessary condition for the semiperfectness of a finitely generated...