Sections induced from weakly sequentially complete spaces
A measure is called -improving if it acts by convolution as a bounded operator from to L² for some q < 2. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are -improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be -improving.
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...