The present note is devoted to study a functional equation on a locally compact Abelian group group which appears in characterization of probability distributions by the optimality of an estimate. Let X be a second countable locally compact Abelian group, Y = X be its character group, (x; y) be the value of a character y 2 Y at an element x 2 X. Denote by M(X) the convolution semigroup of proba...

For any point xEX, we denote by HXEG the isotropy subgroup of G at xEX. The group G will be a compact Lie group. The space X is assumed to be locally compact, locally connected, finite dimensional and separable metric. We shall use the Alexander-Wallace-Spanier cohomology groups. A subscript c will denote cohomology with compact supports. The coefficient ring J will denote either the integers Z...

If U and V are commuting unitary representations of locally compact abelian groups S and T, new representations of S (perturbations of U) can be obtained from composition with images of U in V. If most of these representations are equivalent to U, U is said to be V stable. We investigate conditions which, together with stability, ensure that U has (uniform) Haar spectrum. The principal applicat...

This paper is centered on an extremely general problem: Problem. Is it consistent (perhaps modulo large cardinals) that a locally compact space X must be the union of countably many ω-bounded subspaces if every closed discrete subspace of X is countable [in other words, if X is ω1-compact]? A space is ω-bounded if every countable subset has compact closure. This is a strengthening of countable ...

We extend the classical Uncertainty Principle to the context of Gelfand pairs. The Gelfand pair setting includes riemannian symmetric spaces, compact topological groups, and locally compact abelian groups. If the locally compact abelian group is Rn we recover a sharp form of the classical Heisenberg uncertainty principle. Section

A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory.