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

We investigate the amenability of skew field extensions of the complex numbers. We prove that all skew fields of finite Gelfand-Kirillov transcendence degree are amenable. However there are both amenable and non-amenable finitely generated skew fields of infinite Gelfand-Kirillov transcendence degree. AMS Subject Classifications: 12E15, 43A07

The discrete decomposition with finite multiplicities of square-integrable cuspforms, and concommitant estimates, are foundational. Lie-group and symmetric-space versions of these results were known [Selberg 1956], [Gelfand-Fomin 1952] in the mid 1950’s. The general assertion was made in [GelfandGraev 1962] and [Gelfand-PS 1963], the latter observing that an adelic formulation can proceed in th...

We consider the initial value Cauchy problem for a class of evolution equations whose Hamiltonian is Weyl quantization homogeneous quadratic form with non-negative definite real part. The solution semigroup shown to be strongly continuous on several spaces: Shubin-Sobolev spaces, Schwartz space, tempered distributions, equal index Beurling type Gelfand-Shilov spaces and their dual ultradistribu...

Abstract We develop a diagrammatic categorification of the polynomial ring Z[x], based on geometrically defined graded algebra. This construction generalizes to some special functions, such as Chebyshev polynomials. Diagrammatic algebras featured in these categorifications lead first topological interpretations Bernstein-Gelfand-Gelfand reciprocity property.

A topological group G together with a compact subgroup K are said to form a Gelfand pair if the set L1(K\G/K) of K-bi-invariant integrable functions on G is a commutative algebra under convolution. The situation where G and K are Lie groups has been the focus of extensive and ongoing investigation. Riemannian symmetric spaces G/K furnish the most widely studied and best understood examples. ([H...

Quadratic algebras associated to pseudo-roots of noncommutative polynomials have been introduced by I. Gelfand, Retakh, and Wilson in connection with studying the decompositions of noncommutative polynomials. Later they (with S. Gelfand and Serconek) shown that the Hilbert series of these algebras and their quadratic duals satisfy the necessary condition for Koszulity. It is proved in this note...