This paper is about the topologies arising from statistical coincidence on locally finite point sets in locally compact Abelian groups G. The first part defines a uniform topology (autocorrelation topology) and proves that, in effect, the set of all locally finite subsets of G is complete in this topology. Notions of statistical relative denseness, statistical uniform discreteness, and statisti...

For a locally compact group G, let A(G) denote its Fourier algebra and Ĝ its dual object, i.e. the collection of equivalence classes of unitary representations of G. We show that the amenability constant of A(G) is less than or equal to sup{deg(π) : π ∈ Ĝ} and that it is equal to one if and only if G is abelian.

We derive necessary conditions for sampling and interpolation of bandlimited functions on a locally compact abelian group in line with the classical results of H. Landau for bandlimited functions on R. Our conditions are phrased as comparison principles involving a certain canonical lattice.

We consider the problem of separability: decide whether a Hermitian operator on a finite dimensional Hilbert tensor product H = H ⊗ · · ·⊗H is separable or entangled. We show that the tensor convolution ( φ . . . φ ) : G → H defined for mappings φ : G → H μ on an almost arbitrary locally compact abelian group G, give rise to formulation of an equivalent problem to the separability one.

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity compo...

A theorem of A. Weil asserts that a topological group embeds as a (dense) subgroup of a locally compact group if and only if it contains a non-empty precompact open set; such groups are called locally precompact. Within the class of locally precompact groups, the authors classify those groups with the following topological properties: (a) Dieudonné completeness; (b) local realcompactness; (c) r...