### A Characterization of a Class of Locally Compact Abelian Groups via Korovkin Theory.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

For a locally compact, abelian group $G$, we study the space ${S}_{0}\left(G\right)$ of functions on $G$ belonging locally to the Fourier algebra and with ${l}^{1}$-behavior at infinity. We give an abstract characterization of the family of spaces $\left\{{S}_{0}\right(G):G$ abelian$\}$ by its hereditary properties.

Let ${\tau}_{X}$ and ${\tau}_{Y}$ be representations of a topological group G on Banach spaces X and Y, respectively. We investigate the continuity of the linear operators Φ: X → Y with the property that $\Phi \circ {\tau}_{X}\left(t\right)={\tau}_{Y}\left(t\right)\circ \Phi $ for each t ∈ G in terms of the invariant vectors in Y and the automatic continuity of the invariant linear functionals on X.

For any connected Lie group G and any Laplacian Λ = X²₁ + ⋯ + X²ₙ ∈ 𝔘𝔤 (X₁,...,Xₙ being a basis of 𝔤) one can define the commutant 𝔅 = 𝔅(Λ) of Λ in the convolution algebra ℒ¹(G) as well as the commutant ℭ(Λ) in the group C*-algebra C*(G). Both are involutive Banach algebras. We study these algebras in the case of a "distinguished Laplacian" on the "Iwasawa part AN" of a semisimple Lie group. One obtains a fairly good description of these algebras by objects derived from the semisimple group....

It is known that the dual of a compact, connected, non-abelian group may contain no infinite central Sidon sets, but always does contain infinite central $p$-Sidon sets for $p\>1.$ We prove, by an essentially constructive method, that the latter assertion is also true for every infinite subset of the dual. In addition, we investigate the relationship between weighted central Sidonicity for a compact Lie group and Sidonicity for its torus.

Let $A$ be a finite subset of an abelian group $G$ and let $P$ be a closed half-plane of the complex plane, containing zero. We show that (unless $A$ possesses a special, explicitly indicated structure) there exists a non-trivial Fourier coefficient of the indicator function of $A$ which belongs to $P$. In other words, there exists a non-trivial character $\chi \in \widehat{G}$ such that ${\sum}_{a\in A}\chi \left(a\right)\in P$.

Let G be a locally compact group and let π be a unitary representation. We study amenability and H-amenability of π in terms of the weak closure of (π ⊗ π)(G) and factorization properties of associated coefficient subspaces (or subalgebras) in B(G). By applying these results, we obtain some new characterizations of amenable groups.