Invariant Linear Forms Research Articles

Topologically invariant linear forms on spaces of abstract distributions on a locally compact abelian group

A special case of the main result proved in this paper is the following. IfG is a locally compact, σ-compact, non-compact connected abelian group, thenL 2 (G)={f−ψ*f:f∈L2(G), ψ∈L 1 (G), ψ≥0 and ∫ G ψ=1}. In this case, any topologically invariant linear form onL 2 (G) is 0.

Translation invariant forms on $L^p(G)(1<p<\infty)$

It is shown that if G is a connected metrizable compact Abelian group and 1<p<∞, any (possibly discontinuous) translation invariant linear form on L p (G) is a scalar multiple of the Haar measure. This result extends the theorem of G.H. Meisters and W.M. Schmidt (J. Funct. Anal. 13 (1972), 407-424) on L 2 (G). Our method permits in fact to consider any superreflexive translation invariant Banach lattice on G, which is the adopted point of view. We study the representation of an element f of this invariant lattice X as a sum of a bounded number of elements of the form g-τ(a)g, where g in X, a in G and τ(a) the corresponding translation operator. Our approach consists in proving the boundedness of certain random convolution operators using interpolation techniques.

Translation-invariant linear forms on 𝐿_{𝑝}(𝐺)

Let G G be a compact group such that the identity representation of G G is not contained in the regular representation on L 2 0 ( G , λ G ) L_2^0(G,{\lambda _G}) of G G with the discrete topology. Then any left translation invariant linear form on L p ( G ) , 1 &gt; p &gt; ∞ {L_p}(G),1 &gt; p &gt; \infty , is continuous and must be a constant times the Haar integral. This shows that many classical matrix groups G G admit only continuous left translation invariant linear forms on L p ( G ) , 1 &gt; p &gt; ∞ {L_p}(G),1 &gt; p &gt; \infty .