Invariant Weighted Algebras ℒ p w (G)

Abstract

Criterion of (Shilov) regularity for weighted algebras L w

Lebesgue weighted L p -algebra on locally compact groups

“…We can easily show that the net λ(V ) −1 χ V , where V runs over the net of all relatively compact neighborhoods of the identity, is a left approximate identity for L 1,p ω (G). Recall that the weight function ω is quasi submultiplicative if and only if L 1 (G, ω) L p (G, ω) L p (G, ω); [13,Theorem 3.1]. We want to use this and show that for such a weight function ω, L 1,p ω (G) has a bounded approximate identity if and only if G is discrete.…”

“…Consider, as in [13], the weight function ω| [n+1/n 2 ,n+1] = 1 + n 2 and ω n + 1/(2n 2 ) = 1 + |n| Acta Mathematica Hungarica 133, 2011 defined on the additive group R, and extend ω piecewise linearly. Then L 2 (R, ω) is a Banach algebra but ω is not quasi submultiplicative.…”

Example of a weighted algebra Lpw(G) on an uncountable discrete group

“…The results of this section allow us to extend to the class of amenable groups Theorem 1.1 in [7], claiming that σ -compactness is necessary for the existence of weighted algebras. This extension is carried out in Section 3.…”

“…representable as a countable union of compact sets). It is also proved [7] that for abelian groups σ -compactness is necessary for containment in this class.…”

“…It has long been known that the real line belongs to this class [10] (one may take, e.g., w(t) = 1 + t 2 for any p > 1), as well as Z and R n . In [7] it is proved that this class contains all σ -compact groups (i.e. representable as a countable union of compact sets).…”