Multipliers of weighted semigroups and associated Beurling Banach algebras

Abstract

Given a weighted discrete abelian semigroup (S, ω), the semigroup M ω (S) of ω-bounded multipliers as well as the Rees quotient M ω (S)/S together with their respective weightsω andω q induced by ω are studied; for a large class of weights ω, the quotient 1 (M ω (S),ω)/ 1 (S, ω) is realized as a Beurling algebra on the quotient semigroup M ω (S)/S; the Gel'fand spaces of these algebras are determined; and Banach algebra properties like semisimplicity, uniqueness of uniform norm and regularity of associated Beurling algebras on these semigroups are investigated. The involutive analogues of these are also considered. The results are exhibited in the context of several examples.

The Multiplier Algebra of a Beurling Algebra

“…For the general discrete case 1 (S , ω) (in particular, 1 (Q + , ω)), it was proclaimed in 2000 in [4, page 536] that 'presumably the golden age for the study of these algebras lies in the future'. The present paper along with [2] is our response to this (see also [5]). For S = Z + , this gives Banach algebras of power series for which we refer the reader to [4].…”

“…A weight ω on S represents a frequency function or a norm on S . Taking (S , ω) as an intrinsic object, a study of multipliers on (S , ω) has been initiated in [2]. The subsemigroup M ω (S ) of M(S ) consists of multipliers α on S which are ω-bounded in the sense that ω(α(s)) ≤ Kω(s) (s ∈ S ) for some K > 0.…”

“…The Beurling algebra 1 (M ω (S ), ω) is analogously defined. The interrelation between the Banach algebra structure of 1 (S , ω) and the structure of (S , ω) is a fascinating aspect of harmonic analysis [2,5].…”

The Relationship between Two Involutive SemigroupsSandSTIs Defined by a Left MultiplierT

“…A semigroup S is faithful if it is left and right faithful. If S is faithful then : S → S is multiplier if and only if = ( ) for all , ∈ S [6].…”