Multipliers of weighted semigroups and associated Beurling Banach algebras (Q1930303)

Let \((S,\omega)\) be a weighted discrete abelian semigroup. The \(\omega\)-bounded multipliers on \(S\) form a semigroup \(M_\omega(S)\), on which \(\omega\) induces a natural weight \[ \tilde{\omega}(T):=\sup\left\{\frac{\omega(Ts)}{\omega(s)}:\;s\in S\right\}\quad(T\in M_\omega(S)). \] The authors investigate to what extend various natural (Banach algebraic) properties on \((S,\omega)\) and on its associated Beurling algebra \(\ell^1(S,\omega)\) determine or are determined by the corresponding properties on \((M_\omega(S),\tilde{\omega})\) and on its associated Beurling algebra \(\ell^1(M_\omega(S),\tilde{\omega})\); the case of involutive weighted semigroups with associated \(*\)-Beurling algebras is also considered. To name just a few, the properties studied include semisimplicity, uniqueness of (\(C^*\)-)norm, and (\(*\)-)regularity.