The Operator Norm on Weighted Discrete Semigroup Algebras ${\ell }^1(S,\omega )$

Abstract

Let $\omega$ be a weight on a right cancellative semigroup $S$. Let $\Vert \cdot {\Vert }_{\omega }$ be the weighted norm on the weighted discrete semigroup algebra ${\ell }^1(S,\omega )$. In this paper, we prove that the weight $\omega$ satisfies F-property if and only if the operator norm $\Vert \cdot {\Vert }_{\omega op}$ of $\Vert \cdot {\Vert }_{\omega }$ is exactly equal to another weighted norm $\Vert \cdot {\Vert }_{{\tilde{\omega }}_1}$ [Theorem 2.5 ($iii$)]. Though its proof is elementary, the result is unexpectedly surprising. In particular, $\Vert \cdot {\Vert }_{1op}$ is same as $\Vert \cdot {\Vert }_1$ on ${\ell }^1(S)$. Moreover, various examples are discussed to understand the relating among $\Vert \cdot {\Vert }_{\omega op}$, $\Vert \cdot {\Vert }_{\omega }$, and ${\ell }^1(S,\omega )$.

7 pages