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 $\|\cdot \|_{\omega }$ 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 $\| \cdot \|_{\omega op}$ of $\| \cdot \|_{\omega }$ is exactly equal to another weighted norm $\| \cdot \|_{\widetilde {\omega }_1}$. Though its proof is elementary, the result is unexpectedly surprising. In particular, the operator norm $\| \cdot \|_{1 op}$ is same as the $\ell ^1$- norm $\| \cdot \|_1$ on $\ell ^1(S)$. Moreover, various examples are discussed to understand the relation among $\| \cdot \|_{\omega op}$, $\| \cdot \|_{\omega }$, and $\ell ^1(S, \omega )$.