The operator norm on weighted discrete semigroup algebras ℓ¹(𝑆,𝜔)

Let ω \omega be a weight on a right cancellative semigroup S S . Let ‖ ⋅ ‖ ω \|\cdot \|_{\omega } be the weighted norm on the weighted discrete semigroup algebra ℓ 1 ( S , ω ) \ell ^1(S, \omega ) . In this paper, we prove that the weight ω \omega satisfies F-property if and only if the operator norm ‖ ⋅ ‖ ω o p \| \cdot \|_{\omega op} of ‖ ⋅ ‖ ω \| \cdot \|_{\omega } is exactly equal to another weighted norm ‖ ⋅ ‖ ω ~ 1 \| \cdot \|_{\widetilde {\omega }_1} . Though its proof is elementary, the result is unexpectedly surprising. In particular, the operator norm ‖ ⋅ ‖ 1 o p \| \cdot \|_{1 op} is same as the ℓ 1 \ell ^1 - norm ‖ ⋅ ‖ 1 \| \cdot \|_1 on ℓ 1 ( S ) \ell ^1(S) . Moreover, various examples are discussed to understand the relation among ‖ ⋅ ‖ ω o p \| \cdot \|_{\omega op} , ‖ ⋅ ‖ ω \| \cdot \|_{\omega } , and ℓ 1 ( S , ω ) \ell ^1(S, \omega ) .