The operator norm on weighted discrete semigroup algebras $\ell ^1(S,\omega )$

Let ω \omega be a weight on right cancellative semigroup alttext="upper S"> S encoding="application/x-tex">S . alttext="double-vertical-bar dot double-vertical-bar Subscript omega"> ‖ ⋅<!-- ⋅ <mml:msub> encoding="application/x-tex">\|\cdot \|_{\omega } the weighted norm discrete algebra alttext="script l Superscript 1 Baseline left-parenthesis upper S comma omega right-parenthesis"> ℓ<!-- ℓ <mml:mn>1 stretchy="false">( , stretchy="false">) encoding="application/x-tex">\ell ^1(S, \omega ) In this paper, we prove that satisfies F-property if and only operator o p"> o p encoding="application/x-tex">\| \cdot op} of is exactly equal to another overTilde Sub 1"> ~<!-- ~ </mml:mover> \|_{\widetilde {\omega }_1} Though its proof elementary, result unexpectedly surprising. particular, \|_{1 same as ^1 - \|_1 ^1(S) Moreover, various examples are discussed understand relation among ,