Inner invariant extensions of Dirac measures on compactly cancellative topological semigroups

Abstract

Let ${\cal S}$ be a left compactly cancellative foundation semigroup with identity $e$ and $M_a({\cal S})$ be its semigroup algebra. In this paper, we give a characterization for the existence of an inner invariant extension of $\delta_e$ from $C_b({\cal S})$ to a mean on $L^\infty({\cal S},M_a({\cal S}))$ in terms of asymptotically central bounded approximate identities in $M_a({\cal S})$. We also consider topological inner invariant means on $L^\infty({\cal S},M_a({\cal S}))$ to study strict inner amenability of $M_a({\cal S})$ and their relation with strict inner amenability of ${\cal S}$.