Inner invariant means on locally compact topological semigroups

Abstract

Let ${\cal S}$ be a locally compact semigroup and $M_a({\cal S})$ be its semigroup algebra. In this paper, we investigate inner invariant means on $L^\infty({\cal S},M_a({\cal S}))$ of all $M_a( {\cal S})$-measurable complex-valued bounded functions on ${\cal S}$ and its closed subspace $C_b({\cal S})$, the space of all bounded continuous complex-valued functions on ${\cal S}$. We also study topological inner invariant means on certain closed subspaces $X$ of $L^\infty({\cal S},M_a({\cal S}))$ and their relation with inner invariant means on $X$.