Inner invariant extensions of Dirac measures on compactly cancellative topological semigroups

Let S be a left compactly cancellative foundation semigroup with identity e and M-a(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(S) to a mean on L-infinity(S, M-a(S)) in terms of asymptotically central bounded approximate identities in M-a(S). We also consider topological inner invaxiant means on L-infinity(S, M-a(S)) to study strict inner amenability of M-a(S) and their relation with strict inner amenability of S.