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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
$L^{\infty}(S, M_{a}(S))$ in terms of asymptotically central
bounded approximate identities in $M_{a}(S)$. We also consider
topological inner invariant means on $L^{\infty}(S, M_{a}(S))$ to
study strict inner amenability of $M_{a}(S)$ and their relation
with strict inner amenability of $S$.
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