Approximate identities in spaces of all absolutely continuous measures on locally compact semigroups (Q1780018)

Let \(S\) be a locally compact (topological) semigroup, \(M(S)\) be the Banach algebra of all bounded regular Borel measures on \(S\) with total variation norm and convolution as multiplication and \(M_a(S)\) be the space of two-sided absolutely continuous measures, a norm closed subalgebra of \(M(S)\). The main result of the paper proves that if \(S\) has the identity \(e\) and satisfies the condition \((L)\) (i.e. whenever \(U\) is open in \(S\) and \(a\in S\), then \(aU\) is open in \(S\)) then \(M_a(S)\) is closed under absolute continuity and has an approximate identity. This result is related to and similar with the A. C. Baker and J. W. Baker's results concerning algebras of measures on locally compact semigroups.