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Given a foundation locally compact Hausdorff topological semigroup S , we consider on M a ( S )* the τ c -topology, i.e. the weak topology under all right multipliers induced by measures in M a ( S ). For such an arbitrary S the τ c -topology is not weaker than the weak*-topology and not stronger than the norm topology on M a ( S )*. However, a further investigation shows that for compact S the norm topology and τ c -topology coincide on every norm bounded subset of M a ( S ). Among the other results we mention that except for discrete S the τ c -topology is always different from the norm-topology. Finally, we give some results about τ c -almost periodic functionals.
