Topological invariant means on almost periodic functionals: Solution to problems by Dales–Lau–Strauss and Daws

Abstract:

Let $\mathcal {G}$ be a locally compact group, and denote by $\mathrm {WAP} (\mathrm {M}(\mathcal {G}))$ and $\mathrm {AP}(\mathrm {M}(\mathcal {G}))$ the spaces of weakly almost periodic, respectively, almost periodic functionals on the measure algebra $\mathrm {M}(\mathcal {G})$. Problem 3 in [H.G. Dales, A.T.-M. Lau, D. Strauss, Second duals of measure algebras, Dissertationes Math. (Rozprawy Mat.) 481 (2012), 1–121] asks if $\mathrm {WAP}(\mathrm {M}(\mathcal {G}))$ and $\mathrm {AP}(\mathrm {M}(\mathcal {G}))$ admit topological invariant means, and if yes, whether they are unique. The questions regarding existence had already been raised in [M. Daws, Characterising weakly almost periodic functionals on the measure algebra, Studia Math. 204 (2011), no. 3, 213–234]. We answer all these problems in the affirmative.