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


Author: Matthias Neufang
Journal: Proc. Amer. Math. Soc. 145 (2017), 3595-3598
MSC (2010): Primary 22D15, 43A10
DOI: https://doi.org/10.1090/proc/13671
Published electronically: April 26, 2017
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Abstract: Let $ \mathcal {G}$ be a locally compact group, and denote by $ \textup {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 $ \textup {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.


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Matthias Neufang
Affiliation: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6, Canada – and – Laboratoire de Mathématiques Paul Painlevé (UMR CNRS 8524), Université Lille 1 – Sciences et Technologies, UFR de Mathé- matiques, 59655 Villeneuve d’Ascq Cedex, France
Email: mneufang@math.carleton.ca; matthias.neufang@math.univ-lille1.fr

DOI: https://doi.org/10.1090/proc/13671
Received by editor(s): September 10, 2016
Published electronically: April 26, 2017
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2017 American Mathematical Society