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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Approximation in Banach space representations of compact groups
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by M. Filali and M. Sangani Monfared PDF
Proc. Amer. Math. Soc. 148 (2020), 5159-5170 Request permission


Let $\pi \colon G \longrightarrow \mathcal B(E)$ be a continuous representation of a compact group $G$ on a Banach space $E$. We prove that the set of vectors $\pi (h)x$, as $h$ runs through the set $T(G)$ of all trigonometric polynomials on $G$, and $x$ runs through $E$, spans an invariant dense linear subspace of $E$. We prove the existence of a topological direct sum decomposition $E=\bigoplus _{\theta \in \widehat G}E_\theta$ for $E$, where each $E_\theta$ is a closed $\pi$-invariant subspace of $E$. If $\lambda _p\colon M(G)\longrightarrow \mathcal B(L^p(G))$, $p\in (1,\infty )$, is the left regular representation of the measure algebra $M(G)$ and $B\subset PM_p(G)$ is a homogeneous Banach space, we show that $B\cap \lambda _p(T(G))$ is norm dense in $B$. Since Hilbert space techniques are not available, new machinery is developed in the paper for the proofs.
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Additional Information
  • M. Filali
  • Affiliation: Department of Mathematical Sciences, University of Oulu, Oulu 90014, Finland
  • MR Author ID: 292620
  • Email:
  • M. Sangani Monfared
  • Affiliation: Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario, N9B 3P4, Canada
  • MR Author ID: 711290
  • Email:
  • Received by editor(s): February 10, 2020
  • Published electronically: September 11, 2020
  • Additional Notes: The first author is grateful for the hospitality and partial support from the Department of Mathematics and Statistics at Windsor University.
    The second author was supported by an NSERC grant.
  • Communicated by: Adrian Ioana
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 5159-5170
  • MSC (2010): Primary 43A77, 46H15, 43A20, 22D10, 22D20
  • DOI:
  • MathSciNet review: 4163829