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Approximation in Banach space representations of compact groups


Authors: M. Filali and M. Sangani Monfared
Journal: Proc. Amer. Math. Soc. 148 (2020), 5159-5170
MSC (2010): Primary 43A77, 46H15, 43A20, 22D10, 22D20
DOI: https://doi.org/10.1090/proc/15247
Published electronically: September 11, 2020
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Abstract: 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: mahmoud.filali@oulu.fi

M. Sangani Monfared
Affiliation: Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario, N9B 3P4, Canada
MR Author ID: 711290
Email: monfared@uwindsor.ca

DOI: https://doi.org/10.1090/proc/15247
Keywords: Compact groups, Banach space representations, homogeneous Banach spaces, approximation by trigonometric polynomials, measure algebra, pseudo measures.
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
Article copyright: © Copyright 2020 American Mathematical Society