## Approximation in Banach space representations of compact groups

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- by M. Filali and M. Sangani Monfared
- Proc. Amer. Math. Soc.
**148**(2020), 5159-5170 - 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.## References

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## Bibliographic 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
- 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: https://doi.org/10.1090/proc/15247
- MathSciNet review: 4163829