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Approximation in Banach space representations of compact groups.
- Source :
-
Proceedings of the American Mathematical Society . Dec2020, Vol. 148 Issue 12, p5159-5170. 12p. - Publication Year :
- 2020
-
Abstract
- Let π : G → B(E) be a continuous representation of a compact group G on a Banach space E. We prove that the set of vectors π (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 = ⊕θ ∈ GEθ for E, where each Eθ is a closed π-invariant subspace of E. If λp : M(G) → B(Lp(G)), p ∈(1,∞), is the left regular representation of the measure algebra M(G) and B ⊂ PMp(G) is a homogeneous Banach space, we show that B λp(T(G)) is norm dense in B. Since Hilbert space techniques are not available, new machinery is developed in the paper for the proofs. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 148
- Issue :
- 12
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 146510618
- Full Text :
- https://doi.org/10.1090/proc/15247