Approximation in Banach space representations of compact groups.

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]