Group actions and direct sum decompositions of $L^ p$ spaces

Abstract:

Let $G$ be a locally compact group of measure preserving transformations on a $\sigma$-finite measure space $\left ( {X,\mathcal {B},m} \right )$, and let $S$ be a subset of ${M^1}\left ( G \right )$. Let $1 < p < \infty ,{I_p} = \left \{ {f:f \in {L^p}\left ( m \right ){\text { and}}{{\text { }}_g}f = f,{\text {for all }}g \in G} \right \}$, let ${I_p}\left ( S \right ) = \left \{ {f:f \in {L^p}\left ( m \right ){\text { and }}\mu * f = f{\text { for all }}\mu \in S} \right \}$, and let ${K_p}\left ( S \right )$ be the closed subspace of ${L^p}\left ( m \right )$ generated by functions of the form $\mu * f - f$, for $f \in {L^p}\left ( m \right )$ and $\mu \in S$. Conditions are given on $S$ which ensure that ${I_p} = {I_p}\left ( S \right )$, and this is used to express ${L^p}\left ( m \right )$ as a direct sum of ${I_p}$ and ${K_p}\left ( S \right )$.