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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Group actions and direct sum decompositions of $ L\sp p$ spaces

Author: Rodney Nillsen
Journal: Proc. Amer. Math. Soc. 106 (1989), 975-985
MSC: Primary 43A15; Secondary 28D15
MathSciNet review: 972237
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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)$.

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Keywords: Groups, invariant measures, convolution, direct sums, $ {L^p}$ spaces, ergodicity
Article copyright: © Copyright 1989 American Mathematical Society

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