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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Group algebra modules. III

Authors: S. L. Gulick, T.-S. Liu and A. C. M. van Rooij
Journal: Trans. Amer. Math. Soc. 152 (1970), 561-579
MSC: Primary 46.80; Secondary 22.00
Part IV: Trans. Amer. Math. Soc. (2) (1970), 581-596
MathSciNet review: 0270171
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Abstract: Let $ \Gamma $ be a locally compact group and $ K$ a Banach space. The left $ {L^1}(\Gamma )$ module $ K$ is by definition absolutely continuous under the composition $ \ast $ if for $ k \in K$ there exist $ f \in {L^1}(\Gamma ),k' \in K$ with $ k = f \ast k'$. If the locally compact Hausdorff space $ X$ is a transformation group over $ \Gamma $ and has a measure quasi-invariant with respect to $ \Gamma $, then $ {L^1}(X)$ is an absolutely continuous $ {L^1}(\Gamma )$ module--the main object we study. If $ Y \subseteq X$ is measurable, let $ {L_Y}$ consist of all functions in $ {L^1}(X)$ vanishing outside $ Y$. For $ \Omega \subseteq \Gamma $ not locally null and $ B$ a closed linear subspace of $ K$, we observe the connection between the closed linear span (denoted $ {L_\Omega } \ast B$) of the elements $ f \ast k$, with $ f \in {L_\Omega }$ and $ k \in B$, and the collection of functions of $ B$ shifted by elements in $ \Omega $. As a result, a closed linear subspace of $ {L^1}(X)$ is an $ {L_Z}$ for some measurable $ Z \subseteq X$ if and only if it is closed under pointwise multiplication by elements of $ {L^\infty }(X)$. This allows the theorem stating that if $ \Omega \subseteq \Gamma $ and $ Y \subseteq X$ are both measurable, then there is a measurable subset $ Z$ of $ X$ such that $ {L_\Omega } \ast {L_Y} = {L_Z}$. Under certain restrictions on $ \Gamma $, we show that this $ Z$ is essentially open in the (usually stronger) orbit topology on $ X$. Finally we prove that if $ \Omega $ and $ Y$ are both relatively sigma-compact, and if also $ {L_\Omega } \ast {L_Y} \subseteq {L_Y}$, then there exist $ {\Omega _1}$ and $ {Y_1}$ locally almost everywhere equal to $ \Omega $ and $ Y$ respectively, such that $ {\Omega _1}{Y_1} \subseteq {Y_1}$; in addition we characterize those $ \Omega $ and $ Y$ for which $ {L_\Omega } \ast {L_\Omega } = {L_\Omega }$ and $ {L_\Omega } \ast {L_Y} = {L_Y}$.

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Keywords: Transformation group, quasi-invariant measure, absolutely continuous measure, approximate identity, factorable, orbit topology, vanishing algebra, group algebra module
Article copyright: © Copyright 1970 American Mathematical Society