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Transactions of the American Mathematical Society

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



Group algebra modules. IV

Authors: S. L. Gulick, T.-S. Liu and A. C. M. van Rooij
Journal: Trans. Amer. Math. Soc. 152 (1970), 581-596
MSC: Primary 46.80; Secondary 22.00
Part III: Trans. Amer. Math. Soc. (2) (1970), 561-579
MathSciNet review: 0270171
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Abstract: Let $ \Gamma $ be a locally compact group, $ \Omega $ a measurable subset of $ \Gamma $, and let $ {L_\Omega }$ denote the subspace of $ {L^1}(\Gamma )$ consisting of all functions vanishing off $ \Omega $. Assume that $ {L_\Omega }$ is a subalgebra of $ {L^1}(\Gamma )$. We discuss the collection $ {\Re _\Omega }(K)$ of all module homomorphisms from $ {L_\Omega }$ into an arbitrary Banach space $ K$ which is simultaneously a left $ {L^1}(\Gamma )$ module. We prove that $ {\Re _\Omega }(K) = {\Re _\Omega }({K_0}) \oplus {\Re _\Omega }({K_{\text{abs} }})$, where $ {K_0}$ is the collection of all $ k \in K$ such that $ fk = 0$, for all $ f \in {L^1}(\Gamma )$, and where $ {K_{\text{abs} }}$ consists of all elements of $ K$ which can be factored with respect to the module composition. We prove that $ {\Re _\Omega }({K_0})$ is the collection of linear continuous maps from $ {L_\Omega }$ to $ {K_0}$ which are zero on a certain measurable subset of $ X$. We reduce the determination of $ {\Re _\Omega }({K_{\text{abs} }})$ to the determination of $ {\Re _\Gamma }({K_{\text{abs} }})$. Denoting the topological conjugate space of $ K$ by $ {K^ \ast }$, we prove that $ {({K_{\text{abs} }})^ \ast }$ is isometrically isomorphic to $ {\Re _\Omega }({K^ \ast })$. Finally, we discuss module homomorphisms $ R$ from $ {L_\Omega }$ into $ {L^1}(X)$ such that for each $ f \in {L_\Omega },Rf$ vanishes off $ Y$.

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Keywords: Transformation group, quasi-invariant measure, absolutely continuous measure, approximate identity, factorable, group algebra module, module homomorphism, absolutely continuous module, order-free module, subsemigroup, left-translation invariant, orbit topology
Article copyright: © Copyright 1970 American Mathematical Society