## Group algebra modules. III

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- by S. L. Gulick, T.-S. Liu and A. C. M. van Rooij
- Trans. Amer. Math. Soc.
**152**(1970), 561-579 - DOI: https://doi.org/10.1090/S0002-9947-1970-99932-7
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Part IV: Trans. Amer. Math. Soc. (2) (1970), 581-596

## 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 ${L_\Omega } \ast {L_Y} = {L_Y}$.## References

- N. Bourbaki,
*Éléments de mathématique. XXV. Première partie. Livre VI: Intégration. Chapitre 6: Intégration vectorielle*, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1281, Hermann, Paris, 1959 (French). MR**0124722** - Paul J. Cohen,
*Factorization in group algebras*, Duke Math. J.**26**(1959), 199–205. MR**104982** - S. L. Gulick, T. S. Liu, and A. C. M. van Rooij,
*Group algebra modules. I*, Canadian J. Math.**19**(1967), 133–150. MR**222662**, DOI 10.4153/CJM-1967-008-4 - S. L. Gulick, T. S. Liu, and A. C. M. van Rooij,
*Group algebra modules. II*, Canadian J. Math.**19**(1967), 151–173. MR**222663**, DOI 10.4153/CJM-1967-009-0 - Edwin Hewitt and Kenneth A. Ross,
*Abstract harmonic analysis. Vol. I*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR**551496** - Teng-sun Liu,
*Invariant subspaces of some function spaces*, Quart. J. Math. Oxford Ser. (2)**14**(1963), 231–239. MR**151577**, DOI 10.1093/qmath/14.1.231 - Teng-sun Liu, Arnoud van Rooij, and Ju-kwei Wang,
*Transformation groups and absolutely continuous measures. II*, Nederl. Akad. Wetensch. Proc. Ser. A 73=Indag. Math.**32**(1970), 57–61. MR**0260980** - Teng-sun Liu and Arnoud van Rooij,
*Sums and intersections of normed linear spaces*, Math. Nachr.**42**(1969), 29–42. MR**273370**, DOI 10.1002/mana.19690420103 - Walter Rudin,
*Measure algebras on abelian groups*, Bull. Amer. Math. Soc.**65**(1959), 227–247. MR**108689**, DOI 10.1090/S0002-9904-1959-10322-0 - Arthur B. Simon,
*Vanishing algebras*, Trans. Amer. Math. Soc.**92**(1959), 154–167. MR**123919**, DOI 10.1090/S0002-9947-1959-0123919-9

## Bibliographic Information

- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**152**(1970), 561-579 - MSC: Primary 46.80; Secondary 22.00
- DOI: https://doi.org/10.1090/S0002-9947-1970-99932-7
- MathSciNet review: 0270171