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 be a locally compact group and a Banach space. The left module is by definition absolutely continuous under the composition if for there exist with . If the locally compact Hausdorff space is a transformation group over and has a measure quasi-invariant with respect to , then is an absolutely continuous module--the main object we study. If is measurable, let consist of all functions in vanishing outside . For not locally null and a closed linear subspace of , we observe the connection between the closed linear span (denoted ) of the elements , with and , and the collection of functions of shifted by elements in . As a result, a closed linear subspace of is an for some measurable if and only if it is closed under pointwise multiplication by elements of . This allows the theorem stating that if and are both measurable, then there is a measurable subset of such that . Under certain restrictions on , we show that this is essentially open in the (usually stronger) orbit topology on . Finally we prove that if and are both relatively sigma-compact, and if also , then there exist and locally almost everywhere equal to and respectively, such that ; in addition we characterize those and for which and .

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1970-99932-7

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