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On the relation between left thickness and topological left thickness in semigroups


Author: James C. S. Wong
Journal: Proc. Amer. Math. Soc. 86 (1982), 471-476
MSC: Primary 43A07
DOI: https://doi.org/10.1090/S0002-9939-1982-0671218-7
MathSciNet review: 671218
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Abstract: In this paper, we establish an interesting relation between left thickness and topological left thickness in semigroups by showing that a Borel subset $ T$ of a locally compact semigroup $ S$ is topological left thick in $ S$ iff a certain subset $ {M_T}$ associated with $ T$ is left thick in a semigroup $ {S_1}$ containing $ S$, or equivalent, iff $ {M_T}$ contains a left ideal of $ {S_1}$. Our results contain a topological analogue of a result of H. Junghenn in [Amenability of function spaces on thick subsemigroups, Proc. Amer. Math. Soc. 75 (1979), 37-41]. However, even in the case of discrete semigroups, our results are more general and in a way more natural than those of Junghenn's. Furthermore, the fact that $ {M_T}$ is left thick iff it contains a left ideal in $ {S_1}$ is quite surprising, since in general, a left thick subset need not contain a left ideal although the converse is always true.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0671218-7
Keywords: Locally compact semigroups, convolution measure algebras, topological left thickness and left thickness, invariant means
Article copyright: © Copyright 1982 American Mathematical Society

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