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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the dimension of left invariant means and left thick subsets


Author: Maria Klawe
Journal: Trans. Amer. Math. Soc. 231 (1977), 507-518
MSC: Primary 43A07
MathSciNet review: 0447970
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Abstract: If S is a left amenable semigroup, let $ \dim \langle Ml(S)\rangle $ denote the dimension of the set of left invariant means on S. Theorem. If S is left amenable, then $ \dim \langle Ml(S)\rangle = n < \infty $ if and only if S contains exactly n disjoint finite left ideal groups. This result was proved by Granirer for S countable or left cancellative. Moreover, when S is infinite, left amenable, and either left or right cancellative, we show that $ \dim \langle Ml(S)\rangle $ is at least the cardinality of S. An application of these results shows that the radical of the second conjugate algebra of $ {l_1}(S)$ is infinite dimensional when S is a left amenable semigroup which does not contain a finite ideal.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0447970-5
PII: S 0002-9947(1977)0447970-5
Keywords: Semigroup, left thick, invariant means, radical, finite left ideal group
Article copyright: © Copyright 1977 American Mathematical Society