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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
DOI: https://doi.org/10.1090/S0002-9947-1977-0447970-5
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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  • [1] Ching Chou, Minimal sets and ergodic measures for $ \beta {\mathbf{N}}\backslash {\mathbf{N}}$, Illinois J. Math. 13 (1969), 777-788. MR 40 #2814. MR 0249569 (40:2814)
  • [2] -, On the size of the set of left invariant means on a semigroup, Proc. Amer. Math. Soc. 23 (1969), 199-205. MR 40 #710. MR 0247444 (40:710)
  • [3] -, The exact cardinality of the set of invariant means on a group, Proc. Amer. Math. Soc. 55 (1976), 103-106. MR 0394036 (52:14842)
  • [4] P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847-870. MR 26 #622. MR 0143056 (26:622)
  • [5] M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509-544. MR 19, 1067. MR 0092128 (19:1067c)
  • [6] L. R. Fairchild, Extreme invariant means without minimal support, Trans. Amer. Math. Soc. 172 (1972), 83-93. MR 46 #7799. MR 0308685 (46:7799)
  • [7] E. Granirer, On amenable semigroups with a finite-dimensional set of invariant means. I, II, Illinois J. Math. 7 (1963), 32-48; ibid. 7 (1963), 49-58. MR 26 #1744, #1745. MR 0144197 (26:1744)
  • [8] -, A theorem on amenable semigroups, Trans. Amer. Math. Soc. 111 (1964), 367-379. MR 29 #3870. MR 0166597 (29:3870)
  • [9] -, Extremely amenable semigroups, Math. Scand. 17 (1965), 177-197. MR 33 #5760. MR 0197595 (33:5760)
  • [10] E. Granirer and M. Rajagopalan, A note on the radical of the second conjugate algebra of a semigroup algebra, Math. Scand. 15 (1964), 163-166. MR 32 #2924. MR 0185457 (32:2924)
  • [11] E. Hewitt and K. Ross, Abstract harmonic analysis, Vol. 1, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #158.
  • [12] I. S. Luthar, Uniqueness of the invariant mean on an abelian semigroup, Illinois J. Math. 3 (1959), 28-44. MR 21 #2184. MR 0103414 (21:2184)
  • [13] Theodore Mitchell, Constant functions and left invariant means on semigroups, Trans. Amer. Math. Soc. 119 (1965), 244-261. MR 33 #1743. MR 0193523 (33:1743)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0447970-5
Keywords: Semigroup, left thick, invariant means, radical, finite left ideal group
Article copyright: © Copyright 1977 American Mathematical Society

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