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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If *S* is a left amenable semigroup, let denote the dimension of the set of left invariant means on *S*. Theorem. *If S is left amenable, then* *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 is at least the cardinality of *S*. An application of these results shows that the radical of the second conjugate algebra of is infinite dimensional when *S* is a left amenable semigroup which does not contain a finite ideal.

**[1]**Ching Chou,*Minimal sets and ergodic measures for 𝛽𝑁\𝑁*, Illinois J. Math.**13**(1969), 777–788. MR**0249569****[2]**Ching Chou,*On the size of the set of left invariant means on a semi-group*, Proc. Amer. Math. Soc.**23**(1969), 199–205. MR**0247444**, https://doi.org/10.1090/S0002-9939-1969-0247444-1**[3]**Ching Chou,*The exact cardinality of the set of invariant means on a group*, Proc. Amer. Math. Soc.**55**(1976), no. 1, 103–106. MR**0394036**, https://doi.org/10.1090/S0002-9939-1976-0394036-3**[4]**Paul Civin and Bertram Yood,*The second conjugate space of a Banach algebra as an algebra*, Pacific J. Math.**11**(1961), 847–870. MR**0143056****[5]**Mahlon M. Day,*Amenable semigroups*, Illinois J. Math.**1**(1957), 509–544. MR**0092128****[6]**Lonnie Fairchild,*Extreme invariant means without minimal support*, Trans. Amer. Math. Soc.**172**(1972), 83–93. MR**0308685**, https://doi.org/10.1090/S0002-9947-1972-0308685-2**[7]**E. Granirer,*On amenable semigroups with a finite-dimensional set of invariant means. I*, Illinois J. Math.**7**(1963), 32–48. MR**0144197****[8]**Edmond Granirer,*A theorem on amenable semigroups*, Trans. Amer. Math. Soc.**111**(1964), 367–379. MR**0166597**, https://doi.org/10.1090/S0002-9947-1964-0166597-7**[9]**E. Granirer,*Extremely amenable semigroups*, Math. Scand.**17**(1965), 177–197. MR**0197595**, https://doi.org/10.7146/math.scand.a-10772**[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**0185457**, https://doi.org/10.7146/math.scand.a-10740**[11]**E. Hewitt and K. Ross,*Abstract harmonic analysis*, Vol. 1, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR**28**#158.**[12]**Indar S. Luthar,*Uniqueness of the invariant mean on an abelian semigroup*, Illinois J. Math.**3**(1959), 28–44. MR**0103414****[13]**Theodore Mitchell,*Constant functions and left invariant means on semigroups*, Trans. Amer. Math. Soc.**119**(1965), 244–261. MR**0193523**, https://doi.org/10.1090/S0002-9947-1965-0193523-8

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
43A07

Retrieve articles in all journals with MSC: 43A07

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