## Functional analytic properties of topological semigroups and $n$-extreme amenability

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- by Anthony To-ming Lau
- Trans. Amer. Math. Soc.
**152**(1970), 431-439 - DOI: https://doi.org/10.1090/S0002-9947-1970-0269772-9
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## Abstract:

Let $S$ be a topological semigroup, $\operatorname {LUC} (S)$ be the space of left uniformly continuous functions on $S$, and $\Delta (S)$ be the set of multiplicative means on $\operatorname {LUC} (S)$. If $( \ast )\operatorname {LUC} (S)$ has a left invariant mean in the convex hull of $\Delta (S)$, we associate with $S$ a*unique*finite group $G$ such that for any maximal proper closed left translation invariant ideal $I$ in $\operatorname {LUC} (S)$, there exists a linear isometry mapping $\operatorname {LUC} (G)/I$ one-one onto the set of bounded real functions on $G$. We also generalise some recent results of T. Mitchell and E. Granirer. In particular, we show that $S$ satisfies $( \ast )$ iff whenever $S$ is a jointly continuous action on a compact hausdorff space $X$, there exists a nonempty finite subset $F$ of $X$ such that $sF = F$ for all $s \in S$. Furthermore, a discrete semigroup $S$ satisfies $( \ast )$ iff whenever $\{ {T_s};s \in S\}$ is an antirepresentation of $S$ as linear maps from a norm linear space $X$ into $X$ with $||{T_s}|| \leqq 1$ for all $s \in S$, there exists a finite subset $\sigma \subseteq S$ such that the distance (induced by the norm) of $x$ from ${K_X} = \text {linear span}$ of $\{ x - {T_s}x;x \in X,s \in S\}$ in $X$ coincides with distance of $O(\sigma ,x) = \{ (1/|\sigma |)\sum \nolimits _{a \in \sigma } {{T_{at}}(x);t \in S\} }$ from 0 for all $x \in X$.

## References

- F. F. Bonsall, J. Lindenstrauss, and R. R. Phelps,
*Extreme positive operators on algebras of functions*, Math. Scand.**18**(1966), 161–182. MR**209863**, DOI 10.7146/math.scand.a-10789 - Mahlon M. Day,
*Amenable semigroups*, Illinois J. Math.**1**(1957), 509–544. MR**92128** - Mahlon M. Day,
*Semigroups and amenability*, Semigroups (Proc. Sympos., Wayne State Univ., Detroit, Mich., 1968) Academic Press, New York, 1969, pp. 5–53. MR**0265502** - Nelson Dunford and Jacob T. Schwartz,
*Linear Operators. I. General Theory*, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR**0117523** - Edmond Granirer,
*A theorem on amenable semigroups*, Trans. Amer. Math. Soc.**111**(1964), 367–379. MR**166597**, DOI 10.1090/S0002-9947-1964-0166597-7 - E. Granirer,
*Extremely amenable semigroups*, Math. Scand.**17**(1965), 177–197. MR**197595**, DOI 10.7146/math.scand.a-10772 - E. Granirer,
*Extremely amenable semigroups. II*, Math. Scand.**20**(1967), 93–113. MR**212551**, DOI 10.7146/math.scand.a-10825 - Edmond E. Granirer,
*Functional analytic properties of extremely amenable semigroups*, Trans. Amer. Math. Soc.**137**(1969), 53–75. MR**239408**, DOI 10.1090/S0002-9947-1969-0239408-3 - E. Granirer and Anthony T. Lau,
*Invariant means on locally compact groups*, Illinois J. Math.**15**(1971), 249–257. MR**277667** - Frederick P. Greenleaf,
*Invariant means on topological groups and their applications*, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto-London, 1969. MR**0251549** - Edwin Hewitt and Kenneth A. Ross,
*Abstract harmonic analysis. Vol. I*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR**551496** - Anthony To-ming Lau,
*Topological semigroups with invariant means in the convex hull of multiplicative means*, Trans. Amer. Math. Soc.**148**(1970), 69–84. MR**257260**, DOI 10.1090/S0002-9947-1970-0257260-5 - E. S. Ljapin,
*Semigroups*, Translations of Mathematical Monographs, Vol. 3, American Mathematical Society, Providence, R.I., 1963. MR**0167545** - Theodore Mitchell,
*Fixed points and multiplicative left invariant means*, Trans. Amer. Math. Soc.**122**(1966), 195–202. MR**190249**, DOI 10.1090/S0002-9947-1966-0190249-2 - Theodore Mitchell,
*Function algebras, means, and fixed points*, Trans. Amer. Math. Soc.**130**(1968), 117–126. MR**217577**, DOI 10.1090/S0002-9947-1968-0217577-8 - Theodore Mitchell,
*Topological semigroups and fixed points*, Illinois J. Math.**14**(1970), 630–641. MR**270356** - I. Namioka,
*On certain actions of semi-groups on $L$-spaces*, Studia Math.**29**(1967), 63–77. MR**223863**, DOI 10.4064/sm-29-1-63-77
J. Sorenson,

*Existence of measures that are invariant under a semi-group of transformations*, Thesis, Purdue University, Lafayette, Ind., 1966.

## Bibliographic Information

- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**152**(1970), 431-439 - MSC: Primary 22.05; Secondary 46.00
- DOI: https://doi.org/10.1090/S0002-9947-1970-0269772-9
- MathSciNet review: 0269772