Inner amenable semigroups I
Joseph Ling · Journal of the Mathematical Society of Japan · 1997
Let $S$ be a semigroup.Let $m(S)$ denote the Banach space of bounded real-valued functions on $S$ .A mean on $m(S)$ is a bounded linear functional $\mu$ on $m(S)$ such that for all $f\in m(S),$ $\inf\{f(s):s\in S\}\leqq\mu(f)\leqq\sup\{f(s):s\in S\}$ .An equivalent formulation is that $||\mu||=\mu(1)=1$ , where 1 denotes the constant function on $S$ with value 1.A mean $\mu$ on $m(S)$ is said to be a left [respec- tively, right] invanant mean if $\mu(l_{s}f)$ [respectively, $\mu(r_{s}f)$ ] $=\mu(f)$ for all $s\in S$ and $f\in m(S)$ , where $l_{s}f$ and $r_{s}f$ are defined on $S$ by $(l_{s}f)(t)=f(st)$ and $(r_{s}f)(t)$ $=f(ts),$ $t\in S$ .A mean that is both a left invariant mean $(LIM)$ and a right invariant mean (RIM) is called an invariant mean.A semigroup which admits [respectively left, right] invariant means is called [respectively left, nght] amenable.We refer the reader to [4] for an introductory exposition on amenable semigroups.Many results we use without mention in this article concerning amenable semigroups can be found in this reference.We say that a mean $\mu$ on $m(S)$ is an inner invanant mean if $\mu(l_{s}f)=\mu(r_{s}f)$ for all $s\in S$ and $f\in m(S)$ .A semigroup $S$ which admits inner invariant means is called inner amenable.It follows immediately that every amenable semigroup is inner amenable.In particular, commutative semigroups are inner amenable.Another example of inner amenable semigroups is the class of semigroups with nonempty centres.Indeed, if $a\in S$ commutes with all $s\in S$ , then the point mean (or Dirac measure) $p_{a}$ defined by $p_{a}(f)=f(a)$ for all $f\in m(S)$ is an inner invariant mean on $m(S)$ .We note that if $S$ is a group, then an inner invariant mean is a mean for which $\mu(_{\sim}^{\gamma_{g}}f)=\mu(f)$ for all $s\in Sandf\in m(S)$ , where $(\mathfrak{T}_{s}f)(t)=f(sts^{-1})$ $=(f\circ\sigma_{*})(t)$