Day's fixed point theorem (Theorem 1.3.1; Lecture on amenability; Volker Runde)
Let $G$ be a locally compact group. The following are equivalent:
- $G$ is amenable.
- If $G$ acts (from left side) affinely on a compact, convex subset $K$ of a locally convex vector space $E$, i.e. $$g\cdot(tx+(1-t)y)=t(g\cdot x)+(1-t)(g\cdot y)\qquad (g\in G,x,y\in K,t\in [0,1])$$ such that $$G\times K\to K,\qquad (g,x)\mapsto g\cdot x$$ is separately continuous, then there is $x\in K$ such that $g\cdot x=x$ for all $g\in G$.
My question:
If $G$ is discrete amenable and $G$ acts form both sides affinely and $$g\cdot(a\cdot h)=(g\cdot a)\cdot h,\qquad (g,h\in G, a\in K),$$ and $$e\cdot a=a\cdot e=a, \qquad (a\in K),$$ where $e\in G$ is identity of $G$; then could we say that, there is $x\in K$ such that $x\cdot g=g\cdot x=x$ ??
If it is not true in general, which conditions can assure that it properly (I need infinite group)?