Using tempered Følner sequences one may show a pointwise ergodic theorem for amenable groups.

(see http://www.aimsciences.org/journals/pdfsnews.jsp?paperID=2413&mode=full)

Is there a similar result for semigroups? Where by taking a special kind of Følner sequence you can achieve this. (Either for $L^{1}$ or $L^{2}$)