Similarity problems, Folner sets and isometric representations of amenable semigroups

We revisit Sz.-Nagy's criteria for similarity of Hilbert space bounded linear operators to isometries or unitaries and present new ones. We also discuss counterparts of the Dixmier-Day theorem concerning bounded representations of amenable groups and semigroups. We highlight the role of Folner sets in similarity problems in both settings of unimodular, $\sigma$-compact, amenable groups and in discrete semigroups possessing the Strong Folner condition (SFC).