  The paper deals with continuous homomorphisms $S \ni s \mapsto T_s \in L(E)$
of amenable semigroups $S$ into the algebra $L(E)$ of all bounded linear
operators on a Banach space $E$. For a closed linear subspace $F$ of $E$,
sufficient conditions are given under which there exists a projection $P \in
L(E)$ onto $F$ that commutes with all $T_s$. And when $E$ is a Hilbert space,
sufficient conditions are given for the existence of an invertible operator $R
\in L(E)$ such that all $R T_s R^{-1}$ are isometries. Also certain results on
extending intertwining operators, renorming as well as on operators on
hereditarily indecomposable Banach spaces are offered.
