Positive Reynolds operators and generating derivations

It is shown that the spectrum of a positive Reynolds operator on C-0(X) is contained in the disc centered at a with radius 1/2. Moreover, every positi ve Reynolds operator T with dense range is injective. In this case, the ope rator D = I - T-1 is a densely defined derivation, which generates a one-pa rameter semigroup of algebra homomorphisms. This semigroup yields an integr al representation of T. Along the way, it is proved that a densely defined closed derivation D generates a semigroup if, and only if, R(1, D) exists a nd is a positive operator.