Extensions of semigroups of operators

Let T be a representation of an abelian semigroup S on a Banach space X. We identify a necessary and sufficient condition, which we name superexpansiv eness, for T to have an extension to a representation U on a Banach space Y containing X such that each U(t) (t is an element of S) has a contractive inverse, Although there are many such extensions (Y, U) in general, there i s a unique one which has a certain universal property. The spectrum of this extension coincides with the unitary part of the spectrum of T, so various results in spectral theory of group represent at ions can be extended to s uperexpansive representations.