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 superexpansiveness, for T to have an extension to a representation U on a Danach space Y containing X such that each U(t) (t ∈ S) has a contractive inverse. Although there are many such extensions (Y, U) in general, there is a unique one which has a certain universal property. The spectrum of this extension coincides with the unitary part of the spectrum of ...