  When S is a discrete subsemigroup of a discrete group G such that G = S^{-1}
S, it is possible to extend circle-valued multipliers from S to G; to dilate
(projective) isometric representations of S to (projective) unitary
representations of G; and to dilate/extend actions of S by injective
endomorphisms of a C*-algebra to actions of G by automorphisms of a larger
C*-algebra. These dilations are unique provided they satisfy a minimality
condition. The (twisted) semigroup crossed product corresponding to an action
of S is isomorphic to a full corner in the (twisted) crossed product by the
dilated action of G. This shows that crossed products by semigroup actions are
Morita equivalent to crossed products by group actions, making powerful tools
available to study their ideal structure and representation theory. The
dilation of the system giving the Bost-Connes Hecke C*-algebra from number
theory is constructed explicitly as an application: it is the crossed product
corresponding to the multiplicative action of the positive rationals on the
additive group of finite adeles.
