1999/11/18 by Marcelo Laca, Laca, Marcelo · 1 citation
Mathematics · #46L55 #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:46L55
paper · pdf · doi:10.48550/arxiv.math/9911135
11 pages
arxiv created 1999/11/18 · arxiv updated 2009/11/30
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.