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From Endomorphisms to Automorphisms and Back: Dilations and Full Corners
Author(s) -
Laca Marcelo
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799008492
Subject(s) - mathematics , crossed product , endomorphism , automorphism , pure mathematics , group (periodic table) , discrete group , semigroup , algebra over a field , action (physics) , chemistry , physics , organic chemistry , quantum mechanics
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 C 0 (A f )⋊Q* + , corresponding to the multiplicative action of the positive rationals on the additive group A f of finite adeles.