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DILATIONS OF INTERACTION GROUPS THAT EXTEND ACTIONS OF ORE SEMIGROUPS

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  30 January 2018

FERNANDO ABADIE
FERNANDO ABADIE*
Affiliation:
Centro de Matemática-FC, Universidad de la República. 11 400, Iguá 4225, Montevideo, Uruguay email fabadie@cmat.edu.uy
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Abstract

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We show that if a unital injective endomorphism of a $C^{\ast }$ -algebra admits a transfer operator, then both of them are compressions of mutually inverse automorphisms of a bigger algebra. More generally, every interaction group – in the sense of Exel – extending an action of an Ore semigroup by injective unital endomorphisms of a $C^{\ast }$ -algebra, admits a dilation to an action of the corresponding enveloping group on another unital $C^{\ast }$ -algebra, of which the former is a $C^{\ast }$ -subalgebra: the interaction group is obtained by composing the action with a conditional expectation. The dilation is essentially unique if a certain natural condition of minimality is imposed, and it is faithful if and only if the interaction group is also faithful.

Keywords

transfer operatorconditional expectationinteraction group

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L55: Noncommutative dynamical systems

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 104 , Issue 2 , April 2018 , pp. 145 - 161
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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