DILATIONS OF INTERACTION GROUPS THAT EXTEND ACTIONS OF ORE SEMIGROUPS

Abstract

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.