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DILATIONS OF INTERACTION GROUPS THAT EXTEND ACTIONS OF ORE SEMIGROUPS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 104 / Issue 2 / April 2018
- Published online by Cambridge University Press:
- 30 January 2018, pp. 145-161
- Print publication:
- April 2018
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- Article
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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.
