Dilations of semigroup crossed products as crossed products of dilations

TL;DR

This paper demonstrates that the minimal automorphic dilation of a semigroup crossed product with inner endomorphisms is the group crossed product of the original dilation, extending Laca's construction and applying to recent examples.

Contribution

It establishes a new connection between semigroup and group crossed products via minimal automorphic dilations, generalizing previous results by Laca.

Findings

The minimal automorphic dilation of the semigroup crossed product is the group crossed product of the original dilation.

Applications include recent examples studied by Cuntz and the second author.

Provides a framework for understanding dilations in semigroup dynamical systems.

Abstract

Laca constructed a minimal automorphic dilation for every semigroup dynamical system arising from an action of an Ore semigroup by injective endomorphisms of a unital $C^*$-algebra. Here we show that the semigroup crossed product with its action by inner endomorphisms given by the implementing isometries has as minimal automorphic dilation the group crossed product of the original dilation. Applications include recent examples studied by Cuntz and the second named author.