The dynamical Cuntz semigroup and ideal-free quotients of Cuntz semigroups

TL;DR

This paper introduces a new framework for quotients of W- and Cu-semigroups, including the dynamical Cuntz semigroup, enhancing understanding of their structure under group actions and crossed products.

Contribution

It develops a theory of quotients beyond ideals using normal pairs and defines the dynamical Cuntz semigroup as a universal object for group actions on W-semigroups.

Findings

Defined normal pairs for quotients of W-semigroups

Introduced the dynamical Cuntz semigroup as a universal object

Applied the theory to analyze crossed product C*-algebras

Abstract

We develop a theory of general quotients for W- and Cu-semigroups beyond the case of quotients by ideals. To this end, we introduce the notion of a normal pair, which allows us to take quotients of W-semigroups in a similar way as normal subgroups arise as kernels of group homomorphisms. We use this to define the dynamical Cuntz semigroup as the universal object induced from an action of a group G on a W-semigroup. In the C*-algebraic framework, under mild assumptions, the universality of this dynamical invariant helps us tap into the structure of the Cuntz semigroup of crossed product C*-algebras.