Partial C*-dynamical systems and the ideal structure of partial reduced crossed products

Abstract

In this thesis, we study C*-algebras stemming from partial C*-dynamical systems. We develop equivariant injective envelopes associated to such systems, which allow us to obtain canonical connections to enveloping actions as well as results on the ideal structure of partial crossed products.

We extend the theory of equivariant injective envelopes pioneered by Hamana in the 1980s to partial C*-dynamical systems. To do so, we introduce the category of generalized unital partial actions by allowing for partial *-automorphisms acting on families of special hereditary subalgebras. Utilizing properties of injective envelopes and the notion of an injective unitization of partial C*-dynamical systems, we argue that it suffices to consider unital objects in our category. This also allows us to connect our theory to Abadie’s notion of enveloping actions leading to a canonical relationship of their G-injective envelopes.

Utilizing properties of injective envelopes, we introduce novel non-triviality conditions for partial *-automorphisms inspired by global C*-dynamics. We contrast this notion with existing conditions in the literature. Lastly, we study a non-commutative generalization of stabilizer subgroups for pseudo-Glimm ideals. In particular, we show that for Glimm ideals in the G-injective envelope, these stabilizer subgroups are in fact amenable — a result which is crucial for our main theorems regarding the ideal structure of partial reduced crossed products.

Finally, our main application of the theory of G-injective envelopes is a characterization of the ideal intersection property for partial C*-dynamical systems subject to a cohomological condition as a generalization of the result for global group actions. To state this generalization, we utilize the dynamical conditions introduced previously and generalize the notion of equivariant pseudo-expectations to partial C*-dynamical systems. We also give a sufficient intrinsic condition in terms of non-commutative uniformly recurrent partial subsystems utilizing pseudo-Glimm ideals. As a consequence of our results, we obtain a full characterization of the ideal intersection property for partial actions on commutative C*-algebras in terms of freeness of the partial action on the spectrum of the G-injective envelope.