arXiv subjects
Publications and source records attributed to Matthew Gillespie.
Given a coaction $\delta$ of a locally compact group $G$ on a $\mathrm{C}^*$-algebra $A$, we study the relationship between two different forms of coaction invariance of ideals of $A$ and the ideals of the corresponding crossed product $\mathrm{C}^*$-algebra $A \rtimes_{\delta} G$. In particular, we characterize when these two notions of invariance are equivalent.
Given a regular $\mathrm{C}^{*}$-algebraic locally compact quantum group $(S_r,\Delta)$ with universal quantum group $(S_f,\Delta_f)$, a $\mathrm{C}^{*}$-algebra $A$, and a sufficiently well-behaved full coaction $S_f \overset{\alpha}{\curvearrowright} A$, we construct natural lattice isomorphisms from the strongly coaction invariant ideals of $A$ to the strongly coaction invariant ideals of full and reduced crossed product $\mathrm{C}^{*}$-algebras as an application of the `ladder technique' developed by the author, S. Kaliszewski, John Quigg and Dana P. Williams. In particular, these lattice isomorphisms are determined by either the maximality or normality of the coaction $\alpha$. This result directly generalizes a recent theorem proven by the aforementioned authors for locally compact groups, which in turn generalized a theorem of Elliot Gootman and Aldo Lazar for amenable groups.
We study topological quivers $Q$ admitting a free and proper action by a locally compact group $G$ together with their associated $C^*$-algebras. On the topological side, we provide a complete classification of topological quivers which admit such actions in terms of $G$-bundles over the vertex orbit space and an appropriate isomorphism of bundles over the edge orbits. Following the work by Deaconu, Kumjian, and Quigg on topological graphs, we construct an isomorphism between $C^*(Q/G)$ and Rieffel's fixed-point algebra $C^*(Q)^\alpha$, which is known to be Morita equivalent to $C^*(Q)\rtimes_rG$. Unlike the work with topological graphs, we use previously developed functoriality techniques to identify the isomorphism. We also examine many concrete examples of such group actions, including some exclusive to topological quivers, and the associated Morita equivalences.
We present a new method of establishing a bijective correspondence - in fact, a lattice isomorphism - between action- and coaction-invariant ideals of C*-algebras and their crossed products by a fixed locally compact group. It is known that such a correspondence exists whenever the group is amenable; our results hold for any locally compact group under a natural form of coaction invariance.