Jonathan H. Brown

Cartan Subalgebras in C*-Algebras of Haus dorff étale Groupoids

The reduced C*-algebra of the interior of the isotropy in any Hausdorff étale groupoid G embeds as a C*-subalgebra M of the reduced C*-algebra of G. We prove that the set of pure states of M with unique extension is dense, and deduce that any representation of the reduced C*-algebra of G that is injective on M is faithful. We prove that there is a conditional expectation from the reduced C*-algebra of G onto M if and only if the interior of the isotropy in G is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, M is a Cartan subalgebra. We prove that for a large class of groupoids G with abelian isotropy—including all Deaconu–Renault groupoids associated to discrete abelian groups—M is a maximal abelian subalgebra. In the specific case of k-graph groupoids, we deduce that M is always maximal abelian, but show by example that it is not always Cartan.

Decomposing the C * -algebras of groupoid extensions

We decompose the full and reduced C*-algebras of an extension of a groupoid by the circle into a direct sum of twisted groupoid C*-algebras.

The Socle and Semisimplicity of a Kumjian–Pask Algebra

The Kumjian–Pask algebra KP(Λ) is a graded algebra associated to a higher-rank graph Λ and is a generalization of the Leavitt path algebra of a directed graph. We analyze the minimal left ideals of KP(Λ), and identify its socle as a graded ideal by describing its generators in terms of a subset of vertices of the graph. We characterize when KP(Λ) is semisimple, and obtain a complete structure theorem for a semisimple Kumjian–Pask algebra. As a consequence of this structure theorem, every semisimple Kumjian–Pask algebra can be obtained as a Leavitt path algebra of a directed graph.

The Brauer semigroup of a groupoid and a symmetric imprimitivity theorem

In this paper we define a monoid called the equivariant Brauer semigroup for a locally compact Hausdorff groupoid E whose elements consist of Morita equivalence classes of E-dynamical systems. This construction generalizes both the equivariant Brauer semigroup for transformation groups and the equivariant Brauer group for a groupoid. We show that groupoid equivalence induces an isomorphism of equivariant Brauer semigroups and that this isomorphism preserves the Morita equivalence classes of the respective crossedproducts, thus generalizing Raeburns symmetric imprimitivity theorem.

Centers of algebras associated to higher-rank graphs

The Kumjian–Pask algebras are path algebras associated to higher-rank graphs, and generalize the Leavitt path algebras. We study the center of a simple Kumjian–Pask algebra and characterize commutative Kumjian–Pask algebras