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

By Jonathan H. Brown, Gabriel Nagy, Sarah Reznikoff, Aidan Sims, Dana P. Williams. 2015

47 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3323c670206253f14b9a4cdd8ab3749b2897e93b

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In Computational Theory and Mathematics

1. Enrique Pardo
2. Aidan Sims
3. L. O. Clark
4. Charles Starling
5. R. Exel
6. Jack Spielberg
7. Mark V. Lawson

In Mathematical Physics

1. Enrique Pardo
2. Aidan Sims
3. R. Exel
4. L. O. Clark
5. Charles Starling
6. Ben Steinberg
7. Mark V. Lawson
8. P. Ara

Citations in the corpus

• Simplicity of algebras associated to non-Hausdorff groupoids (2018, background): “2 of [3] which we restate for convenience.”