17 research outputs found
Cartan subalgebras in C*-algebras of Hausdorff etale groupoids
The reduced -algebra of the interior of the isotropy in any Hausdorff
\'etale groupoid embeds as a -subalgebra of the reduced
-algebra of . We prove that the set of pure states of with unique
extension is dense, and deduce that any representation of the reduced
-algebra of that is injective on is faithful. We prove that there
is a conditional expectation from the reduced -algebra of onto if
and only if the interior of the isotropy in is closed. Using this, we prove
that when the interior of the isotropy is abelian and closed, is a Cartan
subalgebra. We prove that for a large class of groupoids with abelian
isotropy---including all Deaconu--Renault groupoids associated to discrete
abelian groups--- is a maximal abelian subalgebra. In the specific case of
-graph groupoids, we deduce that is always maximal abelian, but show by
example that it is not always Cartan.Comment: 14 pages. v2: Theorem 3.1 in v1 incorrect (thanks to A. Kumjain for
pointing out the error); v2 shows there is a conditional expectation onto
iff the interior of the isotropy is closed. v3: Material (including some
theorem statements) rearranged and shortened. Lemma~3.5 of v2 removed. This
version published in Integral Equations and Operator Theor
Twisted k-graph algebras associated to Bratteli diagrams
Given a system of coverings of k-graphs, we show that the cohomology of the
resulting (k+1)-graph is isomorphic to that of any one of the k-graphs in the
system. We then consider Bratteli diagrams of 2-graphs whose twisted
C*-algebras are matrix algebras over noncommutative tori. For such systems we
calculate the ordered K-theory and the gauge-invariant semifinite traces of the
resulting 3-graph C*-algebras. We deduce that every simple C*-algebra of this
form is Morita equivalent to the C*-algebra of a rank-2 Bratteli diagram in the
sense of Pask-Raeburn-R{\o}rdam-Sims.Comment: 28 pages, pictures prepared using tik
Subsets of vertices give Morita equivalences of Leavitt path algebras
DOI:
10.1017/S000497271700024