A generalized Cuntz–Krieger uniqueness theorem for higher-rank graphs

Brown, Jonathan H.; Nagy, Gabriel; Reznikoff, Sarah

doi:10.1016/j.jfa.2013.08.020

Cited by 24 publications

(64 citation statements)

References 15 publications

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“…First an element a ∈ KP R (Λ) is said to be normal if aa * = a * a. The proof of the following proposition follows exactly as the one given in [6,Proposition 4.1]. Note that in [6, Proposition 4.1], they write P α for the projection s α s α * .…”

Section: The Cycline Subalgebramentioning

confidence: 96%

“…Remark 5.1. In this remark we recall some properties of F α,β and T Λ , which are given in [6,Section 5].…”

Section: General Uniqueness Theorem For Kumjian-pask Algebrasmentioning

confidence: 99%

“…An infinite path in E is called essentially aperiodic if it is either aperiodic or of the form µccc · · · , where c is a cycle without entry (see [9,Definition 3.5 (B)] for more details). In [6,Example 5.4], it is seen that T E is the set of essentially aperiodic paths of E.…”

Section: General Uniqueness Theorem For Kumjian-pask Algebrasmentioning

confidence: 99%

“…In [6], a more general version of the Cuntz-Krieger uniqueness theorem is proved in the C * -algebraic setting that has no additional hypotheses on the homomorphism or the graph. Here we translate to Kumjian-Pask algebras the analytic result given in [6]: a representation of KP R (Λ) is injective if and only if it is injective on a distinguished subalgebra, called the cycline subalgebra.…”

Section: Introductionmentioning

confidence: 99%

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The cycline subalgebra of a Kumjian-Pask algebra

Clark

Canto

Nasr-Isfahani

2016

Proc. Amer. Math. Soc.

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Section: The Cycline Subalgebramentioning

confidence: 96%

“…Remark 5.1. In this remark we recall some properties of F α,β and T Λ , which are given in [6,Section 5].…”

Section: General Uniqueness Theorem For Kumjian-pask Algebrasmentioning

confidence: 99%

Section: General Uniqueness Theorem For Kumjian-pask Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The cycline subalgebra of a Kumjian-Pask algebra

Clark

Canto

Nasr-Isfahani

2016

Proc. Amer. Math. Soc.

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“…such that the following hold: (2) . Elements satisfying u = u 2 ∈ G are called units of G and the set of all such units is denoted G (0) ⊂ G and called the unit space of G. There are maps r, s :…”

Section: Invariant States In Theétale Groupoid Frameworkmentioning

confidence: 99%

Traces Arising From Regular Inclusions

Crytser¹,

Nagy²

2016

J. Aust. Math. Soc.

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Abstract. We study the problem of extending a state on an abelian C * -subalgebra to a tracial state on the ambient C * -algebra. We propose an approach that is well-suited to the case of regular inclusions, in which there is a large supply of normalizers of the subalgebra. Conditional expectations onto the subalgebra give natural extensions of a state to the ambient C * -algebra; we prove that these extensions are tracial states if and only if certain invariance properties of both the state and conditional expectations are satisfied. In the example of a groupoid C * -algebra, these invariance properties correspond to invariance of associated measures on the unit space under the action of bisections. Using our framework, we are able to completely describe the tracial state space of a Cuntz-Krieger graph algebra. Along the way we introduce certain operations called graph tightenings, which both streamline our description and provides connections to related finiteness questions in graph C * -algebras. Our investigation has close connections with the so-called unique state extension property and its variants.

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

Brown

Nagy

Reznikoff

et al. 2016

Integr. Equ. Oper. Theory

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Abstract. The reduced C * -algebra of the interior of the isotropy in any Hausdorf etale 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.

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A generalized Cuntz–Krieger uniqueness theorem for higher-rank graphs

Cited by 24 publications

References 15 publications

The cycline subalgebra of a Kumjian-Pask algebra

The cycline subalgebra of a Kumjian-Pask algebra

Traces Arising From Regular Inclusions

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

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