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CONDITION (K) FOR BOOLEAN DYNAMICAL SYSTEMS

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  27 May 2021

TOKE MEIER CARLSEN Open the ORCID record for TOKE MEIER CARLSEN [Opens in a new window]  and
EUN JI KANG Open the ORCID record for EUN JI KANG [Opens in a new window]
TOKE MEIER CARLSEN
Affiliation:
Department of Sciences and Technology, University of the Faroe Islands, Vestara Bryggja 15, FO-100Tórshavn, Faroe Islands e-mail: toke.carlsen@gmail.com
EUN JI KANG*
Affiliation:
Research Institute of Mathematics, Seoul National University, Seoul08826, Korea
*
e-mail: kkang33@snu.ac.kr
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Abstract

We generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system $({{\mathcal {B}}},{{\mathcal {L}}},\theta )$ with countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ satisfies Condition (K) if and only if every ideal of its $C^*$ -algebra is gauge-invariant, if and only if its $C^*$ -algebra has the (weak) ideal property, and if and only if its $C^*$ -algebra has topological dimension zero. As a corollary we prove that if the $C^*$ -algebra of a locally finite Boolean dynamical system with ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ countable either has real rank zero or is purely infinite, then $({{\mathcal {B}}}, {{\mathcal {L}}}, \theta )$ satisfies Condition (K). We also generalize the notion of maximal tails from directed graph to Boolean dynamical systems and use this to give a complete description of the primitive ideal space of the $C^*$ -algebra of a locally finite Boolean dynamical system that satisfies Condition (K) and has countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ .

Keywords

C*-algebras of Boolean dynamical systemsCondition (K)gauge-invariant idealsgraph C*-algebrasideal propertymaximal tailprimitive ideal spacereal rank zerotopological dimension zeroultrafilter cycle

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L55: Noncommutative dynamical systems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 2 , April 2022 , pp. 145 - 169
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Copyright
© 2021 Australian Mathematical Publishing Association Inc

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Footnotes

Communicated by Aidan Sims

Research partially supported by NRF-2017R1D1A1B03030540.

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