Journal of Operator Theory

Volume 93, Issue 2, Spring 2025 pp. 511-539.

A Cuntz-Krieger uniqueness theorem for $C^*$-algebras of relative generalized Boolean dynamical systems

Authors: Toke Meier Carlsen (1), Eun Ji Kang (2)
Author institution: (1) Department of Science and Technology, University of the Faroe Islands, Vestara Bryggja 15, FO-100 Torshavn, The Faroe Islands
(2) Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea

Summary: We prove a version of the Cuntz-Krieger uniqueness theorem for $C^*$-algebras of arbitrary relative generalized Boolean dynamical systems. We then describe properties of a $C^*$-algebra of a relative generalized Boolean dynamical system when the underlying Boolean dynamical system satisfies condition (K). We also define a notion of minimality of a Boolean dynamical system and give sufficient and necessary conditions for the minimality. Using these results, we characterize the generalized Boolean dynamical systems whose $C^*$-algebra is simple.

DOI: http://dx.doi.org/10.7900/jot.2023jun19.2451
Keywords: spectral synthesis, Toeplitz algebras, Fock spaces

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