Journal of Operator Theory

Volume 96, Issue 1, Summer 2026 pp. 215-256.

$K$-theory of $C^*$-algebras arising from commuting Hilbert bimodules and invariant ideals

Authors: Astrid an Huef (1), Abraham C.S. Ng (2), Aidan Sims (3)
Author institution: (1) School of Mathematics and Statistics, Victoria University of Wellington, Wellington 6140, New Zealand
(2) School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
(3) School of Mathematics and Statistics, University of New South Wales, NSW 2052, Australia

Summary: We study the $K$-theory of the Cuntz-Nica-Pimsner $C^*$-algebra of a rank-two product system that is an extension determined by an invariant ideal of the coefficient algebra. We use a construction of Deaconu and Fletcher that describes the Cuntz-Nica-Pimsner $C^*$-algebra of the product system in terms of two iterations of Pimsner's original construction of a $C^*$-algebra from a right-Hilbert bimodule. We apply our results to the product system built from two commuting surjective local homeomorphisms of a totally disconnected space, where the Cuntz-Nica-Pimsner $C^*$-algebra is isomorphic to the $C^*$-algebra of the associated rank-two Deaconu-Renault groupoid. We then apply a theorem of Spielberg about stable finiteness of an extension to obtain sufficient conditions for stable finiteness of the $C^*$-algebra of the Deaconu-Renault groupoid.

DOI: http://dx.doi.org/10.7900/jot.2025mar30.2493
Keywords: extension of a $C^*$-algebra, $K$-theory, Hilbert bimodule, product system, Pimsner exact sequence, stably finite $C^*$-algebra, Deaconu-Renault groupoid

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