Isomorphisms in K-theory from isomorphisms in groupoid homology theories

Miller, Alistair

doi:10.1090/tran/9373

Isomorphisms in K-theory from isomorphisms in groupoid homology theories
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by Alistair Miller;

Trans. Amer. Math. Soc. 378 (2025), 3349-3391

DOI: https://doi.org/10.1090/tran/9373

Published electronically: January 30, 2025

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Abstract:

We prove that for torsion-free amenable ample groupoids, an isomorphism in groupoid homology induced by an étale correspondence yields an isomorphism in the K-theory of the associated $\mathrm {C}^\ast$-algebras. We apply this to extend X. Li’s K-theory formula for left regular inverse semigroup $\mathrm {C}^\ast$-algebras. These results are obtained by developing the functoriality of the ABC spectral sequence.

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Isomorphisms in K-theory from isomorphisms in groupoid homology theories
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Abstract: