Essential Cartan subalgebras of $C^*$-algebras
DOI:
https://doi.org/10.7146/math.scand.a-160113Abstract
We define essential Cartan pairs of $C^*$-algebras generalising the definition of Renault [Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. (2008), no. 61, 29–63] and show that such pairs are given by essential twisted groupoid $C^*$-algebras as defined by Kwaśniewski and Meyer [Essential crossed products by inverse semigroup actions: Simplicity and pure infiniteness, Doc. Math. 26 (2021), 271–335]. We show that the underlying twisted groupoid is effective, and is unique up to isomorphism among twists over effective groupoids giving rise to the essential Cartan pair. We also show that for twists over effective groupoids giving rise to such pairs, the automorphism group of the twist is isomorphic to the automorphism group of the induced essential Cartan pair via explicit constructions.
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