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CARTAN SUBALGEBRAS IN DIMENSION DROP ALGEBRAS

Part of: Selfadjoint operator algebras Combinatorics

Published online by Cambridge University Press:  13 June 2019

Selçuk Barlak  and
Sven Raum Open the ORCID record for Sven Raum [Opens in a new window]
Selçuk Barlak
Affiliation:
Abteilung für Mathematik und Ihre Didaktik, Auf dem Campus 1b, DE-24943Flensburg, Germany (selcuk.barlak@uni-flensburg.de)
Sven Raum
Affiliation:
Stockholm University, Department of Mathematics, Kräftriket 6, SE-106 91Stockholm, Sweden (raum@math.su.se)
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Abstract

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We completely classify Cartan subalgebras of dimension drop algebras with coprime parameters. More generally, we classify Cartan subalgebras of arbitrary stabilised dimension drop algebras that are non-degenerate in the sense that the dimensions of their fibres in the endpoints are maximal. Conjugacy classes by an automorphism are parametrised by certain congruence classes of matrices over the natural numbers with prescribed row and column sums. In particular, each dimension drop algebra admits only finitely many non-degenerate Cartan subalgebras up to conjugacy. As a consequence of this parametrisation, we can provide examples of subhomogeneous $\text{C}^{\ast }$-algebras with exactly $n$ Cartan subalgebras up to conjugacy. Moreover, we show that in many dimension drop algebras two Cartan subalgebras are conjugate if and only if their spectra are homeomorphic.

Keywords

Cartan subalgebraC∗-diagonaldimension drop algebrasubhomogeneous C∗-algebrasenumeration

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L85: Noncommutative topology 05A15: Exact enumeration problems, generating functions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 3 , May 2021 , pp. 725 - 755
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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