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CARTAN SUBALGEBRAS IN DIMENSION DROP ALGEBRAS

Published online by Cambridge University Press:  13 June 2019


Selçuk Barlak
Affiliation:
Abteilung für Mathematik und Ihre Didaktik, Auf dem Campus 1b, DE-24943 Flensburg, Germany (selcuk.barlak@uni-flensburg.de)
Sven Raum
Affiliation:
Stockholm University, Department of Mathematics, Kräftriket 6, SE-106 91 Stockholm, Sweden (raum@math.su.se)
Corresponding

Abstract

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.


Type
Research Article
Creative Commons
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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