Cartan subalgebras in dimension drop algebras

TL;DR

This paper classifies Cartan subalgebras in dimension drop algebras, revealing finite conjugacy classes and linking conjugacy to spectrum homeomorphism, advancing understanding of their structure.

Contribution

It provides a complete classification of Cartan subalgebras in dimension drop algebras, including parametrization and conjugacy criteria.

Findings

Finite conjugacy classes of Cartan subalgebras in these algebras.

Conjugacy determined by spectrum homeomorphism in many cases.

Explicit examples of subhomogeneous C*-algebras with a fixed number of Cartan subalgebras.

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 C*-algebras with exactly n Cartan subalgebras up to conjugacy. Moreover, we show that in many dimension drop algebras, Cartan subalgebras are conjugate if and only if their spectrum is homeomorphic.