Cartan Subalgebras of Dimension Drop Interval Algebras and Their Inductive Limits

Abstract

We describe the conjugacy of non-degenerate Cartan subalgebras of the dimension drop interval algebras $M_m(\widetilde{\mathbb I}_p)$ in terms of double cosets of permutation groups, and describe their spectra by finite bi-partite multigraphs. All Cartan subalgebras are non-degenerate precisely when the two parameters are coprime. We also give splitting interval examples with prescribed finite numbers of Cartan conjugacy classes. Passing to inductive limits, we construct a unital simple AD algebra $\mathcal{A}$ of real rank zero and stable rank one, with $K_0(\mathcal{A})\cong\mathbb Z[1/3]$ and $K_1(\mathcal{A})\cong\mathbb Z/2\mathbb Z$. It contains infinitely many mutually isomorphic but pairwise nonconjugate $\mathrm{C}^*$-diagonals. Their spectra are all homeomorphic to the dense amalgam of circles, while orbit multiplicities in the Weyl groupoids distinguish their embeddings. The same AD algebra also contains infinitely many pairwise nonisomorphic Cartan subalgebras.