arXiv subjects
Publications and source records attributed to Philipp Sibbel.
We construct C*-diagonals in all UCT Kirchberg algebras. Together with known constructions of C*-diagonals in the stably finite setting by Xin Li, this shows that every classifiable C*-algebra (in the sense of the Elliott classification programme) has a C*-diagonal. In fact, we show that every UCT Kirchberg algebra admits an untwisted principal étale groupoid model with a finite-dimensional unit space.
We show that every stable UCT Kirchberg algebra has a principal étale groupoid model, and thus contains a C$^*$-diagonal. Every unital UCT Kirchberg algebra $A$ for which $[1_A]_0$ has infinite order in $K_0(A)$ is also covered by our methods. In particular, we obtain a principal étale groupoid model for the Cuntz algebra $\mathcal{O}_\infty$.
We prove that there exists a C*-diagonal with Cantor spectrum in the Cuntz algebra $\mathcal{O}_k$ for $2 \le k < \infty$. Our method generalises to an uncountable family of UCT Kirchberg algebras with distinct K-theory. Moreover, we construct principal étale groupoid models for these Cuntz algebras and UCT Kirchberg algebras.
We compute the dynamic asymptotic dimension of the principal groupoid models for the Cuntz algebras $\mathcal{O}_k$ for $2 \leq k < \infty$ that have arisen from work of Winter and the authors. Our method generalises to a wide class of Deaconu-Renault groupoids. As an application of our results, we prove that $\mathcal{O}_2$ has infinitely many non-conjugate C$^*$-diagonals with Cantor spectrum, and we generalise this result to other Cuntz algebras by combining the main result with work of Kopsacheilis-Winter and Brown-Clark-Sierakowski-Sims.
We prove the existence of a C*-diagonal in the Cuntz algebra O_2 with spectrum homeomorphic to the Cantor space.