A stably finite analogue of the Cuntz algebra O2

Abstract

The Elliott Programme seeks classification of simple, separable, nuclear $C^*$-algebras via a functor based on $K$-theory. There are a handful of $C^*$-algebras, including the Cuntz algebras $\mathcal{O}_2$ and $\mathcal{O}_\infty$, that play particularly important roles in the programme. It is principally in this context that the Jiang-Su algebra $\mathcal{Z}$ is regarded as an analogue of $\mathcal{O}_\infty$, and this thesis proposes an analogue of $\mathcal{O}_2$ in a similar fashion. More specifically, we construct a simple, nuclear, stably projectionless $C^*$-algebra $W$ which has trivial $K$-theory and a unique tracial state, and we prove that $W$ shares some of the properties of the $C^*$-algebras named above. In particular, we show that every trace-preserving endomorphism of $W$ is approximately inner, and that $W$ admits a trace-preserving embedding into the central sequences algebra $M(W)\cap W'$. While we do not quite prove that $W\otimes W \cong W$, we show how this can be deduced from a conjectured generalization of an existing classification theorem. Assuming this conjecture, we also show that $W$ is absorbed tensorially by a large class of $C^*$-algebras with trivial $K$-theory. Finally, we provide presentations of both $\mathcal{Z}$ and $W$ as universal $C^*$-algebras, leading us to suggest that, in addition to its position as a stably finite analogue of $\mathcal{O}_2$, $W$ may be also thought of, both intrinsically and extrinsically, as a stably projectionless analogue of $\mathcal{Z}$