A Stably Finite Analogue of the Cuntz Algebra O 2

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 O2 and O∞, that play particularly important roles in the programme. It is principally in this context that the Jiang-Su algebra Z is regarded as an analogue of O∞, and this thesis proposes an analogue of O2 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 tracepreserving embedding into the central sequences algebra M(W ) ∩W ′. While we do not quite prove that W ⊗ W ∼= 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 Z and W as universal C∗-algebras, leading us to suggest that, in addition to its position as a stably finite analogue of O2, W may be also thought of, both intrinsically and extrinsically, as a stably projectionless analogue of Z.