We construct a simple, nuclear, stably projectionless C*‐algebra W which has trivial K‐theory and a unique tracial state, and we investigate the extent to which W might fit into the hierarchy of strongly self‐absorbing C*‐algebras as an analogue of the Cuntz algebra 𝒪2. In this context, we show that every non‐degenerate endomorphism of W is approximately inner and we construct a trace‐preserving embedding of W into the central sequences algebra M(W)∞∩W′. We conjecture that W⊗ W ≅ W and we note some implications of this, for example, that W would be absorbed tensorially by a certain class of nuclear, stably projectionless C*‐algebras. Finally, we explain why W may play some role in the classification of such algebras.

Abstract. We prove that in a simple, unital, exact, Z-stable C * -algebra of stable rank one, the distance between the unitary orbits of self-adjoint elements with connected spectrum is completely determined by spectral data. This fails without the assumption of Z-stability.

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We use order zero maps to express the Jiang-Su algebra Z as a universal C -algebra on countably many generators and relations, and we show that a natural deformation of these relations yields the stably projectionless algebra W studied by Kishimoto, Kumjian and others. Our presentation is entirely explicit and involves only -polynomial and order relations.

We realise the algebra W, the algebra Z 0 and the algebras Z 0 ⊗ A, where A is a unital UHF algebra as Fraïssé limits of suitable classes of structures. In doing so, we show that such algebras are generic objects without the use of any classification result.