  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 O_2. In this context, we show that every
nondegenerate endomorphism of W is approximately inner and we construct a
trace-preserving embedding of W into the central sequences algebra M(W)_\infty
\cap W'. We conjecture that W\otimes W is isomorphic to 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.
