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 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.
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