A simple ${{C}^{*}}$-algebra is constructed for which the Murray-von Neumann equivalence classes of projections, with the usual addition—induced by addition of orthogonal projections—form the additive semigroup
$$\left\{ 0,\,2,\,3,\ldots \right\}.$$
(This is a particularly simple instance of the phenomenon of perforation of the ordered ${{K}_{0}}$-group, which has long been known in the commutative case—for instance, in the case of the four-sphere—and was recently observed by the second author in the case of a simple ${{C}^{*}}$-algebra.)