Cancellation Does not Imply Stable Rank One

A unital C * ‐algebra A is said to have cancellation of projections if the semigroup D ( A ) of Murray–von Neumann equivalence classes of projections in matrices over A is cancellative. It has long been known that stable rank one implies cancellation for any A , and some partial converses have been established. In this paper it is proved that cancellation does not imply stable rank one for simple, stably finite C * ‐algebras. 2000 Mathematics Subject Classification 46L80 (primary), 46L85 (secondary).