Approximately central matrix units and the structure of noncommutative tori

The property of approximate divisibility for C*-algebras is introduced and studied. Simple approximately divisible C*-algebras are shown to have nice nonstable K-theory properties. Non- rational noncommutative tori are shown to be approximately divisible. It follows that every simple noncommutative torus (in particular, every irrational rotation algebra) has stable rank one and real rank zero.