We discuss noncommutative solitons on a noncommutative torus and their appl
ication to tachyon condensation. In the large B limit, they can be exactly
described by the Powers-Rieffel projection operators known in the mathemati
cal literature. The resulting soliton spectrum is consistent with T duality
and is surprisingly interesting. It is shown that an instability arises fo
r any D-branes, leading to the decay into many smaller D-branes. This pheno
menon is the consequence of the fact that the K-homology for a type II von
Neumann factor is labeled by R.