It appears that there is such a condition for so-called Rickart C*-algebras (to each element a in the algebra there is a selfadjoint idempotent generating the left annihilator of a). This condition is mentioned in the first paragraph of the paper ["Polar decomposition in Rickart C*-algebras" by Dmitry Goldstein][1].

I don't expect there will be a trivial abstract characterization of the property in general. 

  [1]: http://dmle.cindoc.csic.es/pdf/PUBLICACIONSMATEMATIQUES_1995_39_01_01.pdf