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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.

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