A very nice feature of W*-algebras is the following:

once you have an element $a$ of a W*-algebra $M$, and $a=u|a|$ (the polar decomposition), then $u\in M$.

It seems that it carries over to AW*-algebras without pain. This is simply because (A)W*-algebras have lots of projections, unlike general C*-algebras.

Can one give an abstract characterisation of C*-algebras with the above mentioned property?