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?



Polar decomposition in C*-algebras
There exist further generalizations but they do not go very far from AW\*-algebras. See


[Link](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-137/issue-1/Three-quavers-on-unitary-elements-in-C-algebras/pjm/1102650543.full)
and


<http://dmle.cindoc.csic.es/pdf/PUBLICACIONSMATEMATIQUES_1995_39_01_01.pdf>


