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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?

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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 expect there will be a trivial abstract characterization of the property in general.

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