Some affine geometric aspects of operator algebras (Q1822732)

In earlier papers the authors introduced the concept of strongly facially symmetric Banach spaces. Examples of such Banach spaces are the preduals of \(JBW^*\)-triples and the duals of \(JB^*\)-triples. It follows that the preduals of \(W^*\)-algebras are also strongly facially symmetric. The authors show that the predual \(A_*\) of a \(W^*\)-algebra A has certain additional properties and go on to give a new proof of the polar decomposition theorem for elements of \(A_*\). The beauty of their approach is that it does not rely upon order-theoretic properties of the self-adjoint part of \(A_*\) but merely upon the structure of the unit ball in the complex Banach space \(A_*\). It is therefore susceptible to generalization.