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Affine geometric proofs of the Banach Stone theorems of Kadison and Kaup. (English) Zbl 0738.47029
R. V. Kadison, Ann. Math. (2) 54, 325–338 (1951; Zbl 0045.06201), proved that a surjective linear isometry \(T\) between two unital \(C^*\)-algebras \(A\), \(B\) is of the form \(Tx=u\cdot\rho(x)\), \(x\in A\), where \(u\) is a unitary element of \(B\) and \(\rho: A\to B\) is a Jordan isomorphism. Using the complicated machinery of infinite dimensional holomorphy, W. Kaup, Math. Ann. 228, 39–64 (1977; Zbl 0335.58005), extended this result proving that every surjective linear isometry \(T\) between two \(JB^*\)-triples is a \(JB^*\)-triple isomorphism.
The aim of this paper is to give an elementary proof of Kaup’s result based on the affine geometric properties of faces (= extremal convex subsets) in the state space together with analogs of standard operator tools as spectral, polar and Jordan decompositions, biduals and a theorem of Effros.

MSC:
47C15 Linear operators in \(C^*\)- or von Neumann algebras
46E40 Spaces of vector- and operator-valued functions
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
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