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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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