an:00007176
Zbl 0738.47029
Dang, T.; Friedman, Y.; Russo, B.
Affine geometric proofs of the Banach Stone theorems of Kadison and Kaup
EN
Rocky Mt. J. Math. 20, No. 2, 409-428 (1990).
00156208
1990
j
47C15 46E40 58B20
Banach Stone theorem; Jordan isomorphism; infinite dimensional holomorphy; surjective linear isometry; \(JB^*\)-triples; \(JB^*\)-triple isomorphism; affine geometric properties of faces; extremal convex subsets; state space; spectral, polar and Jordan decompositions; biduals; theorem of Effros
\textit{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, \textit{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.
Costic?? Must????a (Cluj-Napoca)
Zbl 0045.06201; Zbl 0335.58005