Surjective isometries between real JB*-triples

Abstract

In [ 19 ], R. Kadison proved that every surjective linear isometry $\Phi{:}\, {\C A} \to {\C B}$ between two unital C*-algebras has the form $$\Phi (x) = u T (x), \hbox{ $x\in {\C A}$,}$$ where $u$ is a unitary element in ${\C B}$ and $T$ is a Jordan *-isomorphism from ${\C A}$ onto ${\C B}$ . This result extends the classical Banach–Stone theorem [ 3 , 32 ] obtained in the 1930s to non-abelian unital C*-algebras. A. L. Paterson and A. M. Sinclair extended Kadison's result to surjective isometries between C*-algebras by replacing the unitary element $u$ by a unitary element in the multiplier C*-algebra of the range algebra [ 28 ]. Thus, every surjective linear isometry between C*-algebras preserves the triple products as $$\J xyz \,{=}\, 2^{-1} ( x y^* z + z y^* x).$$