Juan Caballero Martínez

Surjective isometries between real JB*-triples

In [18], R. Kadison proved that every surjective linear isometry Φ : A → B between two unital C∗-algebras is of the form Φ(x) = uT (x), x ∈ A, where u is a unitary element in B and T is a Jordan ∗-isomorphism from A to B. This result extends to non-abelian unital C∗-algebras the classical Banach-Stone Theorem [3, 31] obtained in the 1930’s. Kadison’s result was extended to surjective isometries between C∗-algebras by A. L. T. Paterson and A. M. Sinclair [27], by replacing the unitary element u by a unitary element in the multiplier C∗-algebra of the range algebra. In particular, every surjective isometry between C∗-algebras preserves the triple products of the form {x, y, z} = 2−1(xy∗z + zy∗x). In the non-associative case, J. Wright and M. Youngson [34, Theorem 6], established that every unital surjective linear isometry between two unital JB∗-algebras is a Jordan ∗-isomorphism. In 1995, J. M. Isidro and A. Rodŕiguez [17, Theorem 1.9] show that every surjective linear isometry Φ between two JB-algebras is of the form