The the nowadays known as Tingley’s problem asks whether every surjective
isometry $f : S(X) → S(Y)$ between the unit spheres of two normed spaces $X$ and
$Y$ admits an extension to a surjective real linear isometry $T : X → Y$ . The origins
of this problem go back to the paper published by $D$. Tingley in 1987, who made
the first contribution in the study of surjective isometries between the unit spheres
of two (finite dimensional) normed spaces.
A solution to Tingley’s problem has been pursued by many researchers during
the last thirty years. Most of positive answers to this problem correspond to infinite
dimensional Banach spaces which are very close to commutative $C^∗$-algebras and their dual spaces. Quite recently, R. Tanaka revitalized the interest on Tingley’s problem in the case of non-commutative $C^∗$-algebras by providing a positive solution for finite dimensional $C^∗$-algebras.
In this talk we shall present some new results extending Tanaka’s solution to the case of surjective isometries between the unit spheres of the space of compact operators between arbitrary complex Hilbert spaces, two von Neumann algebras, and some other $C^∗$-algebras. Some of the new results are based on the advantage of applying Jordan techniques in the study of Tingley’s problem. More precisely, the results determining the facial structure of the closed unit ball of a $C^∗$-algebra, and more generally, the facial structure of the closed unit ball of a $JB^∗$ -triple, provide new tools to tackle this old problem.
On the extension of isometries
2018. 10. 03. 16:00
Antonio M. Peralta, University of Granada, Spain