No abstract

Let f : S(E) → S(B) be a surjective isometry between the unit spheres of two weakly compact JB * -triples not containing direct summands of rank smaller than or equal to 3. Suppose E has rank greater than or equal to 5. Applying techniques developed in JB * -triple theory, we prove that f admits an extension to a surjective real linear isometry T : E → B. Among the consequences, we show that every surjective isometry between the unit spheres of two compact C * -algebras A and B (and in particular when A = K(H) and B = K(H ′ )) extends to a surjective real linear isometry from A into B. These results provide new examples of infinite dimensional Banach spaces where Tingley's problem admits a positive answer.

Let X be a Banach space. Then there is a locally convex topology for X, the "Right topology," such that a linear map T , from X into a Banach space Y , is weakly compact, precisely when T is a continuous map from X, equipped with the "Right" topology, into Y equipped with the norm topology. When T is only sequentially continuous with respect to the Right topology, it is said to be pseudo weakly compact. This notion is related to Pelczynski's Property (V ).

Let E and B be arbitrary weakly compact JB *

In a first result, we prove that every continuous local triple derivation on a JB * -triple is a triple derivation. We also give an automatic continuity result, that is, we show that local triple derivations on a JB * -triple are continuous even if not assumed a priori to be so. These results provide positive answers to the conjectures posed by Mackey (Bull. London Math. Soc. 45 (2013) 811-824). In particular, every local triple derivation on a C * -algebra is a triple derivation. We also explore the connections between (bounded local) triple derivations and generalized (Jordan) derivations on a C * -algebra.