A solution to Tingley’s problem for isometries between the unit spheres of compact C*-algebras and JB*-triples

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.

Tingley's problem through the facial structure of an atomic JBW⁎-triple

“…Akemann and G.K. Pedersen in [1]. Their characterization played a decisive role in the arguments presented in [28,29,30,26] and [15]. The result of Akemann and Pedersen was extended to the strictly wider setting of JB * -triples by C.M.…”

“…This result constitutes a positive answer to Tingley's isometric extension problem [31] in the setting of B(H)-spaces and atomic von Neumann algebras. Solutions to Tingley's problem for compact operators, compact C * -algebras and weakly compact JB * -triples have been previously obtained in [26,14]. For additional information on the historic background and the state of the art of Tingley's problem the reader is referred to the introduction of [15] and to the monograph [32].…”

A reflection on Tingley's problem and some applications

“…This problem has been dealt in several ways and lots of positive answers have been found, see, e.g., [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] -it is really impressive the development of machinery and technics that this problem has led to.…”

On the Mazur–Ulam property for the space of Hilbert-space-valued continuous functions

“…Behind their simple statements, Tingley's problem and the Mazur-Ulam property are hard problems which remain unsolved even for surjective isometries between the unit spheres of a couple of two dimensional normed spaces (the reader is invited to take a look to the recent papers [55] and [10], where this particular case is treated). Positive solutions to Tingley's problem have been found for surjective isometries ∆ : S(X) → S(Y ) when X and Y are von Neumann algebras [29], compact C * -algebras [48], atomic JBW * -triples [28], spaces of trace class operators [24], spaces of p-Schatten von Neumann operators with 1 ≤ p ≤ ∞ [25], preduals of von Neumann algebras and the self-adjoint parts of two von Neumann algebras [43]. The surveys [18,56], and [46] are appropriate references to the reader in order to check the state-of-the-art of this problem.…”