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, without assuming any restriction on the rank of their direct summands (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.