Low rank compact operators and Tingley's problem

Let E and B be arbitrary weakly compact JB⁎-triples whose unit spheres are denoted by S(E) and S(B), respectively. We prove that every surjective isometry f:S(E)→S(B) admits an extension to a surjective real linear isometry T:E→B. This is a complete solution to Tingley's problem in the setting of weakly compact JB⁎-triples. Among the consequences, we show that if K(H,K) denotes the space of compact operators between arbitrary complex Hilbert spaces H and K, then every surjective isometry f:S(K(H,K))→S(K(H,K)) admits an extension to a surjective real linear isometry T:K(H,K)→K(H,K).