Low rank compact operators and Tingley's problem

Abstract

Let $E$ and $B$ be arbitrary weakly compact JB${}^{\ast }$-triples whose unit spheres are denoted by $S(E)$ and $S(B)$, respectively. We prove that every surjective isometry $f:S(E)\to S(B)$ admits an extension to a surjective real linear isometry $T:E\to B$. This is a complete solution to Tingley's problem in the setting of weakly compact JB${}^{\ast }$-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 $f:S(K(H,K))\to S(K(H,K))$ admits an extension to a surjective real linear isometry $T:K(H,K)\to K(H,K)$.