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Let $f: S(E) \to S(B)$ be a surjective isometry between the unit spheres oftwo weakly compact JB$^*$-triples not containing direct summands of ranksmaller 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 $f$ admitsan extension to a surjective real linear isometry $T: E\to B$. Among theconsequences, we show that every surjective isometry between the unit spheresof 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 whereTingley's problem admits a positive answer.

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

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