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

Abstract

Let $f:S(E)\to S(B)$ be a surjective isometry between the unit spheres of two weakly compact JB${}^{\ast }$-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${}^{\ast }$-triple theory, we prove that $f$ admits an extension to a surjective real linear isometry $T:E\to B$. Among the consequences, we show that every surjective isometry between the unit spheres of two compact C${}^{\ast }$-algebras $A$ and $B$ (and in particular when $A=K(H)$ and $B=K(H^{\prime })$) 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.