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

TL;DR

This paper proves that surjective isometries between unit spheres of certain compact C*-algebras and JB*-triples extend to linear isometries, advancing the understanding of Tingley's problem in infinite-dimensional spaces.

Contribution

It establishes the extension of sphere isometries to linear isometries for compact C*-algebras and JB*-triples with specific rank conditions, providing new solutions to Tingley's problem.

Findings

Surjective isometries extend to linear isometries in specified JB*-triples.

Every surjective sphere isometry between compact C*-algebras extends linearly.

Results apply to infinite-dimensional Banach spaces, confirming Tingley's problem in these cases.

Abstract

Let $f: S(E) \to S(B)$ 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 $T: E\to B$. Among the consequences, we show that every surjective isometry between the unit spheres of 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 where Tingley's problem admits a positive answer.