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.

References in corpus (2)

• Extension of isometries between unit spheres of finite-dimensional polyhedral Banach spaces
• A Note on The Mazur-Ulam Property of Almost-CL-spaces

Cited by in corpus (9)

• A reflection on Tingley's problem and some applications
• Extension of isometries from the unit sphere of a rank-2 Cartan factor
• On the extension of isometries between the unit spheres of a JBW${}^{\ast }$-triple and a Banach space
• On the extension of isometries between the unit spheres of a C*-algebra and B(H)
• The Mazur-Ulam property for commutative von Neumann algebras
• Tingley's problem through the facial structure of an atomic JBW*-triple
• Tingley's problem on uniform algebras
• Linearity of isometries between convex Jordan curves
• Tingley's problem for spaces of trace class operators