On the extension of isometries between the unit spheres of a C*-algebra and B(H)

Abstract

Given two complex Hilbert spaces $H$ and $K$, let $S(B(H))$ and $S(B(K))$ denote the unit spheres of the C${}^{\ast }$-algebras $B(H)$ and $B(K)$ of all bounded linear operators on $H$ and $K$, respectively. We prove that every surjective isometry $f:S(B(K))\to S(B(H))$ admits an extension to a surjective complex linear or conjugate linear isometry $T:B(K)\to B(H)$. This provides a positive answer to Tingley's problem in the setting of $B(H)$ spaces.

Cited by in corpus (2)

• Tingley's problem through the facial structure of an atomic JBW*-triple
• The Mazur--Ulam property in ${\ell }_{\infty }$-sum and $c_0$-sum of strictly convex Banach spaces