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

Abstract

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