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

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