On the unit sphere of positive operators

Given a C${}^{\ast }$-algebra $A$, let $S(A^{+})$ denote the set of those positive elements in the unit sphere of $A$. Let $H_1$, $H_2,$ $H_3$ and $H_4$ be complex Hilbert spaces, where $H_3$ and $H_4$ are infinite-dimensional and separable. In this note we prove a variant of Tingley’s problem by showing that every surjective isometry $\mathrm{\Delta }:S(B(H_1{)}^{+})\to S(B(H_2{)}^{+})$ or (respectively, $\mathrm{\Delta }:S(K(H_3{)}^{+})\to S(K(H_4{)}^{+})$) admits a unique extension to a surjective complex linear isometry from $B(H_1)$ onto $B(H_2))$ (respectively, from $K(H_3)$ onto $B(H_4)$). This provides a positive answer to a conjecture posed by G. Nagy [\emph{Publ. Math. Debrecen}, 2018].