On the unit sphere of positive operators

Abstract

Given a $C^{\ast }$-algebra $A$, let $S\left(A^{+}\right)$ denote the set of 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 article, we prove a variant of Tingley’s problem by showing that every surjective isometry $\Delta :S\left(B\right(H_1{)}^{+})\to S(B\left(H_2{)}^{+}\right)$ (resp., $\Delta :S\left(K\right(H_3{)}^{+})\to S(K\left(H_4{)}^{+}\right)$) admits a unique extension to a surjective complex linear isometry from $B\left(H_1\right)$ onto $B\left(H_2\right)$ (resp., from $K\left(H_3\right)$ onto $K\left(H_4\right)$). This provides a positive answer to a conjecture recently posed by Nagy.