On the unit sphere of positive operators

TL;DR

This paper solves a variant of Tingley's problem for positive operators on certain operator algebras, showing that surjective isometries on the positive unit sphere extend uniquely to linear isometries.

Contribution

It proves that every surjective isometry between positive parts of unit spheres extends uniquely to a linear isometry, confirming a conjecture by G. Nagy.

Findings

Surjective isometries extend uniquely to linear isometries.

Results apply to positive operators on B(H) and K(H).

Provides a positive answer to Nagy's conjecture.

Abstract

Given a C$^*$-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 $\Delta : S(B(H_1)^+)\to S(B(H_2)^+)$ or (respectively, $\Delta : S(K(H_3)^+)\to S(K(H_4)^+)$) 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].