Given a -algebra , let denote the set of positive elements in the unit sphere of . Let , , , and be complex Hilbert spaces, where and are infinite-dimensional and separable. In this article, we prove a variant of Tingley’s problem by showing that every surjective isometry (resp., ) admits a unique extension to a surjective complex linear isometry from onto (resp., from onto ). This provides a positive answer to a conjecture recently posed by Nagy.
"On the unit sphere of positive operators." Banach J. Math. Anal. 13 (1) 91 - 112, January 2019. https://doi.org/10.1215/17358787-2018-0017