Tingley’s problem for positive unit spheres of operator algebras and diametral relations

Abstract:

We answer, in the affirmative, Tingley’s problem for positive unit spheres of (complex) von Neumann algebras. More precisely, let $\Lambda : \mathrm {S}_{\mathcal {A}^+} \to \mathrm {S}_{\mathcal {B}^+}$ be a bijection between the sets of positive norm-one elements of two von Neumann algebras $\mathcal {A}$ and $\mathcal {B}$. We show that if $\Lambda$ is an isometry, then it extends to a bijective complex linear Jordan $^*$-isomorphism from $\mathcal {A}$ onto $\mathcal {B}$. In the case in which $\Lambda$ satisfies the weaker assumption of preserving pairs of points at diametrical distance, namely, \begin{equation*} \|\Lambda (a) - \Lambda (b)\| =1 \quad \text {if and only if} \quad \|a-b\| = 1 \quad (a,b\in \mathrm {S}_{\mathcal {A}^+}), \end{equation*} one can still conclude that $\mathcal {A}$ is complex linear Jordan $^*$-isomorphic to $\mathcal {B}$.

On our way, we also show that if there is an order isomorphism $\Theta :\mathrm {P}_\mathcal {A}\to \mathrm {P}_\mathcal {B}$ between the projection lattices of $\mathcal {A}$ and $\mathcal {B}$ that preserves pairs of points at diametrical distance, that is, \begin{equation*} \|\Theta (p) - \Theta (q)\| =1 \quad \text {if and only if} \quad \|p-q\| = 1\quad (p,q\in \mathrm {P}_\mathcal {A}), \end{equation*} then $\mathcal {A}$ and $\mathcal {B}$ are complex linear Jordan $^*$-isomorphic. If, in addition, either $\mathcal {A}$ has no type $\mathbf {I}_2$ summand, or $\Theta$ is an isometry, then $\Theta$ extends to a complex linear Jordan $^*$-isomorphism from $\mathcal {A}$ onto $\mathcal {B}$.

Actually, the above results are proved in the slightly more general situation that $\mathcal {A}$ and $\mathcal {B}$ are $AW^*$-algebras.