Identifying JBW$^*$-algebras through their spheres of positive elements

Let $\mathfrak{A}$ and $\mathfrak{B}$ be JBW$^*$-algebras with projection lattices $\mathcal{P} (\mathfrak{A})$ and $\mathcal{P} (\mathfrak{B})$, and let $\Theta: \mathcal{P} (\mathfrak{A})\to \mathcal{P}(\mathfrak{B})$ be an order isomorphism. We prove that if $\mathfrak{A}$ does not contain any type $I_2$ direct summand and $\Theta$ preserves points at distance $1$, then $\Theta$ extends to a Jordan $^*$-isomorphism from $\mathfrak{A}$ onto $\mathfrak{B}$. We also establish that if $\mathfrak{A}$ and $\mathfrak{B}$ are two atomic JBW$^*$-algebras of type $I_2$ and $\Theta: \mathcal{P} (\mathfrak{A})\to \mathcal{P}(\mathfrak{B})$ preserves points at distance $\frac{\sqrt{2}}{2}$, then $\mathfrak{A}$ is Jordan $^*$-isomorphic to $\mathfrak{B}$. Furthermore, if $\mathfrak{A}$ and $\mathfrak{B}$ are two general JBW$^*$-algebras such that the type $I_2$ part of $\mathfrak{A}$ is atomic and $\Theta$ is an isometry, we prove the existence of an extension of $\Theta$ to a Jordan $^*$-isomorphism from $\mathfrak{A}$ onto $\mathfrak{B}$. We provide a positive answer to Tingley's problem for positive spheres showing that if $\mathfrak{A}$ and $\mathfrak{B}$ are JBW$^*$-algebras such that the type $I_2$ part of $\mathfrak{A}$ is atomic, then every surjective isometry from the set, $S_{\mathfrak{A}^+}$, of positive norm-one elements of $\mathfrak{A}$ onto the positive norm-one elements of $\mathfrak{B}$ extends to a Jordan $^*$-isomorphism from $\mathfrak{A}$ onto $\mathfrak{B}$. We prove a metric characterization of projections in JBW$^*$-algebras as follows: if $a$ is a norm-one positive element in a JBW$^*$-algebra $\mathfrak{A}$, then $a$ is a projection if, and only if, it satisfies the double sphere property, that is, $$\Big\{c \in S_{\mathfrak{A}^+} : \|c - b\| = 1 \; \text{for all} \; b \in S_{\mathfrak{A}^+} \; \text{with} \; \|b - a\| = 1\Big\} = \{a\}.$$