Pedro Saavedra | Searcharxiv

arXiv2026

A metric characterization of projections among positive norm-One elements in unital C$^*$-algebras

We characterize projections among positive norm-one elements in unital C$^*$-algebras in pure geometric terms determined by the norm of the underlying Banach space. Concretely, let $A$ be a C$^*$-algebra (or a JB$^*$-algebra) whose positive cone and unit sphere are denoted by ${A}^+$ and $\mathrm{S}_{A}$, respectively. The positive portion of the unit sphere in $A$, denoted by $\mathrm{S}_{{A}^+}$, is the set ${A}^+ \cap \mathrm{S}_{A}$, while the unit sphere of positive norm-one elements around a subset $\mathscr{S}$ in $\mathrm{S}_{A^+}$ is the set $$\hbox{Sph}_{_{\mathrm{S}_{{A}^+}}} (\mathscr{S}) :=\Big\{ x\in \mathrm{S}_{{A}^+} : \|x-s\|=1 \hbox{ for all } s\in \mathscr{S} \Big\}.$$ Assuming that $A$ is unital, we establish that an element $a\in \mathrm{S}_{{A}^+}$ is a projection if, and only if, it satisfies the double sphere property, that is, $ \hbox{Sph}_{_{\mathrm{S}_{{A}^+}}} \left(\hbox{Sph}_{_{\mathrm{S}_{{A}^+}}} \left(\{a\}\right) \right) = \{a\}.$

math.OA↗

arXiv2025

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, $$\Big\{c \in S_{\mathfrak{A}^+} : \|c - b\| = 1 \; \text{for all} \; b \in S_{\mathfrak{A}^+} \; \text{with} \; \|b - a\| = 1\Big\} = \{a\}.$$

math.OA↗