A metric characterization of projections among positive norm-One elements in unital C$^*$-algebras
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914255651995648 |
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| author | Peralta, Antonio M. Saavedra, Pedro |
| author_facet | Peralta, Antonio M. Saavedra, Pedro |
| contents | 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\}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09669 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A metric characterization of projections among positive norm-One elements in unital C$^*$-algebras Peralta, Antonio M. Saavedra, Pedro Operator Algebras Functional Analysis 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\}.$ |
| title | A metric characterization of projections among positive norm-One elements in unital C$^*$-algebras |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2601.09669 |