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

TL;DR

The paper addresses the problem of identifying projections among positive norm-one elements in unital $C^*$-algebras using purely geometric, norm-based criteria. It introduces and leverages the double-sphere property, showing that a positive norm-one element $p$ is a projection if and only if $ \mathrm{Sph}_{_{\mathrm{S}_{A^+}}}(\mathrm{Sph}_{_{\mathrm{S}_{A^+}}}(p))=\{p\}$, with a complete proof in the unital $C^*$-algebra setting. The authors develop the argument through analysis of the maps $\mathrm{U}_p$ and $\mathrm{U}_{p^{\perp}}$, spectral considerations in $A^{**}$, and a refined sphere lemma; they also establish a corresponding auxiliary result for the unit, $\mathrm{Sph}_{_{\mathrm{S}_{A^+}}}(\mathrm{Sph}_{_{\mathrm{S}_{A^+}}}(\mathbf{1}))=\{\mathbf{1}\}$. The final part extends the characterization to unital $JB^*$-algebras, adapting the proof to the Jordan framework via functional calculus, Peirce decomposition, and refined Cauchy–Schwarz arguments, thereby proving that the same double-sphere criterion characterizes projections in this broader setting. Overall, the work provides a purely geometric criterion for detecting projections, strengthening links between norm geometry and operator-algebraic structure with potential implications for isometries and geometric analysis in operator algebras.

Abstract

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\}.$

Theorems & Definitions (21)

• Theorem 2.1
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5
• proof
• proof : Proof of Theorem \ref{['t chracterization of projections']}