Abstract
Let $E$ and $P$ be subsets of a Banach space $X$, and let us define the unit sphere around $E$ in $P$ as the set $$Sph(E;P) :=\left\{ x\in P : \|x-b\|=1 \hbox{ for all } b\in E \right\}.$$ Given a $C^*$-algebra $A$ and a subset $E\subset A,$ we shall write $Sph ^{+} (E)$ or $Sph ^{+}_{A} (E)$ for the set $Sph(E;S(A^+)),$ where $S(A^+)$ denotes the unit sphere of $A^+$. We prove that, for every complex Hilbert space $H$, the following statements are equivalent for every positive element $a$ in the unit sphere of $B(H)$:
(a) $a$ is a projection;
(b) $Sph^+_{B(H)} \left( Sph^+_{B(H)}(\{a\}) \right) =\{a\}$.
Citation
Antonio M. Peralta. "Characterizing projections among positive operators in the unit sphere." Adv. Oper. Theory 3 (3) 731 - 744, Summer 2018. https://doi.org/10.15352/aot.1804-1343
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