Characterizing projections among positive operators in the unit sphere

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) $Sph^+_{B(H)} \left( Sph^+_{B(H)}(\{a\}) \right) =\{a\}$‎.‎

‎ We also prove that the equivalence remains true when $B(H)$ is replaced with an atomic von Neumann algebra or with $K(H_2)$‎, ‎where $H_2$ is an infinite-dimensional and separable complex Hilbert space‎.