Characterizing projections among positive operators in the unit sphere

Document Type: Original Article

Author

Universidad de Granada

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. ‎In the setting of compact operators we establish a stronger conclusion by showing that the identity $$Sph^+_{K(H_2)} \left( Sph^+_{K(H_2)}(a) \right) =\left\{ b\in S(K(H_2)^+)‎ : ‎\!\! \begin{array}{c}‎s_{_{K(H_2)}} (a) \leq s_{_{K(H_2)}} (b)‎, ‎\hbox{ and }\\‎ ‎\textbf{1}-r_{_{B(H_2)}}(a)\leq \textbf{1}-r_{_{B(H_2)}}(b)‎ ‎\end{array}‎\right\},$$
holds for every $a$ in the unit sphere of $K(H_2)^+$‎, ‎where $r_{_{B(H_2)}}(a)$ and $s_{_{K(H_2)}} (a)$ stand for the range and support projections of $a$ in $B(H_2)$ and $K(H_2)$‎, ‎respectively‎.

Keywords