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 $$Sph(E;P):=\left\{x\in P:\Vert x-b\Vert =1\; for\; allb\in E\right\}.$$ Given a C${}^{\ast }$-algebra $A$, and a subset $E\subset A,$ we shall write $Sp{h}^{+}(E)$ or $Sp{h}_A^{+}(E)$ for the set $Sph(E;S(A^{+})),$ where $S(A^{+})$ stands for the set of all positive operators in the unit sphere of $A$. We prove that, for an arbitrary complex Hilbert space $H$, then a positive element $a$ in the unit sphere of $B(H)$ is a projection if and only if $Sp{h}_{B(H)}^{+}\left(Sp{h}_{B(H)}^{+}(\{a\})\right)=\{a\}$. 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 prove a stronger conclusion by showing that the identity $$Sp{h}_{K(H_2)}^{+}\left(Sp{h}_{K(H_2)}^{+}(a)\right)=\left\{b\in S(K(H_2{)}^{+}):\begin{array}{c}{s}_{{}_{K(H_2)}}(a)\le s_{{}_{K(H_2)}}(b),and\text{1}-r_{{}_{B(H_2)}}(a)\le \text{1}-r_{{}_{B(H_2)}}(b)\end{array}\right\},$$ 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.

arXiv admin note: substantial text overlap with arXiv:1711.05652