Characterizing projections among positive operators in the unit sphere

TL;DR

This paper characterizes projections among positive operators in the unit sphere of various operator algebras using a double spherical projection condition, extending results from Hilbert spaces to von Neumann and compact operator algebras.

Contribution

It provides a new characterization of projections among positive operators in the unit sphere for complex Hilbert spaces, atomic von Neumann algebras, and compact operators, with a stronger result for compact operators.

Findings

Characterization of projections via double spherical conditions in B(H)

Extension of characterization to atomic von Neumann algebras

Stronger description for compact operators involving support and range projections

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