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Summer 2018 Characterizing projections among positive operators in the unit sphere
Antonio M. Peralta
Adv. Oper. Theory 3(3): 731-744 (Summer 2018). DOI: 10.15352/aot.1804-1343

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.

Citation

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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

Information

Received: 1 April 2018; Accepted: 11 April 2018; Published: Summer 2018
First available in Project Euclid: 27 April 2018

zbMATH: 06902464
MathSciNet: MR3795112
Digital Object Identifier: 10.15352/aot.1804-1343

Subjects:
Primary: 47A05
Secondary: 46L05 , 47L30

Keywords: ‎bounded linear operator , compact linear operator , projection , unit sphere around a subset‎

Rights: Copyright © 2018 Tusi Mathematical Research Group

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Vol.3 • No. 3 • Summer 2018
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