The SHAI property for the operators on L^p

2021/02/08 by William B. Johnson, Johnson, William B., N. Christopher Phillips +3
Mathematics · #46E30 #47L20 #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2102.03966

openalex publication_date 2021/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAI property. The condition is satisfied for L^p (0, 1) for 1 < p < ∞, spaces with symmetric bases that have finite cotype, and the Schatten p-spaces for 1 < p < ∞.

Related

• On the Banach Problem on Surjections
• Separably injective C*-algebras
• On Bhatia-Šemrl Property, Strong Subdifferentiability and Essential Norm of Operators on Banach Spaces
• Biseparating maps between operator algebras
• Kernels of surjections from \cal L₁-spaces with an application to Sidon sets