Property (M), M-ideals, and almost isometric structure of Banach\n spaces

1993/10/11 by N. J. Kalton, Kalton, Nigel J., Dirk Werner +1 · 2 citations
Mathematics · #46B #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.math/9310216

openalex publication_date 1993/10/11 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28

Abstract

We study M-ideals of compact operators by means of the property~(M)\nintroduced in citeKal-M. Our main result states for a separable Banach space\nX that the space of compact operators on X is an M-ideal in the space of\nbounded operators if (and only if) X does not contain a copy of \ℓ₁,\nhas the metric compact approximation property, and has property~(M). The\ninvestigation of special versions of property~(M) leads to results on almost\nisometric structure of some classes of Banach spaces. For instance, we give a\nsimple necessary and sufficient condition for a Banach space to embed almost\nisometrically into an \ℓ_p-sum of finite-dimensional spaces resp. into\nc₀, and for 2<p< iy we prove that a subspace of L_p embeds almost\nisometrically into \ℓ_p if and only if it does not contain a subspace\nisomorphic to \ℓ₂.\n

Cited by

• Weak null maximum and integration of families of multiplication operators
• Weak cluster points of maximizing sequences on Banach spaces satisfying \boldsymbol(M_p)

Related

• Compactness and an approximation property related to an operator ideal
• Operators ideals and approximation properties
• Almost isometric ideals in Banach spaces
• M-ideals of compact operators and Norm attaining operators
• Geometry of the space of compact operators endowed with the numerical radius norm