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Contractive perturbations in JB*‐triples
Author(s) -
Fernández-Polo Francisco J.,
Moreno Juan Martínez,
Peralta Antonio M.
Publication year - 2012
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdr048
Subject(s) - mathematics , compact space , unit sphere , subspace topology , pure mathematics , banach space , ball (mathematics) , rank (graph theory) , norm (philosophy) , unit (ring theory) , finite element method , mathematical analysis , combinatorics , physics , mathematics education , political science , law , thermodynamics
An element x in the closed unit ball of a Banach space X is said to be geometrically compact (resp., geometrically weakly compact) if the set cp (2) ({ x }) of all its second contractive perturbations is norm‐compact (respectively, weakly compact). We say that x has finite geometric rank if the closed linear subspace generated by cp (2) ({ x }) is finite‐dimensional. We prove that an element in the closed unit ball of a JB*‐triple, E , is geometrically weakly compact if and only if it is a weakly compact element in E . Consequently, x lies in the socle of E whenever it has finite geometric rank. This generalizes a previous contribution by Anoussis and Katsoulis [‘Compact operators and the geometric structure of C*‐algebras’, Proc. Amer. Math. Soc. 124 (1996) 2115–2122] in the setting of C*‐algebras.