Contractive perturbations in JB*‐triples

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.