Grothendieck's Inequalities for Real and Complex JBW*‐Triples

We prove that, if M > 4 ( 12 3 ) and ɛ > 0, if V and W are complex JBW*‐triples (with preduals V * and W * , respectively), and if U is a separately weak*‐continuous bilinear form on V × W , then there exist norm‐one functionals ϕ 1 , ϕ 2 ∈ V * and ψ 1 , ψ 2 ∈ W * satisfying | U ( x , y ) | ⩽ M ∥ U ∥ ( ∥ x ∥ φ 2 2 ε 2∥ x ∥ φ 1 2 ) 1 2( ∥ y ∥ ψ 2 2 ε 2∥ y ∥ ψ 1 2 ) 1 2for all ( x, y ) ∈ V × W . Here, for a norm‐one functional ϕ on a complex JB*‐triple V , |·| ϕ stands for the prehilbertian seminorm on V associated to ϕ given by ∥ x ∥ φ 2 : = φ { x , x , z } for all x ∈ W , where z ∈ V ** satisfies ϕ z = | z | = 1. We arrive at this form of ‘Grothendieck's inequality’ through results of C.‐H. Chu, B. Iochum, and G. Loupias, and an amended version of the ‘little Grothendieck's inequality’ for complex JB*‐triples due to T. Barton and Y. Friedman. We also obtain extensions of these results to the setting of real JB*‐triples. 2000 Mathematical Subject Classification : 17C65, 46K70, 46L05, 46L10, 46L70.