We prove that, if M>4false(123false) 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 ϕ1, ϕ2 ∈ V* and ψ1, ψ2 ∈ W* satisfying false|Ufalse(x,yfalse)false|⩽Mthinmathspace∥U∥false(∥x∥φ22ε2thinmathspace∥x∥φ12false)12false(∥y∥ψ22ε2thinmathspace∥y∥ψ12false)12 for 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 falsefalse∥x∥φ2:=φfalsefalse{x,x,zfalsefalse} 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.
The notion of weights on (topological) *-algebras is defined and studied. The primary purpose is to define the notions of admissibility and approximate admissibility of weights, and to investigate when a weight is admissible or approximately admissible. The results obtained are applied to vector weights and tracial weight on unbounded operator algebras, as well as to weights on smooth subalgebras of a C *-algebra.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.