Little Grothendieck’s theorem for real JB*-triples

We prove that given a real JB*-triple E, and a real Hilbert space H , then the set of those bounded linear operators T from E toH , such that there exists a norm one functionalφ ∈ E∗ and corresponding pre-Hilbertian semi-norm ‖.‖φ on E such that ‖T (x)‖ ≤ 4 √ 2‖T‖ ‖x‖φ for all x ∈ E, is norm dense in the set of all bounded linear operators from E toH . As a tool for the above result, we show that ifA is a JB-algebra and T : A → H is a bounded linear operator then there exists a state φ ∈ A∗ such that ‖T (x)‖ ≤ 2 √ 2‖T‖φ(x2) for all x ∈ A. Mathematics Subject Classification (2000): 17C65, 46K70, 46L05, 46L10, 46L70