Little Grothendieck's theorem for real JB*-triples

We prove that given a real JB*-triple E, and a real Hilbert space H, then t he set of those bounded linear operators T from E to H, such that there exi sts a norm one functional phi is an element of E* and corresponding pre-Hil bertian semi-norm parallel to.parallel to (phi) on E such that parallel toT(x)parallel to less than or equal to 4 root2 parallel toT paral lel to parallel tox parallel to (phi) for all x is an element of E, is norm dense in the set of all bounded linea r operators from E to H. As a tool for the above result, we show that if A is a JB-algebra and T : A --> H is a bounded linear operator then there exi sts a state phi is an element of A* such that parallel toT(x)parallel to less than or equal to 2 root2 parallel toT paral lel to phi (x(2)) for all x is an element of A.