Little Grothendieck`s theorem for real JB*-triples

Abstract.

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 to H, such that there exists a norm one functional $\varphi \in E^*$ and corresponding pre-Hilbertian semi-norm $\|.\|_{\varphi}$ on E such that

$\|T(x)\| \leq 4 \sqrt{2} \|T\| \|x\|_{\varphi}$

for all $x\in E$, is norm dense in the set of all bounded linear operators from E to H. As a tool for the above result, we show that if A is a JB-algebra and $T: A \rightarrow H$ is a bounded linear operator then there exists a state $\varphi \in A^*$ such that

$\| T(x) \| \leq 2 \sqrt{2} \|T\| \varphi ( x^2)$

for all $x\in A$.