# Little Grothendieck`s theorem for real JB*-triples

DOI: 10.1007/PL00004878

- Cite this article as:
- Peralta, A. Math Z (2001) 237: 531. doi:10.1007/PL00004878

- 1 Citations
- 36 Downloads

## 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 \(x\in A\).