Mathematische Zeitschrift

, Volume 237, Issue 3, pp 531–545

Little Grothendieck`s theorem for real JB*-triples


  • Antonio M. Peralta
    • Dept. Análisis Matemático, Ftad. de Ciencias, Universidad de Granada, 18071 Granada, Spain (e-mail:
Original article

DOI: 10.1007/PL00004878

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


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 \(x\in A\).

Mathematics Subject Classification (2000): 17C65, 46K70, 46L05, 46L10, 46L70
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© Springer-Verlag Berlin Heidelberg 2001