Grothendieck's inequalities for real and complex \(\text{JBW}^ *\)-triples. (Q2766431)

In aiming to extend the results of \textit{T. Barton} and \textit{Y. Friedman} [J. Lond. Math. Soc., II. Ser. 36, 513--523 (1987; Zbl 0652.46036)] and \textit{C.-H. Chu}, \textit{B. Iochum} and \textit{G. Loupias} [Math. Ann. 284, 41--53 (1989; Zbl 0648.46059)] from \(\text{JBW}^*\)-triples to real \(\text{JBW}^*\)-triples, the authors discovered two apparent gaps in the proof of Theorem 1.4 of the former paper.NEWLINENEWLINEThe first part of the present paper examines the status of these. The authors were able to show that for a pair \(A\) and \(B\) of \(\text{JB}^*\)-triples every bounded bilinear form \(\phi\) on \(A\times B\) has a separately weak\(^*\)-continuous extension to \(A^{**}\times B^{**}\), a result stated without proof by Barton and Friedman. They were unable to determine whether or not Theorem 1.4 was true or not, but were able to show that, under the same circumstances as above, the set of bounded bilinear forms on \(A\times B\) the separately weak\(^*\)-continuous extensions to \(A^{**}\times B^{**}\) of which attain their norms at points of the unit balls of \(A^{**}\) and \(B^{**}\) is norm-dense in the space of bounded bilinear forms on \(A\times B\). It is not true that, in general, the two sets coincide.NEWLINENEWLINEThe second part of the paper is concerned with proving slightly weaker versions of the so-called little Grothendieck inequality for \(\text{JBW}^*\)-triples than those claimed by Barton and Friedman. The authors go on to apply their result to \(\text{JBW}^*\)-algebras, and then prove similar non-trivial results for real \(\text{JBW}^*\)-triples and real \(\text{JBW}^*\)-algebras. In the third part, the authors show the existence of a Grothendieck equality for separately weak\(^*\)-continuous bilinear forms of the product of two \(\text{JBW}^*\)-triples and of two real \(\text{JBW}^*\)-triples.NEWLINENEWLINEIn the final section, the authors show how their results bring new information about various topologies on \(\text{JBW}^*\)-triples and their preduals.