Grothendieck's theorem and factorization of operators in Jordan triples

Grothendieck's Inequalities for Real and Complex JBW*-Triples

“…We have been able to verify Assertion (i), but only by applying the fact, later proved by C.-H. Chu, B. Iochum and G. Loupias [8,Lemma 5], that every bounded linear operator from a complex JB Ã -triple to the dual of another complex JB Ã -triple factors through a complex Hilbert space. Actually, this fact is also claimed in the Barton±Friedman paper (see [ ;…”

“…Since the Barton±Friedman proof of their claim actually shows that the inequality (1.1) holds (for suitable norm-one functionals J P E à and w P F à ) whenever the separately weak à -continuous extension of V given by Lemma 1 attains its norm at a pair of complete tripotents, the next theorem follows from Lemma 2. ) appears in the Chu±Iochum±Loupias paper already quoted (see [8,Theorem 6]). However, such a proof relies on the Barton± Friedman version of the so-called`Little Grothendieck Theorem' for complex JB à -triples [3, Theorem 1.3], and the Barton±Friedman argument for this result also has a gap (see [26]).…”

“…Section 1 of the present paper is devoted to reviewing the main results in [3]. In fact we observe some gaps in the proofs of those results (some of which are also subsumed in [8]), and provide the reader with quick partial amendments of such gaps by applying theorems of J. Lindenstrauss [24] and V. Zizler [37] (see Theorems 1 and 2, respectively).…”

“…This paper relies on the important works of T. Barton and Y. Friedman [3] and C.-H. Chu, B. Iochum and G. Loupias [8] on the generalization of`Grothendieck's inequalities' to complex JB Ã -triples. Of course, the Barton±Friedman±Chu± Iochum±Loupias techniques are strongly related to those of A. Grothendieck [15], G. Pisier (see [27,28,29]), and U. Haagerup [16], leading to the classical Grothendieck inequalities' for C Ã -algebras.…”

Little Grothendieck`s theorem for real JB*-triples

“…The "Grothendieck's Theorem" for (complex) JB*-algebras (which is a verbatim extension of Haagerup's proof for C*-algebras [H2]), is stated by Chu, Iochum and Loupias in [CIL,Theorem 2. ].…”