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Conditional expectation and bicontractive projections on Jordan $$C^ *$$- algebras and their generalizations. (English) Zbl 0598.46044
A JB$${}^*$$-triple is a complex Banach space equipped with a compatible triple product structure. The class contains all Jordan $$C^*$$-algebras (also called JB$${}^*$$-algebras), including the exceptional one, as well as all $$C^*$$-algebras. According to W. Kaup [Math. Scand. 54, 95- 100 (1984; Zbl 0578.46066)], if P is a contractive projection on a $$JB^*$$-triple U, then P(U) is a $$JB^*$$-triple and $$P\{PabPc\}=P\{PaPbPc\}$$ for a,b,c in U.
By using these results of Kaup and their own previous work [Math. Scand. 52, 279-311 (1983; Zbl 0547.46048); J. Reine Angew. Math. 356, 67-89 (1985; Zbl 0547.46049)], the authors obtain additional structure of P(U) and prove a new conditional expectation formula. Specifically it is shown that P(U) is isomorphic to a JB$${}^*$$-subtriple of the bidual U”, which is a $$JBW^*$$-triple by work of Dineen [Math. Scand., to appear] and Barton-Timoney [Math. Scand., to appear]; and that $$P\{PaPbc\}=P\{PaPbPc\}$$ holds for a,b,c in U. These reslts are then used to solve the bicontractive projection problem for $$JB^*$$-triples: a contractive projection P on a $$JB^*$$-triple U is bicontractive if and only if $$P=(I+\theta)$$ where $$\theta$$ is an involutive isometry of U. This last result includes all known results on bicontractive projections on Banach spaces supporting an algebraic structure, i.e. $$C^*$$- algebras, $$J^*$$-algebras, JB algebras.

##### MSC:
 46L70 Nonassociative selfadjoint operator algebras 17C65 Jordan structures on Banach spaces and algebras 17A40 Ternary compositions 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras
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