Conditional expectation and bicontractive projections on Jordan \(C^ *\)- algebras and their generalizations
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 \textit{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 \textit{Dineen} [Math. Scand., to appear] and \textit{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.
scientific article; zbMATH DE number 3964735
scientific article