On continuous Peirce decompositions, Schur multipliers and the perturbation of triple functional calculus (Q5947139)

The usual odd continuous functional calculus on a \(\text{JB}^*\)-triple \(E\) is extended to arbitrary bounded odd functions defined on the triple spectrum \(\text{Sp}(a)\) of \(a\in E\). In contrast to the continuous case, the values of \(f(a)\) no longer is contained in \(E\), but in its bidual \(E^{**}\). A representation of every element \(a\in E\) as an (infinite) linear combination of an orthogonal family of tripotents \(e_\lambda\in E^{**}\) is then obtained and used to define the Peirce spectrum \(\Sigma(a)\), the notion of Schur multiplier and its associated symbol. The authors study for odd \(C^1\)-functions \(f\) of \(\mathbb{R}\) into \(\mathbb{C}\) differentiability of \(f_E\), and establish various results where the derivative \(df_E(a)\) exists and is a Schur multiplier, with the divided difference as symbol. Finally, they prove that the class of all odd \(C^1\)-functions on \(\mathbb{R}\) giving a \(C^1\)-triple functional calculus on every \(\text{JB}^*\)-triple coincides with the class of all odd \(C^1\)-functions giving a \(C^1\) Hermitian operator functional calculus on every complex Hilbert space. Most of these results carry over to real \(\text{JB}^*\)-triples.