Modules, annihilators and module derivations of \(JB^*\)-algebras (Q1911869)

Let \(A\) be a Jordan Banach algebra. A Jordan \(A\)-module \(X\) is said to be a Banach Jordan \(A\)-module if it is a Banach space and there exists a real number \(M\geq 0\) such that \(|a\circ x|\leq M|a|\;|x|\), for any \(a\in A\), \(x\in X\). For every submodule \(S\) of \(X\) the linear subspace \(\{a\in A: R_a (S)= 0\}\) is denoted by \(R(S)\) and the subset \(\{a\in A: U_a (S) =0\}\) by \(\tau (S)\). Here \(U_a =2R_a^2- R_{a^2}\), \(R_a\) being the linear operator of multiplication by \(a\) in the Jordan algebra \(A\oplus X\) obtained by split null extension of \(A\) and \(X\) by the bilinear maps of the module. First the authors prove that if \(A\) is a \(JB\)-algebra then the following assertions hold: i) The largest ideal of \(A\) contained in \(R(S)\) agrees with \(\tau (S)\cap R(S)\); ii) \(\tau (S)\) is hereditary and it contains the segment joining two arbitrary points of it; iii) if \(X\) agrees with the Banach-Jordan \(A\)-module \(A^*\) then \(\tau (S)\) is an ideal of \(A\). The linear map \(D: A\to X\) is called a module derivation if \(D(a\circ b)= a\circ D(b)+ (Da)\circ b\) for any \(a,b\in A\). The authors study the continuity of these maps when \(A\) is a \(JB^*\)-algebra. So in this case \(D\) is continuous iff the selfadjoint elements of \(\tau (\sigma)\) are a real subspace of \(A\), being \(\sigma\) the set of the elements \(x\in X\) such that there exists a sequence \(\{a_n \}\subset A\) such that \(a_n\to 0\) and \(Da_n\to x\). More results are given.