Factorization theorems in complete Jordan algebras
Let \(J\) be a Jordan-Banach algebra. According to the terminology of the authors, a net \(\{e_i\}_{i\in I}\) in \(J\) is said to be a bounded approximate unit for \(J\) if \(\sup\{|e_i|: i\in I\}<\infty\) and, for every \(x\) in \(J\), we have \(x=\lim \{x.e_i\}_{i\in I}=\lim \{U_{e_i}(x)\}_{i\in I}\). We recall that, for \(y\) in \(J\), the operator \(U_y\) on \(J\) is defined by means of the equality \(U_y(x):= 2y.(y.x)- y^2.x\) for all \(x\) in \(J\). We recall also that, if \(A\) is an associative Banach algebra, and if \(J\) denotes the Jordan-Banach algebra obtained by replacing the associative product \(xy\) of \(A\) by its Jordan product \(x.y:= {1\over 2}(xy+yx)\), then, for any \(x,y\) in \(J\), we have \(U_y(x)= yxy\), so that every bounded approximate unit for \(A\) (in the associative sense) becomes a bounded approximate unit for \(J\) in the meaning explained above. The main result of the paper is the following Jordan variant of a factorization theorem due to Cohen and Hewitt. If there exists a bounded approximate unit for \(J\), then, for every \(x\) in \(J\), there exist \(y,z\) in \(J\) satisfying \(x=U_y(z)\). Some additional metric information about the elements \(y\) and \(z\) above is provided.
Jordan variant of a factorization theorem due to Cohen and Hewitt
scientific article; zbMATH DE number 894158