Additive Jordan isomorphisms of nest algebras on normed spaces (Q1406524)

Let \(X\) be a (real or complex) Banach space and \(B(X)\) the set of all bounded linear operators on \(X\). A chain \({\mathcal N}\) of closed subspaces of \(X\) is called a net if it contains the trivial subspaces \(\{0\}\) and \(X\), and if it is closed under intersection and closed span. The nest algebra \(\text{alg }{\mathcal N}\) is defined by \(\text{alg }{\mathcal N}= \{T\in B(X): T(E)\subseteq E,E\in{\mathcal N}\}\). In the present paper, the author proves that any additive bijective mapping \(\Phi: \text{alg }{\mathcal N}\to \text{alg }{\mathcal M}\) such that \(\Phi(A^2)= \Phi(A)^2\) (where \({\mathcal N}\) and \({\mathcal M}\) are nets on the same Banach space \(X\)) is either a ring isomorphism or a ring anti-isomorphism. Moreover, if \(X\) is a real or infinite-dimensional complex space, then \(\Phi\) has the usual spatial form.