Jordan isomorphisms of nest algebras

Abstract:

Let $\mathcal {T}(N)$ and $\mathcal {T}(\mathcal {M})$ be two nest algebras. A Jordan isomorphism $\phi$ from $\mathcal {T}(N)$ onto $\mathcal {T}(\mathcal {M})$ is a bijective linear map such that $\phi (T^{2})=\phi (T)^{2}$ for every $T\in \mathcal {T}(\mathcal {N})$. In this note, we prove that every Jordan isomorphism of nest algebras is of the form $T\rightarrow STS^{-1}$ or $T\rightarrow ST^{*}S^{-1}$ and then is, in fact, an isomorphism or an anti-isomorphism.