Isomorphisms of subalgebras of nest algebras

Let $\mathcal T$ be a subalgebra of a nest algebra $\mathcal T(\mathcal N)$. If $\mathcal T$ contains all rank one operators in $\mathcal T(\mathcal N)$, then $\mathcal T$ is said to be large; if the set of rank one operators in $\mathcal T$ coincides with that in the Jacobson radical of $\mathcal T(\mathcal N)$, $\mathcal T$ is said to be radical-type. In this paper, algebraic isomorphisms of large subalgebras and of radical-type subalgebras are characterized. Let $\mathcal N_i$ be a nest of subspaces of a Hilbert space $\mathcal H_i$ and $\mathcal T_i$ be a subalgebra of the nest algebra $\mathcal T(\mathcal N_i)$ associated to $\mathcal N_i$ ($i=1,2$). Let $\phi$be an algebraic isomorphism from $\mathcal T_1$ onto $\mathcal T_2$. It is proved that $\phi$ is spatial if one of the following occurs: (1) $\mathcal T_i$ ($i=1,2$) is large and contains a masa; (2) $\mathcal T_i$ ($i=1,2$) is large and closed; (3) $\mathcal T_i$ ($i=1,2$) is a closed radical-type subalgebra and $\mathcal N_i$ ($i=1,2)$ is quasi-continuous (i.e. the trivial elements of $\mathcal N_i$ are limit points); (4) $\mathcal T_i$ ($i=1,2$) is large and one of $\mathcal N_1$ and $\mathcal N_2$ is not quasi-continuous.