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Isomorphisms of subalgebras of nest algebras
Authors:Fangyan Lu
Affiliation:Department of Mathematics, Suzhou University, Suzhou 215006, People's Republic of China
Abstract:
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.

Keywords:Algebraic isomorphisms   nest algebras   large subalgebras   radical-type algebras   rank one operators   spatially implemented
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