Norm-principal bimodules of nest algebras (Q753116)

Given a complete nest \({\mathcal N}\) of (selfadjoint) projections acting on a separable Hilbert space \({\mathcal H}\), the associated nest algebra is alg \({\mathcal N}=\{A\in {\mathcal L}({\mathcal H}):\) \(AP=PAP\), \(P\in {\mathcal N}\}\) and the quasitriangular algebra is alg \({\mathcal N}+{\mathcal K}\), where \({\mathcal K}\) denotes the ideal of compact oprators on \({\mathcal H}.\) In the paper under review, based in part on the author's doctoral dissertation, the questions of when \({\mathcal L}({\mathcal H})\) is a norm-closed singly- or countably-generated bimodule of a nest algebra or a quasitriangular algebra and when the Calkin algebra \({\mathcal L}({\mathcal H})/{\mathcal K}\) is such a bimodule of alg \({\mathcal N}/{\mathcal K}\) are asked and answered, thus providing important insight into our understanding of the algebraic structure of nest algebras. Not surprisingly, the solution depends on the type of nest involved. Specifically, the infinite nest \({\mathcal N}\) is said to be of order type I provided that exactly one of the projections \(E_{\ell}=\bigvee \{N\in {\mathcal N}:\) dim N\(<\infty \}\) and \(E_ r=\bigvee \{N^{\perp}:\) \(N\in {\mathcal N}\), dim \(N^{\perp}<\infty \}\) is finite dimensional and \(E_{\ell}\) and \((E_ r)^{\perp}\) are both limit points of the nest in the strong operator topology. (A simple example is a nest consisting of a sequence of finite rank projections increasing to the identity, in which case \(E_{\ell}=1\) and \(E_ r=0.)\) The main theorem of the paper is that, for a complete nest in a separable Hilbert space, each of the statements (1) \({\mathcal L}({\mathcal H})\) is a norm-closed principal (singly-generated) bimodule of alg \({\mathcal N},\) (2) \({\mathcal L}({\mathcal H})\) is a norm-closed countably-generated bimodule of alg \({\mathcal N},\) (3) \({\mathcal L}({\mathcal H})\) is a norm-closed principal bimodule of alg \({\mathcal N}+{\mathcal K},\) (4) \({\mathcal L}({\mathcal H})\) is a norm-closed countably-generated bimodule of alg \({\mathcal N}+{\mathcal K},\) (5) the Calkin algebra is a norm-closed principal bimodule of alg \({\mathcal N}/{\mathcal K}\), and (6) the Calkin algebra is a norm-closed countably-generated bimodule of alg \({\mathcal N}/{\mathcal K}\) is equivalent to the condition that the nest is not of order type I. It is also proven that alg \({\mathcal N}+{\mathcal K}\) is always a norm-closed principal bimodule of alg \({\mathcal N}.\) As an important application, the author considers the problem of when the Jacobson radical of a nest algebra alg \({\mathcal N}\) is a norm-closed principal ideal and gives an alternate proof of a result of J. Orr. Namely, for a complete nest \({\mathcal N}\) in a separable Hilbert space \({\mathcal H}\), the Jacobson radical of alg \({\mathcal N}\) is a norm-principal ideal if and only if the nest is countable and has no type I points. (The projection P in \({\mathcal N}\) is a type I point if P\({\mathcal H}\) and \(P^{\perp}{\mathcal H}\) are infinite-dimensional and if the nest \(\{P^{\perp}N\oplus 0:\) \(N\in {\mathcal N}\}\cup \{1\oplus PN:\) \(N\in {\mathcal N}\}\) is an order type I nest in the space \(P^{\perp}{\mathcal H}\oplus P{\mathcal H}\).)