Le \({\mathcal K}\) denote the set of compact operators on a separable Hilbert space \({\mathcal H}\), let \({\mathcal T}({\mathcal N})\) denote the nest algebra associated with a nest \({\mathcal N}\) of subspaces of \({\mathcal H}\), and let \({\mathcal A}({\mathcal N})=({\mathcal T}({\mathcal N})+{\mathcal K})/{\mathcal K}\). The aim of this paper is to give a precise description of an arbitrary isomorphism \(\alpha\) of \({\mathcal A}({\mathcal N})\) onto another such algebra \({\mathcal A}({\mathcal M})\). In an earlier paper [part I, Pac. J. Math. 122, 263-286 (1986; Zbl 0625.47037)], the first author and F. Gilfeather considered the situation when both \({\mathcal N}\) and \({\mathcal M}\) consist of increasing sequences of finite-rank projections converging to the identity. The present paper relies heavily on this special case.
The precise form of \(\alpha\) in the general case is somewhat technical; in outline, there is a unitary element \(u\) of the Calkin algebra and an automorphism \(\alpha_ 0\) of \({\mathcal A}({\mathcal N})\) such that \(\alpha =Ad u\circ \alpha_ 0\). The element \(u\) may be taken to be the image in the Calkin algebra of a small compact perturbation of a partial isometry having a special relationship with \({\mathcal N}\) and \({\mathcal M}\), whilst \(\alpha_ 0\) preserves the nest \({\mathcal N}\) in a special way. As the authors themselves say, the paper is long and technically difficult. In order to help the reader, they give an overview of the proof in an introductory section.