Isomorphisms modulo the compact operators of nest algebras (Q579606)

Let \({\mathcal A}\) and \({\mathcal B}\) be nest algebras on a Hilbert space with finite rank projections. The authors show that the image of \({\mathcal A}\) in the Calkin algebra is isomorphic to the image of \({\mathcal B}\) in the Calkin algebra if and only if there exist integers \(n_ 0\) and \(m_ 0\) with rank \(P_{n_ 0+n}=rank Q_{m_ 0+n}\) for all n, where \(P_ n\) and \(Q_ n\) are the block diagonal projections for the two nests.