Approximate unitary equivalence of continuous nests. (English) Zbl 0599.47066
A short proof is given for the following theorem of N. T. Andersen [J. funct. Analysis 38, 366-400 (1980; Zbl 0451.47050)].
Theorem. Let \({\mathcal N}\) and \({\mathcal M}\) be two continuous nests of projections on a separable Hilbert space. Let \(\epsilon >0\) and let \(\theta\) be an order isomorphism of \({\mathcal N}\) onto \({\mathcal M}\). There is a unitary operator U such that \(\theta\) (N)U-UN is compact for all N in \({\mathcal N}\), and \(\sup \{\| \theta (N)U-UN\|:N\in {\mathcal N}\}<\epsilon\).
Theorem. Let \({\mathcal N}\) and \({\mathcal M}\) be two continuous nests of projections on a separable Hilbert space. Let \(\epsilon >0\) and let \(\theta\) be an order isomorphism of \({\mathcal N}\) onto \({\mathcal M}\). There is a unitary operator U such that \(\theta\) (N)U-UN is compact for all N in \({\mathcal N}\), and \(\sup \{\| \theta (N)U-UN\|:N\in {\mathcal N}\}<\epsilon\).
MSC:
| 47L30 | Abstract operator algebras on Hilbert spaces |
| 46L35 | Classifications of \(C^*\)-algebras |
| 46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |
Keywords:
approximate unitary equivalence; compact perturbations; continuous nests of projections on a separable Hilbert spaceCitations:
Zbl 0451.47050References:
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