Best approximation and quasitriangular algebras

Author:
Timothy G. Feeman

Journal:
Trans. Amer. Math. Soc. **288** (1985), 179-187

MSC:
Primary 47D25; Secondary 41A35, 41A65, 47A66

MathSciNet review:
773055

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Abstract: If is a linearly ordered set of projections on a Hilbert space and is the ideal of compact operators, then is the quasitriangular algebra associated with . We study the problem of finding best approximants in a given quasitriangular algebra to a given operator: given and , is there an in such that ? We prove that if is an operator subalgebra which is closed in the weak operator topology and satisfies a certain condition , then every operator has a best approximant in . We also show that if is an increasing sequence of finite rank projections converging strongly to the identity then satisfies the condition . Also, we show that if is not in then the best approximants in to are never unique.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0773055-9

Keywords:
Quasitriangular operator algebras,
nest algebra,
best approximation

Article copyright:
© Copyright 1985
American Mathematical Society