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Perturbations of AF-Algebras

Published online by Cambridge University Press:  20 November 2018

John Phillips  and
Iain Raeburn
John Phillips
Affiliation:
Dalhousie University, Halifax, Nova Scotia
Iain Raeburn
Affiliation:
University of New South Wales, Kensington, Australia
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Let A and B be C*-algebras acting on a Hilbert space H, and let

where A1 is the unit ball in A and d(a, B1) denotes the distance of a from B1. We shall consider the following problem: if ‖A – B‖ is sufficiently small, does it follow that there is a unitary operator u such that uAu* = B?

Such questions were first considered by Kadison and Kastler in [9], and have received considerable attention. In particular in the case where A is an approximately finite-dimensional (or hyperfinite) von Neumann algebra, the question has an affirmative answer (cf [3], [8], [12]). We shall show that in the case where A and B are approximately finite-dimensional C*-algebras (AF-algebras) the problem also has a positive answer.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 5 , 01 October 1979 , pp. 1012 - 1016
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Bratteli, O., Inductive limits of finite dimensional C*-algebras, Trans. Amer. Math. Soc. 171 (1972), 195–234.Google Scholar
2. Christensen, E., Perturbations of type 1 von Neumann algebras, J. London Math. Soc. 9 (1975), 395–405.Google Scholar
3. Christensen, E., Perturbations of operator algebras, Invent. Math. 43 (1977), 1–13.Google Scholar
4. Christensen, E., Near inclusions of C*-algebras, preprint.Google Scholar
5. Effros, E. and Rosenberg, J., C*-algebras with approximately inner flip, Pac. J. Math. 77 (1978), 417–443.Google Scholar
6. Elliott, G., On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebr. 38 (1976), 29–44.Google Scholar
7. Glimm, J., On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960), 318–340.Google Scholar
8. Johnson, B. E., Perturbations of Banach algebras, Proc. Lond. Math. Soc. (3). 34 (1977), 439–458.Google Scholar
9. Kadison, R. and Kastler, D., Perturbations of von Neumann algebras I. Stability of type, Amer. J. Math. 94 No. 1 (1972), 38–54.Google Scholar
10. Phillips, J. and Raeburn, I., On extensions of A.F.-algebras, Amer. J. Math., to appear.Google Scholar
11. Powers, R., Representations of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. of Math. 86 (1967), 138–171.Google Scholar
12. Raeburn, I. and Taylor, J., Hochschild cohomology and perturbations of Banach algebras, J. Functional Anal. 25 (1977), 258–266.Google Scholar