Abstract
Two C*-subalgebras A and B of the algebra of bounded operators on a Hilbert space are close if the Hausdorff distance between their unitballs is small. I present some results, which show that in many cases close C*-algebras do have many properties in common, and under certain conditions they actually have to be unitarily equivalent via a unitary close to the identity. The study involves a number of questions such as: Which invariants are stable under a small perturbation of a C*-algebra? Are close C*-algebras linearly isomorphic? If the algebras are linearly isomorphic through a mapping which is nearly multiplicative, are the algebras then algebraicly isomorphic? Is an algebra isomorphism of an algebra A onto an algebra B such that φ(a) − a ≤ δ a for some small δ implemented by an invertible near the identity? The following article presents some positive answers and a pair of counter examples.
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© 1988 Kluwer Academic Publishers
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Christensen, E. (1988). Close Operator Algebras. In: Hazewinkel, M., Gerstenhaber, M. (eds) Deformation Theory of Algebras and Structures and Applications. NATO ASI Series, vol 247. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3057-5_9
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DOI: https://doi.org/10.1007/978-94-009-3057-5_9
Publisher Name: Springer, Dordrecht
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