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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Compact perturbations of certain von Neumann algebras

Author: Joan K. Plastiras
Journal: Trans. Amer. Math. Soc. 234 (1977), 561-577
MSC: Primary 47C05; Secondary 46L15
MathSciNet review: 0458241
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Abstract: Let $ \mathcal{E}$ be a sequence of mutually orthogonal, finite dimensional projections whose sum is the identity on a Hilbert space $ \mathcal{H}$. If we denote the commutant of $ \mathcal{E}$ by $ \mathcal{D}(\mathcal{E})$ and the ideal of compact operators on $ \mathcal{H}$ by $ \mathcal{C}(\mathcal{H})$, then it is easily verified that $ \mathcal{D}(\mathcal{E}) + \mathcal{C}(\mathcal{H}) = \{ T + K:T \in \mathcal{D}(\mathcal{E}),K \in \mathcal{C}(\mathcal{H})\} $ is a $ {C^\ast}$-algebra. In this paper we classify all such algebras up to $ ^\ast$-isomorphism and characterize them by examining their relationship to certain quasidiagonal and quasitriangular algebras.

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Keywords: Perturbed block diagonal algebras, $ {C^\ast}$-algebra isomorphism, compact perturbations, quasidiagonal operators, Fredholm index, Hilbert space
Article copyright: © Copyright 1977 American Mathematical Society