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Transactions of the American Mathematical Society

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



Quasidiagonality of direct sums of weighted shifts

Author: Sivaram K. Narayan
Journal: Trans. Amer. Math. Soc. 332 (1992), 757-774
MSC: Primary 47B37; Secondary 47A66
MathSciNet review: 1012511
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Abstract: Let $ \mathcal{H}$ be a separable Hilbert space. A bounded linear operator $ A$ defined on $ \mathcal{H}$ is said to be quasidiagonal if there exists a sequence $ \{ {P_n}\} $ of projections of finite rank such that $ {P_n} \to I$ strongly and $ \left\Vert A{P_n} - {P_n}A\right\Vert \to 0$ as $ n \to \infty $.

We give a necessary and sufficient condition for a finite direct sum of weighted shifts to be quasidiagonal. The condition is stated using a marked graph (a graph with a $ (0)$, $ ( + )$ or $ ( - )$ attached to its vertices) that can be associated with the direct sum.

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Keywords: Quasidiagonality, weighted shifts, marked graphs, crossed products of $ {C^{\ast} }$-algebras
Article copyright: © Copyright 1992 American Mathematical Society

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