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Proceedings of the American Mathematical Society

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Single elements and module isomorphisms of some operator algebra modules

Author: Dong Zhe
Journal: Proc. Amer. Math. Soc. 135 (2007), 191-200
MSC (2000): Primary 47L75
Published electronically: June 22, 2006
MathSciNet review: 2280187
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Abstract: In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in the Alg$ \mathcal{L}$-module $ \mathcal{U}$ is initiated, where $ \mathcal{L}$ is a completely distributive subspace lattice on a Hilbert space $ {\mathcal H}$. Furthermore, as an application of single elements, we study module isomorphisms between norm closed Alg$ \mathcal{N}$-modules, where $ \mathcal{N}$ is a nest, and obtain the following result: Suppose that $ \mathcal{U}, \mathcal{V}$ are norm closed Alg$ \mathcal{N}$-modules and that $ \Phi: \mathcal{U}\rightarrow\mathcal{V}$ is a module isomorphism. Then $ \mathcal{U}=\mathcal{V}$ and there exists a non-zero complex number $ \lambda$ such that $ \Phi (T)=\lambda T, \forall T\in\mathcal{U}$.

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Dong Zhe
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China

Keywords: Single element, module isomorphism, nest algebra module
Received by editor(s): November 3, 2004
Received by editor(s) in revised form: July 30, 2005
Published electronically: June 22, 2006
Additional Notes: This project was partially supported by the National Natural Science Foundation of China (No. 10401030), the Zhejiang Natural Science Foundation (No. M103044) and GFJG
Communicated by: David R. Larson
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.