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Tensor products of $\sigma$-weakly closed nest algebra submodules

Author: Dong Zhe
Journal: Proc. Amer. Math. Soc. 133 (2005), 1629-1637
MSC (2000): Primary 47L75
Published electronically: December 21, 2004
MathSciNet review: 2120262
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Abstract: In this paper we prove that for any unital $\sigma$-weakly closed algebra $\mathcal A$ which is $\sigma$-weakly generated by finite-rank operators in $\mathcal A$, every $\sigma$-weakly closed $\mathcal A$-submodule has $Property\; S_{\sigma}$. In the case of nest algebras, if $\mathcal L_{1},\cdots,\mathcal L_{n}$ are nests, we obtain the following $n$-fold tensor product formula:

\begin{displaymath}\mathcal U_{\phi_{1}}{\overline{\otimes}}\cdots{\overline{\ot... ...{\phi_{n}}= \mathcal U_{\phi_{1}\otimes\cdots \otimes\phi_{n}},\end{displaymath}

where each $\mathcal U_{\phi_{i}}$ is the $\sigma$-weakly closed Alg $\mathcal L_{i}$-submodule determined by an order homomorphism $\phi_{i}$ from $\mathcal L_{i}$ into itself.

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Additional Information

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

Keywords: $Property\; S_{\sigma}$, tensor product, slice map
Received by editor(s): December 17, 2002
Published electronically: December 21, 2004
Additional Notes: This project was partially supported by the National Natural Science Foundation of China (No. 10401030) and the Zhejiang Nature Science Foundation (No. M103044)
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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