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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Jacobson radicals of nest algebras in factors


Author: Xing De Dai
Journal: Proc. Amer. Math. Soc. 119 (1993), 1259-1267
MSC: Primary 46L05; Secondary 46K50, 47D25
DOI: https://doi.org/10.1090/S0002-9939-1993-1160296-4
MathSciNet review: 1160296
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Abstract: Definition. Let $ \beta $ be a nest in a separably acting type $ {\text{I}}{{\text{I}}_\infty }$ factor $ \mathcal{M}$. An element $ P \in \beta \backslash \{ 0,I\} $ is said to be a singular point of $ \beta $ if it satisfies either of the following conditions:

(1) There is a strictly increasing sequence $ \{ {Q_n}\} \subseteq \beta ,\;{\lim _{n \to \infty }}{Q_n} = P$, and $ P - {Q_n}$ is infinite for each $ n \in \mathbb{N}$. Also, there is a projection $ Q \in \beta $ such that $ Q > P$ and $ Q - P$ is finite.

(2) There is a strictly decreasing sequence $ \{ {Q_n}\} \subseteq \beta ,\;{\lim _{n \to \infty }}{Q_n} = P$, and $ {Q_n} - P$ is infinite for each $ n \in \mathbb{N}$. Also, there is a projection $ Q \in \beta $ such that $ Q < P$ and $ P - Q$ is finite.

Main Theorem. Let $ \beta $ be a nest in a separably acting factor $ \mathcal{M}$.

(1) If $ \mathcal{M}$ is of type $ {\text{I}}{{\text{I}}_\infty }$, then a necessary and sufficient condition for the Jacobson radical $ {\mathcal{R}_\beta }$ of $ \operatorname{alg} \beta $ to be a norm-closed singly generated ideal of $ \operatorname{alg} \beta $ is that the nest $ \beta $ is countable and it does not contain a singular point.

(2) If $ \mathcal{M}$ is of type $ {\text{I}}{{\text{I}}_1}$ or type $ {\text{III}}$, then a necessary and sufficient condition for the Jacobson radical $ {\mathcal{R}_\beta }$ of $ \operatorname{alg} \beta $ to be a norm-closed singly generated ideal of $ \operatorname{alg} \beta $ is that the nest $ \beta $ is countable.

(3) In (1) and (2) the single generation is equivalent to countable generation.


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DOI: https://doi.org/10.1090/S0002-9939-1993-1160296-4
Article copyright: © Copyright 1993 American Mathematical Society