Jacobson radicals of nest algebras in factors
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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.References
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Additional Information
- © Copyright 1993 American Mathematical Society
- 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