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 be a nest in a separably acting type factor . An element is said to be a singular point of if it satisfies either of the following conditions:

(1) There is a strictly increasing sequence , and is infinite for each . Also, there is a projection such that and is finite.

(2) There is a strictly decreasing sequence , and is infinite for each . Also, there is a projection such that and is finite.

Main Theorem. *Let* *be a nest in a separably acting factor* .

(1) *If* *is of type* , *then a necessary and sufficient condition for the Jacobson radical* *of* *to be a norm-closed singly generated ideal of* *is that the nest* *is countable and it does not contain a singular point*.

(2) *If* *is of type* *or type* , *then a necessary and sufficient condition for the Jacobson radical* *of* *to be a norm-closed singly generated ideal of* *is that the nest* *is countable*.

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

**[1]**X. Dai,*Norm-principal bimodules of nest algebras*, J. Funct. Anal.**90**(1990), 369-390. MR**1052339 (91f:47056)****[2]**-,*Norm-principal bimodules of nest algebras*II, preprint.**[3]**K. R. Davidson,*Nest algebras*, Pitman Res. Notes in Math. Ser., vol. 191, Longman Sci. Tech., Harlow, 1988. MR**972978 (90f:47062)****[4]**R. V. Kadison and J. R. Ringrose,*Fundamentals of the theory of operator algebras*, vols. I and II, Academic Press, New York, 1983, 1986. MR**719020 (85j:46099)****[5]**Frank Gilfeather and David Larson,*Nest-subalgebras of von Neumann algebras*. Adv. in Math.**46**(1982), 171-199. MR**679907 (84c:47047)****[6]**J. L. Orr,*On generators of the radical of a nest algebra*, J. London Math. Soc. (2)**40**(1989), 547-562. MR**1053621 (91c:47090)****[7]**J. R. Ringrose,*On some algebras of operators*, Proc. London Math. Soc. (3)**15**(1965), 61-83. MR**0171174 (30:1405)**

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DOI:
https://doi.org/10.1090/S0002-9939-1993-1160296-4

Article copyright:
© Copyright 1993
American Mathematical Society