The Jacobson radical of a CSL algebra

Davidson, Kenneth R.; Orr, John Lindsay

doi:10.1090/S0002-9947-1994-1250816-9

The Jacobson radical of a CSL algebra
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by Kenneth R. Davidson and John Lindsay Orr

Trans. Amer. Math. Soc. 344 (1994), 925-947

DOI: https://doi.org/10.1090/S0002-9947-1994-1250816-9

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Abstract:

Extrapolating from Ringrose’s characterization of the Jacobson radical of a nest algebra, Hopenwasser conjectured that the radical of a CSL algebra coincides with the Ringrose ideal (the closure of the union of zero diagonal elements with respect to finite sublattices). A general interpolation theorem is proved that reduces this conjecture for completely distributive lattices to a strictly combinatorial problem. This problem is solved for all width two lattices (with no restriction of complete distributivity), verifying the conjecture in this case.

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