Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Jacobson radical of a CSL algebra


Authors: Kenneth R. Davidson and John Lindsay Orr
Journal: Trans. Amer. Math. Soc. 344 (1994), 925-947
MSC: Primary 47D25
DOI: https://doi.org/10.1090/S0002-9947-1994-1250816-9
MathSciNet review: 1250816
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] C. Apostol and K. R. Davidson, Isomorphisms modulo the compact operators of nest algebras. II, Duke Math. J. 56 (1988), 101-127. MR 932858 (89g:47058)
  • [2] W. B. Arveson, Operator algebras and invariant subspaces, Ann. of Math. (2) 100 (1974), 433-532. MR 0365167 (51:1420)
  • [3] K. R. Davidson, Nest algebras, Pitman Res. Notes in Math., vol. 191, Longman Sci. Tech., London and New York, 1988. MR 972978 (90f:47062)
  • [4] -, Problems in reflexive algebras, Proc. GPOTS Meeting 1987, Rocky Mountain Math. J. 20 (1990), 317-330. MR 1065832 (91h:47042)
  • [5] K. R. Davidson and D. R. Pitts, Compactness and complete distributivity for commutative subspace lattices, J. London Math. Soc. (2) 42 (1990), 147-159. MR 1078182 (91j:47050)
  • [6] A. Hopenwasser, The radical of a reflexive operator algebra, Pacific J. Math. 65 (1976), 375-392. MR 0440383 (55:13258)
  • [7] -, The equation $ Tx = y$ in a reflexive operator algebra, Indiana Univ. Math. J. 29 (1980), 124-126.
  • [8] A. Hopenwasser and D. R. Larson, The carrier space of a reflexive operator algebra, Pacific J. Math. 81 (1979), 417-434. MR 547609 (81c:47046)
  • [9] J. R. Ringrose, On some algebras of operators, Proc. London Math. Soc. (3) 15 (1965), 61-83. MR 0171174 (30:1405)
  • [10] B. Wagner, Weak limits of projections and compactness of subspace lattices, Trans. Amer. Math. Soc. 304 (1987), 515-535. MR 911083 (89h:47065)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47D25

Retrieve articles in all journals with MSC: 47D25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1250816-9
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society