The Jacobson radical of a CSL algebra
Authors:
Kenneth R. Davidson and John Lindsay Orr
Journal:
Trans. Amer. Math. Soc. 344 (1994), 925947
MSC:
Primary 47D25
MathSciNet review:
1250816
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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.
 [1]
Constantin
Apostol and Kenneth
R. Davidson, Isomorphisms modulo the compact operators of nest
algebras. II, Duke Math. J. 56 (1988), no. 1,
101–127. MR
932858 (89g:47058), http://dx.doi.org/10.1215/S0012709488056050
 [2]
William
Arveson, Operator algebras and invariant subspaces, Ann. of
Math. (2) 100 (1974), 433–532. MR 0365167
(51 #1420)
 [3]
Kenneth
R. Davidson, Nest algebras, Pitman Research Notes in
Mathematics Series, vol. 191, Longman Scientific & Technical,
Harlow; copublished in the United States with John Wiley & Sons, Inc.,
New York, 1988. Triangular forms for operator algebras on Hilbert space. MR 972978
(90f:47062)
 [4]
Kenneth
R. Davidson, Problems about reflexive algebras, Proceedings of
the Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987),
1990, pp. 317–330. MR 1065832
(91h:47042), http://dx.doi.org/10.1216/rmjm/1181073109
 [5]
Kenneth
R. Davidson and David
R. Pitts, Compactness and complete distributivity for commutative
subspace lattices, J. London Math. Soc. (2) 42
(1990), no. 1, 147–159. MR 1078182
(91j:47050), http://dx.doi.org/10.1112/jlms/s242.1.147
 [6]
Alan
Hopenwasser, The radical of a reflexive operator algebra,
Pacific J. Math. 65 (1976), no. 2, 375–392. MR 0440383
(55 #13258)
 [7]
, The equation in a reflexive operator algebra, Indiana Univ. Math. J. 29 (1980), 124126.
 [8]
Alan
Hopenwasser and David
Larson, The carrier space of a reflexive operator algebra,
Pacific J. Math. 81 (1979), no. 2, 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]
Bruce
H. Wagner, Weak limits of projections and
compactness of subspace lattices, Trans. Amer.
Math. Soc. 304 (1987), no. 2, 515–535. MR 911083
(89h:47065), http://dx.doi.org/10.1090/S00029947198709110832
 [1]
 C. Apostol and K. R. Davidson, Isomorphisms modulo the compact operators of nest algebras. II, Duke Math. J. 56 (1988), 101127. MR 932858 (89g:47058)
 [2]
 W. B. Arveson, Operator algebras and invariant subspaces, Ann. of Math. (2) 100 (1974), 433532. 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), 317330. 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), 147159. MR 1078182 (91j:47050)
 [6]
 A. Hopenwasser, The radical of a reflexive operator algebra, Pacific J. Math. 65 (1976), 375392. MR 0440383 (55:13258)
 [7]
 , The equation in a reflexive operator algebra, Indiana Univ. Math. J. 29 (1980), 124126.
 [8]
 A. Hopenwasser and D. R. Larson, The carrier space of a reflexive operator algebra, Pacific J. Math. 81 (1979), 417434. MR 547609 (81c:47046)
 [9]
 J. R. Ringrose, On some algebras of operators, Proc. London Math. Soc. (3) 15 (1965), 6183. MR 0171174 (30:1405)
 [10]
 B. Wagner, Weak limits of projections and compactness of subspace lattices, Trans. Amer. Math. Soc. 304 (1987), 515535. MR 911083 (89h:47065)
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DOI:
http://dx.doi.org/10.1090/S00029947199412508169
PII:
S 00029947(1994)12508169
Article copyright:
© Copyright 1994
American Mathematical Society
