The Jacobson radical of a CSL algebra TeX Export

@article{citeulike:586533, abstract = { J. R. Ringrose gave the well-known and useful characterization of the radical of a nest algebra as the closure of the set of operators \$A\$ for which there exists a finite partition \$P\_0=0\$, \$P\_1,\cdots,P\_n=1\$ of the nest, such that the "diagonal elements" \$(P\_i-P\_{i-1})A(P\_i-P\_{i-1})\$ are \$0\$ \$(i=1,\cdots,n)\$. (For references to the previously known results described in this review, see K. R. Davidson \ref[<em> Nest algebras</em>, Longman Sci. Tech., Harlow, 1988; <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=972978\&loc=fromrevtext">MR0972978 (90f:47062)</A>].) A. Hopenwasser conjectured that the analogous result for CSL algebras (using finite sublattices in the place of the finite subnets) was true. In addition to establishing the validity of the conjecture for certain CSLs, Hopenwasser and later Hopenwasser and Larson elucidated the problem, showing that the conjecture is natural and important for the theory of CSL algebras (see Davidson's monograph [op. cit.], where the conjecture was represented by Problem 3 on p. 379 in the list of open problems). <P> The paper under review represents significant progress on the problem, developing powerful methods which cannot be fully described here. The reader should not be misled by the brief description of partial results obtained: the paper makes a profound contribution by fully exposing the combinatorial nature of the problem, at least in the case of completely distributive CSLs. <P> One result is the verification of Hopenwasser's conjecture for the case of a CSL generated by two commuting nests. Also, a conjecture is verified for all completely distributive CSLs which satisfy a combinatorial condition which the authors call "disjointness". This appears to be a very large class of CSLs. However, according to a note added in proof, Ross Willard has constructed an example of a completely distributive CSL which does not have the property of disjointness. Thus the problem of establishing Hopenwasser's conjecture remains open, even in the case of completely distributive CSLs.}, author = {Davidson, Kenneth R. and Orr, John L.}, citeulike-article-id = {586533}, citeulike-linkout-0 = {http://www.ams.org/mathscinet-getitem?mr=1250816}, journal = {Trans. Amer. Math. Soc.}, keywords = {10071261, artin-wedderburn, bicommutant, density, division, jacobson, modulevector, neumann, regularirreducible, ringvon, space, theorem, theoremkaplansky, theoremvon}, mrnumber = {MR1250816}, number = {2}, pages = {925--947}, posted-at = {2007-05-11 22:29:36}, priority = {5}, title = {The Jacobson radical of a CSL algebra}, url = {http://www.ams.org/mathscinet-getitem?mr=1250816}, volume = {344}, year = {1994} }

TY - JOUR ID - citeulike:586533 L3 - citeulike-article-id:586533 N2 - J. R. Ringrose gave the well-known and useful characterization of the radical of a nest algebra as the closure of the set of operators $A$ for which there exists a finite partition $P_0=0$, $P_1,\cdots,P_n=1$ of the nest, such that the "diagonal elements" $(P_i-P_{i-1})A(P_i-P_{i-1})$ are $0$ $(i=1,\cdots,n)$. (For references to the previously known results described in this review, see K. R. Davidson \ref[<em> Nest algebras</em>, Longman Sci. Tech., Harlow, 1988; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=972978&loc=fromrevtext">MR0972978 (90f:47062)</A>].) A. Hopenwasser conjectured that the analogous result for CSL algebras (using finite sublattices in the place of the finite subnets) was true. In addition to establishing the validity of the conjecture for certain CSLs, Hopenwasser and later Hopenwasser and Larson elucidated the problem, showing that the conjecture is natural and important for the theory of CSL algebras (see Davidson's monograph [op. cit.], where the conjecture was represented by Problem 3 on p. 379 in the list of open problems). <P> The paper under review represents significant progress on the problem, developing powerful methods which cannot be fully described here. The reader should not be misled by the brief description of partial results obtained: the paper makes a profound contribution by fully exposing the combinatorial nature of the problem, at least in the case of completely distributive CSLs. <P> One result is the verification of Hopenwasser's conjecture for the case of a CSL generated by two commuting nests. Also, a conjecture is verified for all completely distributive CSLs which satisfy a combinatorial condition which the authors call "disjointness". This appears to be a very large class of CSLs. However, according to a note added in proof, Ross Willard has constructed an example of a completely distributive CSL which does not have the property of disjointness. Thus the problem of establishing Hopenwasser's conjecture remains open, even in the case of completely distributive CSLs. IS - 2 JF - Trans. Amer. Math. Soc. EP - 947 TI - The Jacobson radical of a CSL algebra VL - 344 SP - 925 KW - 10071261 KW - artin-wedderburn KW - bicommutant KW - density KW - division KW - jacobson KW - modulevector KW - neumann KW - regularirreducible KW - ringvon KW - space KW - theorem KW - theoremkaplansky KW - theoremvon AU - Davidson, Kenneth AU - Orr, John PY - 1994/// UR - http://www.ams.org/mathscinet-getitem?mr=1250816 ER -