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On the primitive ideals of nest algebras

Part of: Linear spaces and algebras of operators Radicals and radical properties of rings

Published online by Cambridge University Press:  21 July 2020

John Lindsay Orr
John Lindsay Orr*
Affiliation:
Toll House, Traquair Road, InnerleithenEH44 6PF, UK (me@johnorr.us)
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Abstract

We show that Ringrose's diagonal ideals are primitive ideals in a nest algebra (subject to the continuum hypothesis). This answers an old question of Lance and provides for the first time concrete descriptions of enough primitive ideals to obtain the Jacobson radical as their intersection. Separately, we provide a standard form for all left ideals of a nest algebra, which leads to insights into the maximal left ideals. In the case of atomic nest algebras, we show how primitive ideals can be categorized by their behaviour on the diagonal and provide concrete examples of all types.

Keywords

nest algebraprimitive idealsnetscontinuum hypothesis

MSC classification

Primary: 47L35: Nest algebras, CSL algebras
Secondary: 16N20: Jacobson radical, quasimultiplication
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 3 , August 2020 , pp. 737 - 760
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Copyright
Copyright © The Authors, 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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References

Bonsall, F. F. and Duncan, J., Complete normed algebras, Ergebnisse der Mathematik, Volume 80 (Springer, 1973).CrossRefGoogle Scholar
Davidson, K. R., Nest algebras, Research Notes in Mathematics, Volume 191 (Pitman, Boston, 1988).Google Scholar
Davidson, K. R., Harrison, K. J. and Orr, J. L., Epimorphisms of nest algebras, Internat. J. Math. 6(5) (1995), 657–687.CrossRefGoogle Scholar
Davidson, K. R., Katsoulis, E. and Pitts, D. R., The structure of free semigroup algebras, J. Reine Angew. Math. 533 (2001), 99–125.Google Scholar
Davidson, K. R. and Orr, J. L., The Jacobson radical of a CSL algebra, Trans. Amer. Math. Soc. 334(2) (1994), 925–947.CrossRefGoogle Scholar
Davidson, K. R. and Orr, J. L., Principal bimodules of nest algebras, J. Funct. Anal. 157(2) (1998), 488–533.CrossRefGoogle Scholar
Davidson, K. R. and Power, S. C., Best approximation in C*-algebras, J. Reine Angew. Math. 368 (1986), 43–62.Google Scholar
Donsig, A. P., Semisimple triangular AF algebras, J. Funct. Anal. 111(2) (1993), 323–349.CrossRefGoogle Scholar
Donsig, A. P., Katavolos, A. and Manoussos, A., The Jacobson radical for analytic crossed products, J. Funct. Anal. 187(1) (2001), 129–145.CrossRefGoogle Scholar
Fall, T., Arveson, W. and Muhly, P., Perturbations of nest algebras, J. Operator Theory 1(1) (1979), 137–150.Google Scholar
Katsoulis, E. G., Moore, R. L. and Trent, T. T., Interpolation in nest algebras and applications to operator corona theorems, J. Operator Theory 29(1) (1993), 115–123.Google Scholar
Katsoulis, E. G. and Ramsey, C., Crossed products of operator algebras: applications of Takai duality, J. Funct. Anal. 275(5) (2018), 1173–1207.CrossRefGoogle Scholar
Lance, E. C., Some properties of nest algebras, Proc. Lond. Math. Soc. 19(3) (1969), 45–68.CrossRefGoogle Scholar
Larson, D. R., Nest algebras and similarity transformations, Ann. of Math. 121 (1985), 409–427.CrossRefGoogle Scholar
Marcus, A. W., Spielman, D. A. and Srivastava, N., Interlacing families II: mixed characteristic polynomials and the Kadison–Singer problem, Ann. Math. 182 (2015), 327–350.CrossRefGoogle Scholar
Mastrangelo, L., Muhly, P. S. and Solel, B., Locating the radical of a triangular operator algebra, Math. Proc. Cambridge Philos. Soc. 115(1) (1994), 27–38.CrossRefGoogle Scholar
Orr, J. L., The maximal ideals of a nest algebra, J. Funct. Anal. 124 (1994), 119–134.CrossRefGoogle Scholar
Orr, J. L., Triangular algebras and ideals of nest algebras, Mem. Amer. Math. Soc. 562(117) (1995).Google Scholar
Orr, J. L., The stable ideals of a continuous nest algebra, J. Operator Theory 45 (2001), 377–412.Google Scholar
Orr, J. L., The maximal two-sided ideals of nest algebras, J. Operator Theory 73 (2015), 407–416.CrossRefGoogle Scholar
Orr, J. L. and Pitts, D. R., Factorization of triangular operators and ideals through the diagonal, Proc. Edinb. Math. Soc. 40 (1997), 227–241.CrossRefGoogle Scholar
Peters, J., Semicrossed products of C*-algebras, J. Funct. Anal. 59(3) (1984), 498–534.CrossRefGoogle Scholar
Ringrose, J. R., On some algebras of operators, Proc. Lond. Math. Soc. 15(3) (1965), 61–83.CrossRefGoogle Scholar
Weaver, N., Set theory and C*-algebras, Bull. Symb. Log. 13(1) (2007), 1–20.CrossRefGoogle Scholar