Lie ideals in triangular operator algebras
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- by T. D. Hudson, L. W. Marcoux and A. R. Sourour PDF
- Trans. Amer. Math. Soc. 350 (1998), 3321-3339 Request permission
Abstract:
We study Lie ideals in two classes of triangular operator algebras: nest algebras and triangular UHF algebras. Our main results show that if $\mathcal {L}$ is a closed Lie ideal of the triangular operator algebra $\mathbb {A}$, then there exist a closed associative ideal $\mathcal {K}$ and a closed subalgebra $\mathfrak {D}_{\mathcal {K}}$ of the diagonal $\mathbb {A}\cap \mathbb {A}^*$ so that $\mathcal {K} \subseteq \mathcal {L} \subseteq \mathcal {K}+ \mathfrak {D}_{\mathcal {K}}$.References
- 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
- J. A. Erdos and S. C. Power, Weakly closed ideals of nest algebras, J. Operator Theory 7 (1982), no. 2, 219–235. MR 658610
- C. K. Fong, C. R. Miers, and A. R. Sourour, Lie and Jordan ideals of operators on Hilbert space, Proc. Amer. Math. Soc. 84 (1982), no. 4, 516–520. MR 643740, DOI 10.1090/S0002-9939-1982-0643740-0
- K.-H. Förster and B. Nagy, Lie and Jordan ideals in $B(c_0)$ and $B(l_\rho )$, Proc. Amer. Math. Soc. 117 (1993), no. 3, 673–677. MR 1123652, DOI 10.1090/S0002-9939-1993-1123652-6
- James G. Glimm, On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960), 318–340. MR 112057, DOI 10.1090/S0002-9947-1960-0112057-5
- I. N. Herstein, Topics in ring theory, University of Chicago Press, Chicago, Ill.-London, 1969. MR 0271135
- Alan Hopenwasser and Justin R. Peters, Full nest algebras, Illinois J. Math. 38 (1994), no. 3, 501–520. MR 1269701
- T. D. Hudson, Ideals in triangular AF algebras, Proc. London Math. Soc. (3) 69 (1994), no. 2, 345–376. MR 1281969, DOI 10.1112/plms/s3-69.2.345
- Laurent Marcoux, On the closed Lie ideals of certain $C^*$-algebras, Integral Equations Operator Theory 22 (1995), no. 4, 463–475. MR 1343340, DOI 10.1007/BF01203386
- C. Robert Miers, Closed Lie ideals in operator algebras, Canadian J. Math. 33 (1981), no. 5, 1271–1278. MR 638381, DOI 10.4153/CJM-1981-096-0
- G. J. Murphy, Lie ideals in associative algebras, Canad. Math. Bull. 27 (1984), no. 1, 10–15. MR 725245, DOI 10.4153/CMB-1984-002-0
- J. R. Peters, Y. T. Poon, and B. H. Wagner, Triangular AF algebras, J. Operator Theory 23 (1990), no. 1, 81–114. MR 1054818
- Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
- Stephen C. Power, Limit algebras: an introduction to subalgebras of $C^*$-algebras, Pitman Research Notes in Mathematics Series, vol. 278, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR 1204657
Additional Information
- T. D. Hudson
- Affiliation: Department of Mathematics East Carolina University Greenville, North Carolina, 27858-4353
- Email: tdh@math.ecu.edu
- L. W. Marcoux
- Affiliation: Department of Mathematical Sciences University of Alberta Edmonton, Alberta, Canada T6G 2G1
- MR Author ID: 288388
- Email: L.Marcoux@ualberta.ca
- A. R. Sourour
- Affiliation: Department of Mathematics University of Victoria Victoria, British Columbia, Canada V8W 3P4
- Email: sourour@math.uvic.ca
- Received by editor(s): October 4, 1996
- Additional Notes: This research was supported in part by an NSF grant (to Hudson) and by NSERC (of Canada) grants (to Marcoux and Sourour)
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 3321-3339
- MSC (1991): Primary 47D25, 46K50
- DOI: https://doi.org/10.1090/S0002-9947-98-02117-5
- MathSciNet review: 1458325