Lie ideals in triangular operator algebras

Hudson, T.; Marcoux, L.; Sourour, A.

doi:10.1090/S0002-9947-98-02117-5

Lie ideals in triangular operator algebras
HTML articles powered by AMS MathViewer

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