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Weak amenability of triangular Banach algebras


Authors: B. E. Forrest and L. W. Marcoux
Journal: Trans. Amer. Math. Soc. 354 (2002), 1435-1452
MSC (2000): Primary 46H25, 16E40
DOI: https://doi.org/10.1090/S0002-9947-01-02957-9
Published electronically: December 4, 2001
MathSciNet review: 1873013
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Abstract: Let ${\mathcal A}$ and ${\mathcal B}$be unital Banach algebras, and let ${\mathcal M}$ be a Banach ${\mathcal A},{\mathcal B}$-module. Then ${\mathcal T} = \left[ \begin{array}{cc} {\mathcal A} & {\mathcal M} \ 0 & {\mathcal B} \end{array} \right]$ becomes a triangular Banach algebra when equipped with the Banach space norm $\ensuremath {\Vert}\left[ \begin{array}{cc} a & m \ 0 & b \end{array} \right] \... ...rt} _{{\mathcal M}} + \ensuremath {\Vert} b \ensuremath {\Vert} _{{\mathcal B}}$. A Banach algebra ${\mathcal T}$is said to be $n$-weakly amenable if all derivations from ${\mathcal T}$ into its $n^{\mathrm{th}}$ dual space ${\mathcal T}^{(n)}$ are inner. In this paper we investigate Arens regularity and $n$-weak amenability of a triangular Banach algebra ${\mathcal T}$ in relation to that of the algebras ${\mathcal A}$, ${\mathcal B}$ and their action on the module ${\mathcal M}$.


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  • 1. Bade, W.G., Curtis, P.C. and Dales, H.G., Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. 55 (1987), 359-377. MR 88f:46098
  • 2. Bade, W.G., Dales, H.G. and Lykova, Z.A., Algebraic and strong splittings of extensions of Banach algebras, Mem. Amer. Math. Soc., 137 (1999), No. 656.MR 99g:46059
  • 3. Christensen, E., Derivations of nest algebras, Math. Ann., 229 (1977), 155-161.MR 56:6420
  • 4. Civin, P. and Yood, B., The second conjugate space of a Banach algebra as an algebra, Pacific J. Math., 11 (1961), 847-870.MR 26:622
  • 5. Dales, H.G., Ghahramani, F. and Grønbæk, N., Derivations into iterated duals of Banach algebras, Studia Math., 128 (1998), 19-54. MR 99g:46064
  • 6. Davidson, K.R., Nest Algebras, Pitman Research Notes in Mathematics, no. 191, Longman Scientific and Technical, London, 1988.MR 90f:47062
  • 7. Forrest, B.E. and Marcoux, L.W., Derivations of triangular Banach algebras, Indiana Univ. Math. J. 45 (1996), 441-462.MR 97m:46079
  • 8. Gilfeather, F. and Smith, R.R., Cohomology for operator algebras: joins, Amer. J. Math. 116 (1994), 541-561. MR 95d:46077
  • 9. Haagerup, U., All nuclear $C^{*}$-algebras are amenable, Invent. Math. 74 (1983), 305-319. MR 85g:46074
  • 10. Johnson, B.E., Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127 (1992). MR 51:11130
  • 11. Kadison, R.V., Derivations of operator algebras, Ann. Math., 83 (1966), 280-293. MR 33:1747
  • 12. Murphy, G.J., $C^{*}$-Algebras and Operator Theory, Academic Press, San Diego, 1990. MR 91m:46084
  • 13. Palmer, T.W., Banach algebras and the general theory of $\,^{*}$-algebras, Vol. 1, Algebras and Banach Algebras, Cambridge Univ. Press, 1994. MR 95c:46002
  • 14. Sakai, S., Derivations of $W^{*}$-algebras, Ann. Math., 83 (1966), 273-279.MR 33:1748

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Additional Information

B. E. Forrest
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: beforres@math.uwaterloo.ca

L. W. Marcoux
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Address at time of publication: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: L.Marcoux@ualberta.ca, LWMarcoux@math.uwaterloo.ca

DOI: https://doi.org/10.1090/S0002-9947-01-02957-9
Received by editor(s): October 9, 1998
Received by editor(s) in revised form: July 20, 1999
Published electronically: December 4, 2001
Additional Notes: Research supported in part by NSERC (Canada)
Article copyright: © Copyright 2001 American Mathematical Society

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