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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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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: http://dx.doi.org/10.1090/S0002-9947-01-02957-9
PII: S 0002-9947(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