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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
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 $\mathcal {T} = \begin {bmatrix}\mathcal {A} & \mathcal {M}\\ 0 & \mathcal {B} \end {bmatrix}$ becomes a triangular Banach algebra when equipped with the Banach space norm $\left \Vert \begin {bmatrix} a & m\\ 0 & b \end {bmatrix} \right \Vert = \Vert a \Vert _{\mathcal {A}} + \Vert m \Vert _{\mathcal {M}} + \Vert b \Vert _{\mathcal {B}}$. 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
MR Author ID: 288388
Email: L.Marcoux@ualberta.ca, LWMarcoux@math.uwaterloo.ca

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