## Weak amenability of triangular Banach algebras

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- by B. E. Forrest and L. W. Marcoux PDF
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**354**(2002), 1435-1452 Request permission

## 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)
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1873013