The Banach algebras with generalized matrix representation

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord, Islamic Republic of Iran.

10.22072/wala.2020.122402.1273

Abstract

A Banach algebra $\mathfrak{A}$ has a generalized Matrix representation if there exist the algebras $A, B$, $(A,B)$-module $M$ and $(B,A)$-module $N$ such that $\mathfrak{A}$ is isomorphic to the generalized matrix Banach algebra $\Big[\begin{array}{cc}
A & \ M \\
N & \ B%
\end{array}%
\Big]$.
In this paper, the algebras with generalized matrix representation will be characterized. Then we show that there is a unital permanently weakly amenable  Banach algebra $A$ without generalized matrix representation such that $H^1(A,A)=\{0\}$.
This implies that there is a unital Banach algebra $A$ without any triangular matrix representation such that $H^1(A,A)=\{0\}$  and gives a negative answer to the open question of \cite{D}.

Keywords


[1] W.G. Bade, P.C. Curtis and H.G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. Lond. Math. Soc., 55 (1987), 359-377.
[2] D. Bennis and B. Fahid, Derivations and the first cohomology group of trivial extension algebras, Mediterr. J. Math., 14(150) (2017), https://doi.org/10.1007/s00009-017-0949-z.
[3] G.F. Birkenmeier, J.K. Park and S.T. Rizvi, Extensions of Rings and Modules, Birkhauser, New York, 2013.
[4] J.M. Cohen, C*-Algebras without ldempotents, J. Funct. Anal., 33 (1979), 211-216.
[5] H.G. Dales, Banach Algebras and Automatic Continuity, vol. 24 of London Mathematical Society Monographs, The Clarendon Press, Oxford, UK, 2000.
[6] H.G. Dales and A.T.M. Lau, The Second Duals of Beurling Algebras, Memoirs of the American Mathematical Society, 2005.
[7] H.G. Dales, F. Ghahramani and N. Gr{o}nb{ae}k, Derivations into iterated duals of Banach algebras, Stud. Math., 128(1)(1998), 19-54.
[8] G.B. Folland, A Course in Abstract Harmonic Analysis,  CRC Press, (1995).
[9] H. Lakzian and S. Barootkoob, Biprojectivity and biflatness of bi-amalgamated Banach  algebras, Bull. Iran. Math. Soc., https://doi.org/10.1007/s41980-020-00366-w.
[10] A.T.-M. Lau, Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups, Fundam. Math., 118} (1983), 161-175.
[11] Y. Li and F. Wei, newblock Semi-centralizing maps of generalized matrix algebras, Linear Algebra Appl., 436(5) (2012), 1122-1153.
[12] M. Ramezanpour and S. Barootkoob, Generalized module extension Banach algebras: Derivations and weak amenability, Quaest. Math., (2017), 1-15
[13] A.D. Sands, Radicals and morita contexts, J. Algebra, 24 (1973), 335-345.
[14] Y. Zhang, Weak amenableility of module extension of Banach algebras, Trans. Am. Math. Soc., 354 (2002), 4131-4151.
[15] Y. Zhang, $2m-$Weak amenability of group algebras, J. Math. Anal. Appl., 396 (2012), 412-416.