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TRIVIALITY OF THE GENERALISED LAU PRODUCT ASSOCIATED TO A BANACH ALGEBRA HOMOMORPHISM

  • YEMON CHOI (a1)
Abstract

Several papers have, as their raison d’être, the exploration of the generalised Lau product associated to a homomorphism $T:B\rightarrow A$ of Banach algebras. In this short note, we demonstrate that the generalised Lau product is isomorphic as a Banach algebra to the usual direct product $A\oplus B$ . We also correct some misleading claims made about the relationship between this generalised Lau product and an older construction of Monfared [‘On certain products of Banach algebras with applications to harmonic analysis’, Studia Math. 178(3) (2007), 277–294].

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[1] Abtahi, F. and Ghafarpanah, A., ‘A note on cyclic amenability of the Lau product of Banach algebras defined by a Banach algebra morphism’, Bull. Aust. Math. Soc. 92(2) (2015), 282289.
[2] Abtahi, F., Ghafarpanah, A. and Rejali, A., ‘Biprojectivity and biflatness of Lau product of Banach algebras defined by a Banach algebra morphism’, Bull. Aust. Math. Soc. 91(1) (2015), 134144.
[3] Bhatt, S. J. and Dabhi, P. A., ‘Arens regularity and amenability of Lau product of Banach algebras defined by a Banach algebra morphism’, Bull. Aust. Math. Soc. 87(2) (2013), 195206.
[4] Dabhi, P. A., Jabbari, A. and Haghnejad Azar, K., ‘Some notes on amenability and weak amenability of Lau product of Banach algebras defined by a Banach algebra morphism’, Acta Math. Sin. (Engl. Ser.) 31(9) (2015), 14611474.
[5] Javanshiri, H. and Nemati, M., ‘On a certain product of Banach algebras and some of its properties’, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 15(3) (2014), 219227.
[6] Javanshiri, H. and Nemati, M., ‘The multiplier algebra and BSE-functions for certain product of Banach algebras’, Preprint, 2015, arXiv:1509.00895.
[7] Khoddami, A. R., ‘ n-weak amenability of T-Lau product of Banach algebras’, Chamchuri J. Math. 5 (2013), 5765.
[8] Khoddami, A. R., ‘On Banach algebras induced by a certain product’, Chamchuri J. Math. 6 (2014), 8996.
[9] Monfared, M. S., ‘On certain products of Banach algebras with applications to harmonic analysis’, Studia Math. 178(3) (2007), 277294.
[10] Nemati, M. and Javanshiri, H., ‘Some homological and cohomological notions on T-Lau product of Banach algebras’, Banach J. Math. Anal. 9(2) (2015), 183195.
[11] Pourabbas, A. and Razi, N., ‘Some homological properties of -Lau product algebra’. Preprint, 2014, arXiv:1411.0112.
[12] Pourabbas, A. and Razi, N., ‘Cohomological characterization of -Lau product algebras’. Preprint, 2015, arXiv:1509.01933.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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