Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-17T18:38:58.677Z Has data issue: false hasContentIssue false
  • English
  • Français

BIPROJECTIVITY AND BIFLATNESS OF LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

Part of: Topological algebras, normed rings and algebras, Banach algebras Methods of category theory in functional analysis

Published online by Cambridge University Press:  15 July 2014

F. ABTAHI ,
A. GHAFARPANAH  and
A. REJALI
F. ABTAHI*
Affiliation:
Department of Mathematics, University of Isfahan, PO Box 81746-73441, Isfahan, Iran email f.abtahi@sci.ui.ac.ir, abtahif2002@yahoo.com
A. GHAFARPANAH
Affiliation:
Department of Mathematics, University of Isfahan, PO Box 81746-73441, Isfahan, Iran email ghafarpanah@sci.ui.ac.ir, ghafarpanah1393@gmail.com
A. REJALI
Affiliation:
Department of Mathematics, University of Isfahan, PO Box 81746-73441, Isfahan, Iran email rejali@sci.ui.ac.ir, alirejali12@gmail.com
*
✉abtahi.fatemeh@yahoo.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ${\it\varphi}$ be a homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$. We define a multiplication on the Cartesian product space ${\mathcal{A}}\times {\mathcal{B}}$ and obtain a new Banach algebra ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$. We show that biprojectivity as well as biflatness of ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$ are stable with respect to ${\it\varphi}$.

Keywords

biflatnessbiprojectivitybimodule map𝜃-Lau product

MSC classification

Primary: 46M10: Projective and injective objects
Secondary: 46H25: Normed modules and Banach modules, topological modules (if not placed in or ) 46M18: Homological methods (exact sequences, right inverses, lifting, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 1 , February 2015 , pp. 134 - 144
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Abtahi, F., ‘Generalized biprojectivity and biflatness of abstract Segal algebras’, Banach J. Math. Anal. 8 (2014), 107–117.Google Scholar
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 (2013), 195–206.Google Scholar
Dales, H. G., Banach Algebras and Automatic Continuity (Clarendon Press, Oxford, 2000).Google Scholar
Ghahramani, F. and Zhang, Y., ‘Pseudo-amenable and pseudo-contractible Banach algebras’, Math. Proc. Cambridge Philos. Soc. 142 (2007), 111–123.Google Scholar
Helemskii, A. Ya., The Homology of Banach and Topological Algebras, Mathematics and its Applications (Soviet Series), 41 (Kluwer, Dordrecht, 1989), xx+334. Translated from the Russian by Alan West.Google Scholar
Helemskii, A. Ya., Banach and Locally Convex Algebras (Clarendon Press, Oxford, 1993).CrossRefGoogle Scholar
Johnson, B. E., ‘Approximate diagonals and cohomology of certain annihilator Banach algebras’, Amer. J. Math. 94 (1972), 685–698.Google Scholar
Kelley, J. L., General Topology, Graduate Texts in Mathematics, 27 (Springer, New York, 1975), Reprint of the 1955 edition.Google Scholar
Khoddami, A. R. and Ebrahimi Vishki, H. R., ‘Biflatness and biprojectivity of Lau product of Banach algebras’, Bull. Iranian Math. Soc. 39 (2013), 559–568.Google Scholar
Lau, A. T.-M., ‘Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups’, Fund. Math. 118 (1983), 161–175.Google Scholar
Runde, V., Lectures on Amenability, Lecture Notes in Mathematics, 1774 (Springer, Berlin, 2002).CrossRefGoogle Scholar
Samei, E., Spronk, N. and Stokke, R., ‘Biflatness and pseudo-amenability of Segal algebras’, Canad. J. Math. 62 (2010), 845–869.Google Scholar
Sangani Monfared, M., ‘On certain products of Banach algebras with applications to harmonic analysis’, Studia Math. 178 (2007), 277–294.CrossRefGoogle Scholar
Zhang, Y., ‘Nilpotent ideals in a class of Banach algebras’, Proc. Amer. Math. Soc. 127 (1999), 3237–3242.Google Scholar