Left $\phi$-biprojectivity of some classes of abstract Segal algebras

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Basic Sciences, Ilam University P.O. Box 69315- 516 Ilam, Iran.

10.29252/as.2020.1960

Abstract

In this paper, we investigate left $\phi$-biprojectivity of Segal algebras and abstract Segal algebras. We show that for some abstract Segal algebras with some mild conditions left $\phi$-biprojectivity is equivalent with left $\phi$-contractibility. Also, we characterize left $\phi$-biprojectivity of a Segal algebra $S(G)$ in the terms of compactness of $G,$ where $G$ is a locally compact group. We introduce a class of abstract Segal algebras among Triangular Banach algebras. We show that  some abstract Segal algebras related to triangular Banach algebras are not biprojective.

Keywords


[1] M. Alaghmandan, R. Nasr Isfahani and M. Nemati, Character amenability and contractibility of abstract Segal algebras, Bull. Austral. Math. Soc, 82 (2010), 274-281.
[2] B. E. Forrest and L. E. Marcoux; Derivations of triangular Banach algebras, Indiana Univ. Math. J. 45 (1996), 441-462.
[3] B. E. Forrest and L. E. Marcoux; Weak amenability of triangular Banach algebras, Trans. Amer. Math. Math. J. 354 (2001), 1435-1452.
[4] F. Ghahramani and A. T. Lau, Weak amenability of certain classes of Banach algebra without bounded approximate identity, Math. Proc. Cambridge Philos. Soc 133 (2002), 357-371.
[5] Z. Hu, M. S. Monfared and T. Traynor, On character amenable Banach algebras, Studia Math. 193 (2009), 53-78.
[6] A. Jabbari, T. Mehdi Abad and M. Zaman Abadi, On φ-inner amenable Banach algebras, Colloq. Math. vol 122 (2011) 1-10.
[7] R. Nasr-Isfahani and S. Soltani Renani; Character contractibility of Banach algebras and homological properties of Banach modules, Stud. Math. 202(3), (2011), 205-225.
[8] H. Reiter; L 1 -algebras and Segal Algebras, Lecture Notes in Mathematics 231 (Springer, 1971).
[9] V. Runde, Lectures on amenability, Springer, New York, (2002). [10] A. Sahami On left φ-biprojectivity and left φ-biflatness of certain Banach algebras, U.P.B. Sci. Bull., Series A.Vol. 81, Iss. 4, 2019.
[11] S. S. Salimi, A. Mahmoodi, A. Sahami and M. Rostami, Left φ-biflatness and φ-biprojectivity of certain Banach algebras, Submitted.
[12] H. Samea, Essential amenability of abstract Segal algebras, Bull. Austral. Math. Soc, 79 (2009), 319-325.