Generalized Injectivity of Banach Modules
DOI:
https://doi.org/10.5644/SJM.11.2.06Keywords:
Banach algebra, injective module, character, $\phi$-injective module, locally compact groupAbstract
In this paper, we study the notion of $\phi$-injectivity in the special case that $\phi=0$. For an arbitrary locally compact group $G$, we characterize the 0-injectivity of $L^{1}(G)$ as a left $L^{1}(G)$ module. Also, we show that $L^{1}(G)^{**}$ and $L^{p}(G)$ for $1<p<\infty$ are 0-injective Banach $L^{1}(G)$ modules.
Downloads
References
H. G. Dales, Banach Algebras and Automatic Continuity, Clarendon Press, Oxford, 2000.
H. G. Dales and M. E. Polyakov, Homological properties of modules over group algebras, Proc. London Math. Soc., 89 (2004), 390–426.
H. G. Dales, M. Daws, H. L. Pham and P. Ramsden, Multi-norms and the injectivity of $L^{p}(G)$}, J. London Math. Soc., 86 (3) (2012), 779–809.
M. Essmaili, M. Fozouni and J. Laali, Hereditary properties of character injectivity with application to semigroup algebras, Ann. Funct. Anal., 6 (2) (2015), 162–172.
A. Ya. Helemskii, Banach and Locally Convex Algebras, Clarendon Press, Oxford, 1993.
A. Ya. Helemskii, The Homology of Banach and Topological Algebras, Kluwer Academic Publishers Group, Dordrecht, 1989.
R. Nasr-Isfahani and S. Soltani Renani, Character contractibility of Banach algebras and homological properties of Banach modules, Studia Math., 202 (2011), 205–225.
R. Nasr-Isfahani, Character injectivity and projectivity of Banach modules, Quart. J. Math., 65 (2) (2014), 665–676.
P. Ramsden, Homological properties of modules over semigroup algebras, J. Funct. Anal., 258 (2010), 3988–4009.
P. Ramsden, Homological properties of semigroup algebras, Ph. D. Thesis, University of Leeds, 2008.
M. C. White, Injective modules for uniform algebras, Proc. London Math. Soc., 73 (3) (1996), 155–184.