In this paper, we give a short survey of results and problems concerning the notion of bounded weak approximate identities in Banach algebras. Also, we introduce a new version of approximate identities and give one illuminating example to show the difference.
A. Ulger A.T.M. Lau. Characterization of closed ideals with bounded approximate identities in commutative Banach algebras, complemented subspaces of the group von Neumann algebras and applications. Trans. Amer. Math. Soc., 366(8):4151–5171, (2014).
M. Skantharajah B. Forrest. A note on a type of approximate identity in the Fourier algebra. Proc. Amer. Math. Soc., 120(2):651–652, (1994).
A.T-M. Lau B.E. Forrest, E. Kaniuth and N. Spronk. Ideals with bounded approximate identities in Fourier algebras. J. Funct. Anal., 203:286–304, (2003).
C. D. Lahr C.A. Jones. Weak and norm approximate identities are different. Pacific J. Math., 72(1):99–104, (1977).
A. Ulger E. Kaniuth. The Bochner-Schoenberg-Eberlein property for commutative Banach algebras, especially Fourier and Fourier Stieltjes algebras. Trans. Amer. Math. Soc., 362:4331–4356, (2010).
A. T-M. Lau F. Ghahramani. Weak amenability of certain classes of Banach algebras without bounded approximate identities. Math. Proc. Camb. Phil. Soc., 133(357):357–371, (2002).
M. Fozouni. On character space of the algebra of BSE-functions. Sahand Communications in Mathematical Analysis. To appear.
A. Ulger H.G. Dales. Approximate identities in Banach function algebras. Studia Math., 226:155–187, (2015).
S-E. Takahasi J. Inoue. Constructions of bounded weak approximate identities for Segal algebras on LCA groups. Acta. Sci. Math. (Szeged), 66(1-2):257–271, (2000).
S-E. Takahasi J. Inoue. On characterization of the image of the Gelfand transform of commutative Banach algebras. Math. Nachr., 280:105–126, (2007).
S.-E. Takahasi J. Inoue. Segal algebras in commutative Banach algebras. Rocky Mountain J. Math., 44(2):539–589, (2014).
S. Vaes J. Kustermans. Locally compact quantum groups in the Von neumann algebraic setting. Math. Scand., (92):68–92, (2003).
M. Fozouni J. Laali. On ∆-weak ϕ-amenable Banach algebras. U. P. B. Sci. Bull. Series A, 77(4):165–176, (2015).
M. Fozouni J. Laali. Closed ideals with bounded ∆-weak approximate identities in some certain Banach algebras. Miskolc Math. Notes, 17(1):413–420, (2016).
E. Kaniuth. A Course in Commutative Banach Algebras. Springer Verlag, Graduate texts in mathematics, (2009).
M. Nemati M. Fozouni. BSE property for some certain Segal and Banach algebras. Mediterr j. Math. To apperar.
R. Farrokhzad M. Fozouni. Two types of approximate identities depending on the character spaces of Banach algebras. arXiv:1507.05884.
J.P. Pier. Amenable locally compact groups. Wiley Interscience, New York, (1984).
V. Runde. Operator Fig`a-Talamanca-Herz algebras. Studia Math., 155(2):153–170, (2003).
O. Hatori S.-E. Takahasi. Commutative Banach algebras which satisfy a BochnerSchoenberg-Eberlein-type theorem. Proc. Amer. Math. Soc., 110:149–158, (1990).
M. L. Bami Z. Kamali. Bochner-Schoenberg-Eberlein property for abstract Segal algebras. Proc. Japan Acad. Ser. A, 89:107–110, (2013).
M. L. Bami Z. Kamali. The Bochner-Schoenberg-Eberlein property for L1 (R+). J. Fourier Anal. Appl., 20(2):225–233, (2014).
Fozouni, M. (2019). On Bounded Weak Approximate Identities and a New Version of Them. Analytical and Numerical Solutions for Nonlinear Equations, 4(1), 7-13. doi: 10.22128/gadm.2019.282.1018
MLA
Mohammad Fozouni. "On Bounded Weak Approximate Identities and a New Version of Them", Analytical and Numerical Solutions for Nonlinear Equations, 4, 1, 2019, 7-13. doi: 10.22128/gadm.2019.282.1018
HARVARD
Fozouni, M. (2019). 'On Bounded Weak Approximate Identities and a New Version of Them', Analytical and Numerical Solutions for Nonlinear Equations, 4(1), pp. 7-13. doi: 10.22128/gadm.2019.282.1018
VANCOUVER
Fozouni, M. On Bounded Weak Approximate Identities and a New Version of Them. Analytical and Numerical Solutions for Nonlinear Equations, 2019; 4(1): 7-13. doi: 10.22128/gadm.2019.282.1018