Document Type : Research Paper

Author

Department of Mathematics and Statistics, Faculty of Basic Sciences and Engineering, Gonbad Kavous University, P.O.Box 163, Gonbad Kavous, Iran.

Abstract

Suppose that $A$ is a semi-simple and commutative Banach algebra. In this paper we try to characterize the character space of the Banach algebra $C_{\rm{BSE}}(\Delta(A))$ consisting of all  BSE-functions on $\Delta(A)$ where $\Delta(A)$ denotes the character space of $A$. Indeed, in the case that $A=C_0(X)$ where $X$ is a non-empty locally compact Hausdroff space, we give a complete characterization of $\Delta(C_{\rm{BSE}}(\Delta(A)))$ and in the general case we give a partial answer.  Also, using the Fourier algebra, we show that $C_{\rm{BSE}}(\Delta(A))$ is not a $C^*$-algebra in general. Finally for some subsets $E$ of $A^*$, we define the subspace of BSE-like functions on $\Delta(A)\cup E$ and give a nice application of this space related to Goldstine's theorem.

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###### ##### References
[1] C.D. Aliprantis and K.C. Border, Infinite Dimensional Analysis, Springer-Verlag Berlin Heidelberg, edition 3, 2006.

[2] H.G. Dales, Banach Algebras and Automatic Continuity, Clarendon Press, Oxford, 2000.

[3] Z. Kamali and M.L. Bami, The Bochner-Schoenberg-Eberlein Property for ${L}^{1}(mathbb{R}^{+})$, J. Fourier Anal. Appl., 20 (2014), pp. 225-233.

[4] E. Kaniuth, A Course in Commutative Banach Algebras, Springer Verlag, Graduate texts in mathematics, 2009.

[5] E. Kaniuth and A. Ulger, The Bochner-Schoenberg-Eberlein property for commutative Banach algebras, especially Fourier and Fourier-Stieltjes algebras, Trans. Amer. Math. Soc., 362 (2010), pp. 4331-4356.

[6] J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. Ecole Norm. Sup., 33 (2000), pp. 837–934.

[7] R. Larsen, Functional Analysis: an introduction, Marcel Dekker, New York, 1973.

[8] G.J. Murphy, $C^*$-Algebras and Operator Theory, Academic Press Inc, 1990.

[9] J.P. Pier, Amenable Locally Compact Groups, Wiley Interscience, New York, 1984.

[10] W. Rudin, Fourier Analysis on Groups, Wiley-Interscience, New York, 1962.

[11] S.E. Takahasi and O. Hatori, Commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein-type theorem, Proc. Amer. Math. Soc., 110 (1990), pp. 149-158.

[12] S.E. Takahasi and O. Hatori, Commutative Banach algebras and BSE-inequalities, Math. Japonica, 37 (1992), pp. 607-614.