# American Institute of Mathematical Sciences

March  2021, 20(3): 1171-1186. doi: 10.3934/cpaa.2021011

## The BSE concepts for vector-valued Lipschitz algebras

 1 Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan 81746-73441, IRAN 2 Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan, IRAN

* Corresponding author

Received  August 2020 Revised  October 2020 Published  February 2021

Let $(K,d)$ be a compact metric space, $\mathcal A$ be a commutative semisimple Banach algebra and $0<\alpha\leq 1$. The overall purpose of the present paper is to demonstrate that all BSE concepts of ${\rm Lip}_\alpha(K,\mathcal A)$ are inherited from $\mathcal A$ and vice versa. Recently, the authors proved in the case that $\mathcal A$ is unital, ${\rm Lip}_\alpha(K,\mathcal A)$ is a BSE-algebra if and only if $\mathcal A$ is so. In this paper, we generalize this result for an arbitrary commutative semisimple Banach algebra $\mathcal A$. Furthermore, we investigate the BSE-norm property for ${\rm Lip}_\alpha(K,\mathcal A)$ and prove that ${\rm Lip}_\alpha(K,\mathcal A)$ belongs to the class of BSE-norm algebras if and only if $\mathcal A$ is owned by this class. Moreover, we prove that for any natural number $n$ with $n\geq 2$, if all continuous bounded functions on $\Delta({\rm Lip}_\alpha(K,\mathcal A))$ are $n$-BSE-functions, then $K$ is finite. As a result, we obtain that ${\rm Lip}_{\alpha}(K,\mathcal A)$ is a BSE-algebra of type I if and only if $\mathcal A$ is a BSE-algebra of type I and $K$ is finite. Furthermore, in according to a result of Kaniuth and Ülger, which disapproves the BSE-property for ${\rm lip}_{\alpha}K$, we show that for any commutative semisimple Banach algebra $\mathcal A$, ${\rm lip}_{\alpha}(K,\mathcal A)$ fails to be a BSE-algebra, as well. Finally, we concentrate on the classical Lipschitz algebra ${\rm Lip}_\alpha X$, for an arbitrary metric space (not necessarily compact) $(X,d)$ and $\alpha>0$, when ${\rm Lip}_\alpha X$ separates the points of $X$. In particular, we show that ${\rm Lip}_\alpha X$ is a BSE-algebra, as well as a BSE-norm algebra.

Citation: Fatemeh Abtahi, Zeinab Kamali, Maryam Toutounchi. The BSE concepts for vector-valued Lipschitz algebras. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1171-1186. doi: 10.3934/cpaa.2021011
##### References:
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##### References:
 [1] F. Abtahi, Z. Kamali and M. Toutounchi, The Bochner-Schoenberg-Eberlein property for vector-valued Lipschitz algebras, J. Math. Anal. Appl., 479 (2019), 1172-1181.  doi: 10.1016/j.jmaa.2019.06.073.  Google Scholar [2] S. Bochner, A theorem on Fourier- Stieltjes integrals, Bull. Amer. Math. Soc., 40 (1934), 271-276.  doi: 10.1090/S0002-9904-1934-05843-9.  Google Scholar [3] H. G. Dales, Banach function algebras and BSE-norms, Graduate course during $23^rd$, Banach algebra conference, Oulu, Finland, 2017. Google Scholar [4] W. F. Eberlein, Characterizations of Fourier-Stieltjes transforms, Duke Math. J., 22 (1955), 465-468.   Google Scholar [5] K. Esmaeili and H. Mahyar, The character spaces and $\check{S}$ilov boundaries of vector-valued Lipschitz function algebras, Indian J. Pure Appl. Math., 45 (2014), 977-988.  doi: 10.1007/s13226-014-0099-y.  Google Scholar [6] J. Inoue, T. Miura, H. Takagi and S. E. Takahasi, Classification of semisimple commutative Banach algebras of type I, Nihonkai Math. J., 30 (2019), 1-17.   Google Scholar [7] C. A. Jones and C. D. Lahr, Weak and norm approximate identities are different, Pacific J. Math., 72 (1977), 99-104.   Google Scholar [8] E. Kaniuth and A. Ülger, The Bochner-Schoenberg-Eberlein property for commutative Banach algebras, especially Fourier and Fourier-Stieltjes algebras, Trans. Amer. Math. Soc., 362 (2010), 4331-4356.  doi: 10.1090/S0002-9947-10-05060-9.  Google Scholar [9] R. Larsen., An Introduction to the Theory of Multipliers, Springer-Verlag, New York, 1971.  Google Scholar [10] I. J. Schoenberg, A remark on the preceding note by Bochner, Bull. Amer. Math. Soc., 40 (1934), 277-278.  doi: 10.1090/S0002-9904-1934-05845-2.  Google Scholar [11] D. R. Sherbert, Banach algebras of Lipschitz functions, Pacific J. Math., 13 (1963), 1387-1399.   Google Scholar [12] S. E. Takahasi and O. Hatori, Commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein-type theorem, Proc. Amer. Math. Soc., 110 (1990), 149-158.  doi: 10.2307/2048254.  Google Scholar [13] S. E. Takahasi and O. Hatori, Commutative Banach algebras and BSE-inequalities, Math. Japonica, 37 (1992), 47-52.   Google Scholar
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