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

The BSE concepts for vector-valued Lipschitz algebras

Fatemeh Abtahi 1,, , Zeinab Kamali 2, and  Maryam Toutounchi 1,

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  March 2021 Early access  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.

Keywords: BSE-algebra, BSE-function, BSE-norm, Lipschitz algebra, metric space.
Mathematics Subject Classification: Primary:46J05;Secondary:46J10.
Citation: Fatemeh Abtahi, Zeinab Kamali, Maryam Toutounchi. The BSE concepts for vector-valued Lipschitz algebras. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1171-1186. doi: 10.3934/cpaa.2021011
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.

[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.

[3]

H. G. Dales, Banach function algebras and BSE-norms, Graduate course during $23^rd$, Banach algebra conference, Oulu, Finland, 2017.

[4]

W. F. Eberlein, Characterizations of Fourier-Stieltjes transforms, Duke Math. J., 22 (1955), 465-468. 

[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.

[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. 

[7]

C. A. Jones and C. D. Lahr, Weak and norm approximate identities are different, Pacific J. Math., 72 (1977), 99-104. 

[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.

[9]

R. Larsen., An Introduction to the Theory of Multipliers, Springer-Verlag, New York, 1971.

[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.

[11]

D. R. Sherbert, Banach algebras of Lipschitz functions, Pacific J. Math., 13 (1963), 1387-1399. 

[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.

[13]

S. E. Takahasi and O. Hatori, Commutative Banach algebras and BSE-inequalities, Math. Japonica, 37 (1992), 47-52. 

show all references

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.

[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.

[3]

H. G. Dales, Banach function algebras and BSE-norms, Graduate course during $23^rd$, Banach algebra conference, Oulu, Finland, 2017.

[4]

W. F. Eberlein, Characterizations of Fourier-Stieltjes transforms, Duke Math. J., 22 (1955), 465-468. 

[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.

[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. 

[7]

C. A. Jones and C. D. Lahr, Weak and norm approximate identities are different, Pacific J. Math., 72 (1977), 99-104. 

[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.

[9]

R. Larsen., An Introduction to the Theory of Multipliers, Springer-Verlag, New York, 1971.

[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.

[11]

D. R. Sherbert, Banach algebras of Lipschitz functions, Pacific J. Math., 13 (1963), 1387-1399. 

[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.

[13]

S. E. Takahasi and O. Hatori, Commutative Banach algebras and BSE-inequalities, Math. Japonica, 37 (1992), 47-52. 

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