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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 |
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.
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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