×

BSE properties of some Banach function algebras. (English) Zbl 1488.46094

Summary: In this paper, BSE properties of some Banach function algebras are studied. We show that Lipschitz algebras Lip\(_{\alpha}(X, d)\) and Dales-Davie algebras \(D(X,M)\) are BSE-algebras for certain underlying plane sets \(X\). Moreover, we investigate BSE properties of certain subalgebras of Lip\(_{\alpha}(X, d)\) such as Lip\(_{A}(X, \alpha)\), Lip\(^{n}(X, d)\) and Lip\((X, d, \alpha)\). BSE properties of Bloch type spaces \(\mathcal{B}_{\alpha}\) and Zygmund type spaces \(\mathcal{Z}_{\alpha}\) are also investigated in different cases of \(\alpha\in\mathbb{R}\).

MSC:

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
46J10 Banach algebras of continuous functions, function algebras
46E15 Banach spaces of continuous, differentiable or analytic functions
30H50 Algebras of analytic functions of one complex variable
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abtahi, M.; Honary, T. G., On the maximal ideal space of Dales-Davie algebras on infinitely differentiable functions, Bull. London Math. Soc., 39, 940-948 (2007) · Zbl 1214.46032
[2] Abtahi, M.; Honary, T. G., Properties of certain subalgebras of Dales-Davie algebras, Glasgow Math. J., 49, 225-233 (2007) · Zbl 1131.46035
[3] Abtahi, F.; Kamali, Z.; Toutounchi, M., The Bochner-Schoenberg-Eberlein property for vector-valued Lipschitz algebras, J. Math. Anal. Appl., 479, 1172-1181 (2019) · Zbl 1437.46054
[4] Alimohammadi, D.; Nezamabadi, F., Certain subalgebras of Lipschitz algebras of infinitely differentiable functions and their maximal ideal spaces, Int. J. Nonlinear Anal. Appl., 5, 9-22 (2014) · Zbl 1331.46042
[5] Aliprantis, C. D.; Border, K. C., Infinite Dimensional Analysis (2006), Berlin, Heidelberg: Springer-Verlag, Berlin, Heidelberg · Zbl 1156.46001
[6] Arendt, W.; Nikolski, N., Vector-valued holomorphic functions revisited, Math. Z., 234, 777-805 (2000) · Zbl 0976.46030
[7] Carando, D.; Sevilla-Peris, P., Spectra of weighted algebras of holomorphic functions, Math Z., 263, 887-902 (2009) · Zbl 1191.46023
[8] Conway, J. B., Functions of One Complex Variable (1973), New York: Springer-Verlag, New York · Zbl 0277.30001
[9] Dales, H. G., Banach Algebras and Automatic Continuity (2000), Oxford: London Mathematical Society, Oxford · Zbl 0981.46043
[10] Dales, H. G.; Aiena, P.; Eschmeier, J.; Larsen, K. B.; Willis, G., Introduction to Banach Algebras, Operators and Harmonic Analysis (2003), Oxford: London Mathematical Society, Oxford
[11] Dales, H. G.; Davie, M. A., Quasianalytic Banach function algebras, J. Funct. Anal., 13, 28-50 (1973) · Zbl 0254.46027
[12] Dales, H. G.; McClure, J. P., Completion of normed algebras of polynomials, J. Austral. Math. Soc., 20, 504-510 (1975) · Zbl 0315.46043
[13] Dales, H. G.; Ulger, A., Approximate identities and BSE norms for Banach function algebras, manuscript (2014), Toronto: Fields Institute, Toronto
[14] Duren, P. L., Theory of H^pSpaces (1970), New York, London: Academic Press, New York, London · Zbl 0215.20203
[15] Evard, J. Cl; Jafari, F., A complex Rolle’s theorem, Amer. Math. Monthly, 99, 858-861 (1992) · Zbl 0772.30003
[16] Honary, T. G.; Mahyar, H., Approximation in Lipschitz algebras of infinitely differentiable functions, Bull. Korean Math. Soc., 36, 629-636 (1999) · Zbl 0951.46025
[17] Z. Hosseini and E. Feizi, On weighted Banach spaces of vector-valued holomorohic functions, preprint.
[18] Hosseini, M.; Sady, F., Weighted composition operators on C(X)and Lip_c(X, α), Tokyo J. Math., 35, 71-84 (2012) · Zbl 1252.47032
[19] E. Kaniuth, A Course in Commutative Banach Algebras, Graduate Texts in Mathematics, vol. 246, Springer (2009). · Zbl 1190.46001
[20] Kaniuth, E.; Ülger, A., The Bochner-Schoenberg-Eberlein property for commutative Banach algebras, especially Fourier and Fourier-Stieltjes algebras, Trans. Amer. Math. Soc., 362, 4331-4356 (2010) · Zbl 1228.46046
[21] Larsen, R., An Introduction to the Theory of Multipliers (1971), New York: Springer-Verlag, New York · Zbl 0213.13301
[22] Lester, R.; Helms, L., Potential Theory (2014), London: Springer-Verlag, London · Zbl 1295.31001
[23] Lindstrom, M.; Sanatpour, A. H., Derivative-free characterizations of compact generalized composition operators between Zygmund type spaces, Bull. Aust. Math. Soc., 81, 398-408 (2010) · Zbl 1188.47025
[24] Mahyar, H.; Sanatpour, A. H., Compact and quasicompact homomorphisms between differentiable Lipschitz algebras, Bull. Belg. Math. Soc. Simon Stevin, 17, 485-497 (2010) · Zbl 1213.47042
[25] Mahyar, H.; Sanatpour, A. H., Compact composition operators on certain analytic Lipschitz spaces, Bull. Iran. Math. Soc., 38, 85-99 (2012) · Zbl 1320.47025
[26] O’Farrell, A. G., Polynomial approximation of smooth function, J. London Math. Soc. (2), 28, 496-506 (1983) · Zbl 0497.41003
[27] Shapiro, J. H., Weak topologies on subspaces of C(S), Trans. Amer. Math. Soc., 157, 471-479 (1971) · Zbl 0217.16501
[28] Sherbert, D. R., Banach algebras of Lipschitz functions, Pacific J. Math., 13, 1387-1399 (1963) · Zbl 0121.10203
[29] Takahasi, S-E; Hatori, O., Commutative Banach algebras which satisfy a Bochner-Schenberg-Eberlin-type theorem, Proc. Amer. Math. Soc., 110, 149-158 (1990) · Zbl 0722.46025
[30] Takahasi, S-E; Takahashi, Y.; Hatori, O.; Tanahashi, K., Commutative Banach algebras and BSE-norm, Math. Japon., 46, 273-277 (1997) · Zbl 0897.46044
[31] Weaver, N., Lipschitz Algebras (1999), Singapore: World Scientific, Singapore · Zbl 0936.46002
[32] Zhao, R., Pointwise multipliers from weighted Bergman spaces and Hardy spaces to weighted Bergman spaces, Ann. Acad. Sci. Fenn. Math., 29, 139-150 (2004) · Zbl 1069.47033
[33] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer Science + Business Media Inc. (2005). · Zbl 1067.32005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.