Abstract
For a complex Banach space X with open unit ball \(B_X,\) consider the Banach algebras \(\mathcal {H}^\infty (B_X)\) of bounded scalar-valued holomorphic functions and the subalgebra \(\mathcal {A}_u(B_X)\) of uniformly continuous functions on \(B_X.\) Denoting either algebra by \(\mathcal {A},\) we study the Gleason parts of the set of scalar-valued homomorphisms \(\mathcal {M}(\mathcal {A})\) on \(\mathcal {A}.\) Following remarks on the general situation, we focus on the case \(X = c_0,\) giving a complete characterization of the Gleason parts of \(\mathcal {M}(\mathcal {A}_u(B_{c_0}))\) and, among other things, showing that every fiber in \(\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))\) over a point in \(B_{\ell _\infty }\) contains \(2^c\) discs lying in different Gleason parts.
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References
Aron, R.M., Berner, P.D.: A Hahn–Banach extension theorem for analytic mappings. Bull. Soc. Math. Fr. 106, 3–24 (1978)
Aron, R.M., Carando, D., Gamelin, T.W., Lassalle, S., Maestre, M.: Cluster values of analytic functions on a Banach space. Math. Ann. 353, 293–303 (2012)
Aron, R.M., Cole, B., Gamelin, T.W.: Spectra of algebras of analytic functions on a Banach space. J. Reine Angew. Math. 415, 51–93 (1991)
Aron, R.M., Falcó, J., García, D., Maestre, M.: Analytic structure in fibers. Studia Math. 240(2), 101–121 (2018)
Bear, H.S.: Lectures on Gleason Parts. Lecture Notes in Math. Springer, Berlin (1970)
Boyd, C., Ryan, R.A.: Bounded weak continuity of homogeneous polynomials at the origin. Arch. Math. 71(3), 211–218 (1998)
Choi, Y.S., Falcó, J., García, D., Jung, M., Maestre, M.: Analytic structure in fibers of \(\cal{H}^{\infty } (B_{c_{0}})\). (preprint)
Choi, Y.S., García, D., Kim, S.G., Maestre, M.: Composition, numerical range and Aron–Berner extension. Math. Scand. 103, 97–110 (2008)
Cole, B., Gamelin, T.W., Johnson, W.: Analytic disks in fibers over the unit ball of a Banach space. Mich. Math. J. 39(3), 551–569 (1992)
Davie, A.M., Gamelin, T.W.: A theorem on polynomial-star approximation. Proc. Am. Math. Soc. 106(2), 351–356 (1989)
Dimant, V., Singer, J.: Homomorphisms between algebras of holomorphic functions on the infinite polydisk. (preprint)
Dineen, S.: Complex Analysis on Infinite Dimensional Spaces. Springer, London (1999)
Farmer, J.D.: Fibers over the sphere of a uniformly convex Banach space. Mich. Math. J. 45(2), 211–226 (1998)
Gamelin, T.W.: Uniform Algebras. Prentice-Hall Inc., Englewood Cliffs (1969)
Garnett, J.B.: Bounded Analytic Functions. Graduate Texts in Mathematics, vol. 236, 1st edn. Springer, New York (2007)
Gorkin, P.: Gleason parts and COP. J. Funct. Anal. 83, 44–49 (1989)
Hoffman, K.: Bounded analytic functions and Gleason parts. Ann. Math. 86(1), 74–111 (1967)
König, H.: On the Gleason and Harnack metrics for uniform algebras. Proc. Am. Math. Soc. 22, 100–101 (1969)
Mortini, R.: Gleason Parts and Prime Ideals in \(H^\infty \). Lecture Notes in Math, vol. 1573, pp. 136–138. Springer, Berlin (1994)
Renaud, A.: Quelques propriétés des applications analytiques d’une boule de dimension infinie dans une autre. Bull. Sci. Math. 2, 129–159 (1973)
Stout, E.L.: The Theory of Uniform Algebras. Bogden & Quigley Inc, Tarrytown-on-Hudson (1971)
Suárez, D.: Maximal Gleason parts for \(H^\infty \). Mich. Math. J. 45(1), 55–72 (1998)
Acknowledgements
This work was initiated while the first and fourth authors visited the Departamento de Matemática, Universidad de San Andrés during September of 2016. Both of them wish to thank the hospitality they received during their visit.
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Partially supported by PAI-UdeSA. The first and fourth authors were partially supported by MINECO and FEDER Project MTM2017-83262-C2-1-P. The second and third authors were partially supported by Conicet PIP 11220130100483 and ANPCyT PICT 2015-2299. The fourth author was also supported by Project Prometeo/2017/102 of the Generalitat Valenciana.
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Aron, R.M., Dimant, V., Lassalle, S. et al. Gleason parts for algebras of holomorphic functions in infinite dimensions. Rev Mat Complut 33, 415–436 (2020). https://doi.org/10.1007/s13163-019-00324-z
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DOI: https://doi.org/10.1007/s13163-019-00324-z