Boundaries for Gelfand transform images of Banach algebras of holomorphic functions
DOI:
https://doi.org/10.7146/math.scand.a-134348Abstract
In this paper, we study boundaries for the Gelfand transform image of infinite dimensional analogues of the classical disk algebras. More precisely, given a certain Banach algebra $\mathcal{A}$ of bounded holomorphic functions on the open unit ball $B_X$ of a complex Banach space $X$, we show that the Shilov boundary for the Gelfand transform image of $\mathcal{A}$ can be explicitly described for a large class of Banach spaces. Some possible application of our result to the famous Corona theorem is also briefly discussed.
References
Acosta, M. D., Boundaries for spaces of holomorphic functions on $C(K)$, Publ. Res. Inst. Math. Sci. 42 (2006), no. 1, 27–44. http://projecteuclid.org./euclid.prims/1166642057
Acosta, M. D., Aron, R. M., and Moraes, L., Boundaries for Spaces of Holomorphic Functions on $M$-ideals in Their Biduals, Indiana Univ. Math. J. 58 (2009), no. 6, 2575–2595. https://doi.org/10.1512/iumj.2009.58.3807
Acosta, M. D., and Lourenço, M. L., Šilov boundary for holomorphic functions on some classical Banach spaces, Studia Math. 179 (2007), no. 1, 27–39. https://doi.org/10.4064/sm179-1-3
Acosta, M. D., and Moraes, L. A., On boundaries for spaces of holomorphic functions on the unit ball of a Banach space, Banach spaces and their applications in analysis, 229–239, Walter de Gruyter, Berlin, 2007
Acosta, M. D., Moraes, L. A., and Romero Grados, L., On boundaries on the predual of the Lorentz sequence space, J. Math. Anal. Appl. 336 (2007), no. 1, 470–479. https://doi.org/10.1016/j.jmaa.2007.02.041
Albiac, F., and Kalton, N., Topics in Banach Space Theory, Second edition, Graduate Texts in Mathematics, 233. Springer, Cham, 2016 https://doi.org/10.1007/978-3-319-31557-7
Arenson, E. L., Gleason parts and the Choquet boundary of the algebra of functions on a convex compactum, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113 (1981), 204–207.
Aron, R. M., Weakly uniformly continuous and weakly sequentially continuous entire functions. Advances in holomorphy (Proc. Sem. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), pp. 47–66. North-Holland Math. Stud., 34. North-Holland, Amsterdam, 1979.
Aron, R. M., and Berner, P. D., A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), no. 1, 3–24. http://www.numdam.org/item?id=BSMF_1978__106__3_0
Aron, R. M., Carando, D., Gamelin, T. W., Lassalle, S., and Maestre, M., Cluster values of analytic functions on a Banach space. Math. Ann. 353 (2012), no. 2, 293–303. https://doi.org/10.1007/s00208-011-0681-0
Aron, R. M., Choi, Y. S., Lourenço, M. L., and Paques, O. W., Boundaries for algebras of analytic functions on infinite dimensional Banach spaces, Banach spaces (Mérida, 1992), 15–22, Contemp. Math., 144, Amer. Math. Soc., Providence, RI, 1993. https://doi.org/10.1090/conm/144/1209443
Aron, R. M., Cole, B., and Gamelin, T. W., Spectra of algebras of analytic functions on a Banach space. J. Reine Angew. Math. 415 (1991), 51–93
Aron, R. M., Dimant, V., Lassalle, S., and Maestre, M., Gleason parts for algebras of holomorphic functions in infinite dimensions. Rev. Mat. Complut. 33 (2020), no. 2, 415–436. https://doi.org/10.1007/s13163-019-00324-z
Aron, R. M., and Dineen, S., $Q$-reflexive Banach spaces, Rocky Mountain J. Math. 27 (1997), no. 4, 1009–1025. https://doi.org/10.1216/rmjm/1181071856
Aron, R. M., Hervés, C., and Valdivia, M., Weakly continuous mappings on Banach spaces, J. Functional Analysis 52 (1983), no. 2, 189–204. https://doi.org/10.1016/0022-1236(83)90081-2
Bishop, E., A minimal boundary for function algebras, Pacific J. Math. 9 (1959), 629–642. http://projecteuclid.org./euclid.pjm/1103039103
Carando, D., García, D., Maestre, M., and Sevilla-Peris, P., On the spectra of algebras of analytic functions, Topics in complex analysis and operator theory, 165–198, Contemp. Math., 561, Amer. Math. Soc., Providence, RI, 2012. https://doi.org/10.1090/conm/561/11114
Choi, Y. S., Falcó, J., Garc'ıa, D., Jung, M., and Maestre, M., Analytic structure in fibers of $ H^infty (B_c_0)$, J. Math. Anal. Appl. 488 (2020), no. 2, 124088, 16 pp. https://doi.org/10.1016/j.jmaa.2020.124088
Choi, Y. S., Garc'ıa, D., Kim, S. G., and M. Maestre, M., Norm or numerical radius attaining polynomials on $C(K)$, J. Math. Anal. Appl. 295 (2004), no. 1, 80–96. https://doi.org/10.1016/j.jmaa.2004.03.005
Choi, Y. S., and Han, K. H., Boundaries for algebras of holomorphic functions on Marcinkiewicz sequence spaces, J. Math. Anal. Appl. 323 (2006), no. 2, 1116–1133. https://doi.org/10.1016/j.jmaa.2005.11.028
Choi, Y. S., Han, K. H., and Lee, H. J., Boundaries for algebras of holomorphic functions on Banach spaces, Illinois J. Math. 51 (2007), no. 3, 883–896. http://projecteuclid.org./euclid.ijm/1258131108
Choi, Y. S., Lee, H. J., and Song, S. G., Bishop's theorem and differentiability of a subspace of $C_b(K)$, Israel J. Math. 180 (2010), 93–118. https://doi.org/10.1007/s11856-010-0095-9
Dales, H. G., Banach Algebras and Automatic Continuity, London Mathematical Society Monographs. New Series, 24. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 2000.
Dineen, S., Complex analysis on infinite dimensional spaces, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 1999. https://doi.org/10.1007/978-1-4471-0869-6
Fabian, M., Habala, P., Hàjek, P., Santaluc'ıa, V. M., and Zizler, V., Banach space theory. The basis for linear and nonlinear analysis, Springer, New York, 2011. https://doi.org/10.1007/978-1-4419-7515-7
Gamelin, T. W., Uniform Algebras, Prentice Hall, Inc., Englewoof Cliffs, N. J., 1969.
Gamelin, T. W., Analytic functions on Banach spaces. Complex potential theory (Montreal, PQ, 1993), 187–233, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 439, Kluwer Acad. Publ., Dordrecht, 1994.
Gelfand, I., Raikov, D., and Shilov, G., Commutative normed rings, translated from the Russian, with a supplementary chapter, Chelsea Publishing Co., New York, 1964.
Globevnik, J., On interpolation by analytic maps in infinite dimensions, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 2, 243–252. https://doi.org/10.1017/S030500410005450
Globevnik, J., Boundaries for polydisc algebras in infinite dimensions, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 2, 291–303 https://doi.org/10.1017/S0305004100055705
Gutiérrez, J. M., Weakly continuous functions on Banach spaces not containing $ell _1$, Proc. Amer. Math. Soc. 119 (1993), no. 1, 147–152. https://doi.org/10.2307/2159836
Lee, H. J., Randomized series and geometry of Banach spaces, Taiwanese J. Math. 14 (2010), no. 5, 1837–1848. https://doi.org/10.11650/twjm/1500406019
Lee, H. J., Generalized numerical index for function algebras, J. Funct. Spaces 2019, Art. ID 9080867, 6 pp. https://doi.org/10.1155/2019/9080867
Moraes, L. A., and Romero Grados, L., Boundaries for algebras of holomorphic functions, J. Math. Anal. Appl. 281 (2003), no. 2, 575–586. https://doi.org/10.1016/S0022-247X(03)00150-1
Moraes, L. A., and Romero Grados, L., Boundaries for an algebra of bounded holomorphic functions, J. Korean Math. Soc. 41 (2004), no. 1, 231–242. https://doi.org/10.4134/JKMS.2004.41.1.231
Pełczyński, A., and Semadeni, Z., Spaces of continuous functions (III), Studia Math. 18 (1959), 211–222. https://doi.org/10.4064/sm-18-2-211-22
Phelps, R. R., Lectures on Choquet Theory, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London 1966.
Rosenthal, H., A characterization of Banach spaces containing $ell _1$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413 https://doi.org/10.1073/pnas.71.6.241
Šilov, G. E., On the decomposition of a commutative normed ring into a direct sum of ideals, Mat. Sb. 32 (1954) 353–364 (Russian); Amer. Math. Soc. Transl. (2) 1 (1955), 37–48 (English). https://doi.org/10.1090/trans2/001/03
Stout, E. L., The theory of uniform algebras, Bogden & Quigley, Inc., Publishers, Tarrytown-on-Hudson, N.Y., 1971
