Spectra of algebras of entire functions on Banach spaces

Zagorodnyuk, Andriy

doi:10.1090/S0002-9939-06-08260-8

Spectra of algebras of entire functions on Banach spaces
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by Andriy Zagorodnyuk

Proc. Amer. Math. Soc. 134 (2006), 2559-2569

DOI: https://doi.org/10.1090/S0002-9939-06-08260-8

Published electronically: February 8, 2006

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Abstract:

We obtain an explicit description of the spectrum (set of closed maximal ideals) of $H_b(X),$ algebra of analytic functions on a Banach space $X$ which are bounded on bounded subsets. We show that the spectrum of $H_b(X)$ admits a natural linear structure. Some applications to the algebra of uniformly continuous and bounded analytic functions on the unit ball $B\subset X$ are indicated.

References

Raymundo Alencar, Richard Aron, Pablo Galindo, and Andriy Zagorodnyuk, Algebras of symmetric holomorphic functions on $l_p$, Bull. London Math. Soc. 35 (2003), no. 1, 55–64. MR 1934432, DOI 10.1112/S0024609302001431
Richard M. Aron and Paul D. Berner, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), no. 1, 3–24 (English, with French summary). MR 508947, DOI 10.24033/bsmf.1862
R. M. Aron, B. J. Cole, and T. W. Gamelin, Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415 (1991), 51–93. MR 1096902
R. M. Aron, B. J. Cole, and T. W. Gamelin, Weak-star continuous analytic functions, Canad. J. Math. 47 (1995), no. 4, 673–683. MR 1346158, DOI 10.4153/CJM-1995-035-1
R. M. Aron, P. Galindo, D. García, and M. Maestre, Regularity and algebras of analytic functions in infinite dimensions, Trans. Amer. Math. Soc. 348 (1996), no. 2, 543–559. MR 1340167, DOI 10.1090/S0002-9947-96-01553-X
R. M. Aron and J. B. Prolla, Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980), 195–216. MR 552473, DOI 10.1515/crll.1980.313.195
Peter Biström, Jesús A. Jaramillo, and Mikael Lindström, Polynomial compactness in Banach spaces, Rocky Mountain J. Math. 28 (1998), no. 4, 1203–1226. MR 1681664, DOI 10.1216/rmjm/1181071712
F. Cabello Sánchez, R. García, and I. Villanueva, Extension of multilinear operators on Banach spaces, Extracta Math. 15 (2000), no. 2, 291–334. III Congress on Banach Spaces (Jarandilla de la Vera, 1998). MR 1823895
D. Carando, D. García and M. Maestre, Homomorphisms and composition operators on algebras of analytic functions of bounded type, Adv. Math. To appear.
A. M. Davie and T. W. Gamelin, A theorem on polynomial-star approximation, Proc. Amer. Math. Soc. 106 (1989), no. 2, 351–356. MR 947313, DOI 10.1090/S0002-9939-1989-0947313-8
Djamel Deghoul, Construction de caractères exceptionnels sur une algèbre de Fréchet, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 8, 579–580 (French, with English summary). MR 1101036
Verónica Dimant and Raquel Gonzalo, Block diagonal polynomials, Trans. Amer. Math. Soc. 353 (2001), no. 2, 733–747. MR 1804515, DOI 10.1090/S0002-9947-00-02735-5
Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MR 1705327, DOI 10.1007/978-1-4471-0869-6
P. G. Dixon, Scalar homomorphisms on algebras of infinitely long polynomials with an application to automatic continuity theory, J. London Math. Soc. (2) 19 (1979), no. 3, 488–496. MR 540065, DOI 10.1112/jlms/s2-19.3.488
Pablo Galindo, Domingo García, Manuel Maestre, and Jorge Mujica, Extension of multilinear mappings on Banach spaces, Studia Math. 108 (1994), no. 1, 55–76. MR 1259024, DOI 10.4064/sm-108-1-55-76
Theodore W. Gamelin, Analytic functions on Banach spaces, Complex potential theory (Montreal, PQ, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 439, Kluwer Acad. Publ., Dordrecht, 1994, pp. 187–233. MR 1332962
Domingo García, M. Lilian Lourenço, Manuel Maestre, and Luiza A. Moraes, The spectrum of analytic mappings of bounded type, J. Math. Anal. Appl. 245 (2000), no. 2, 447–470. MR 1758549, DOI 10.1006/jmaa.2000.6762
D. Garcia, M. L. Lourenço, L. A. Moraes, and O. W. Paques, The spectra of some algebras of analytic mappings, Indag. Math. (N.S.) 10 (1999), no. 3, 393–406. MR 1819897, DOI 10.1016/S0019-3577(99)80031-7
Raquel Gonzalo, Multilinear forms, subsymmetric polynomials, and spreading models on Banach spaces, J. Math. Anal. Appl. 202 (1996), no. 2, 379–397. MR 1406236, DOI 10.1006/jmaa.1996.0322
Ernest A. Michael, Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc. 11 (1952), 79. MR 51444
Jorge Mujica, Complex analysis in Banach spaces, North-Holland Mathematics Studies, vol. 120, North-Holland Publishing Co., Amsterdam, 1986. Holomorphic functions and domains of holomorphy in finite and infinite dimensions; Notas de Matemática [Mathematical Notes], 107. MR 842435
Jorge Mujica, Ideals of holomorphic functions on Tsirelson’s space, Arch. Math. (Basel) 76 (2001), no. 4, 292–298. MR 1825009, DOI 10.1007/s000130050571