Regularity and Algebras of Analytic Functions in Infinite Dimensions
HTML articles powered by AMS MathViewer
- by R. M. Aron, P. Galindo, D. García and M. Maestre PDF
- Trans. Amer. Math. Soc. 348 (1996), 543-559 Request permission
Abstract:
A Banach space $E$ is known to be Arens regular if every continuous linear mapping from $E$ to $E’$ is weakly compact. Let $U$ be an open subset of $E$, and let $H_b(U)$ denote the algebra of analytic functions on $U$ which are bounded on bounded subsets of $U$ lying at a positive distance from the boundary of $U.$ We endow $H_b(U)$ with the usual Fréchet topology. $M_b(U)$ denotes the set of continuous homomorphisms $\phi :H_b(U) \to \mathbb {C}$. We study the relation between the Arens regularity of the space $E$ and the structure of $M_b(U)$.References
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- 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. Aron, Y. Choi and J. G. Llavona, Estimates by polynomials, Preprint 1993.
- 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. Aron, B. Cole and T. Gamelin, Weak-star continuous analytic functions, Canad. J. Math. 47 (1995), 673–683.
- R. M. Aron, J. Diestel, and A. K. Rajappa, Weakly continuous functions on Banach spaces containing $l_1$, Banach spaces (Columbia, Mo., 1984) Lecture Notes in Math., vol. 1166, Springer, Berlin, 1985, pp. 1–3. MR 827751, DOI 10.1007/BFb0074685
- R. M. Aron, C. Hervés, and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Functional Analysis 52 (1983), no. 2, 189–204. MR 707203, DOI 10.1016/0022-1236(83)90081-2
- 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
- Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
- Seán Dineen, Complex analysis in locally convex spaces, Notas de Matemática [Mathematical Notes], vol. 83, North-Holland Publishing Co., Amsterdam-New York, 1981. MR 640093
- José Bonet, John Horváth, and Manuel Maestre (eds.), Progress in functional analysis, North-Holland Mathematics Studies, vol. 170, North-Holland Publishing Co., Amsterdam, 1992. MR 1150736
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Alfred Rosenblatt, Sur les points singuliers des équations différentielles, C. R. Acad. Sci. Paris 209 (1939), 10–11 (French). MR 85
- 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
- Gilles Godefroy and Bruno Iochum, Arens-regularity of Banach algebras and the geometry of Banach spaces, J. Funct. Anal. 80 (1988), no. 1, 47–59. MR 960222, DOI 10.1016/0022-1236(88)90064-X
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
- P. Harmand, D. Werner, and W. Werner, $M$-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993. MR 1238713, DOI 10.1007/BFb0084355
- Denny H. Leung, Banach spaces with property $(\textrm {w})$, Glasgow Math. J. 35 (1993), no. 2, 207–217. MR 1220563, DOI 10.1017/S0017089500009769
- D. Leung, Private communication.
- Mikael Lindström and Raymond A. Ryan, Applications of ultraproducts to infinite-dimensional holomorphy, Math. Scand. 71 (1992), no. 2, 229–242. MR 1212706, DOI 10.7146/math.scand.a-12424
- 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
- Raymond A. Ryan, Weakly compact holomorphic mappings on Banach spaces, Pacific J. Math. 131 (1988), no. 1, 179–190. MR 917872, DOI 10.2140/pjm.1988.131.179
- N. Ghoussoub and M. Talagrand, A noncompletely continuous operator on $L_{1}(G)$ whose random Fourier transform is in $c_{0}(\hat G)$, Proc. Amer. Math. Soc. 92 (1984), no. 2, 229–232. MR 754709, DOI 10.1090/S0002-9939-1984-0754709-1
Additional Information
- R. M. Aron
- Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44242
- MR Author ID: 27325
- Email: aron@mcs.kent.edu
- P. Galindo
- Affiliation: Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot (Valencia), Spain
- Email: galindo@vm.ci.uv.es
- D. García
- Email: garciad@vm.ci.uv.es
- M. Maestre
- Email: maestre@vm.ci.uv.es
- Received by editor(s): May 9, 1994
- Additional Notes: The first author was supported in part by US–Spain Joint Committee for Cultural and Educational Cooperation, grant II–C 91024, and by NSF Grant Int-9023951
Supported in part by DGICYT pr. 91-0326 and by grant 93-081; the research of the second author was undertaken in part during the academic year 1993-94 while visiting Kent State University
The third author supported in part by DGICYT pr. 91-0326
The fourth author supported in part by US–Spain Joint Committee for Cultural and Educational Cooperation, grant II–C 91024 and by DGICYT pr. P.B.91-0326 and P.B.91-0538 - © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 543-559
- MSC (1991): Primary 46G20; Secondary 46J10
- DOI: https://doi.org/10.1090/S0002-9947-96-01553-X
- MathSciNet review: 1340167