Regularity and Algebras of Analytic Functions in Infinite Dimensions
Authors:
R. M. Aron, P. Galindo, D. García and M. Maestre
Journal:
Trans. Amer. Math. Soc. 348 (1996), 543559
MSC (1991):
Primary 46G20; Secondary 46J10
MathSciNet review:
1340167
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Abstract 
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Additional Information
Abstract: A Banach space is known to be Arens regular if every continuous linear mapping from to is weakly compact. Let be an open subset of , and let denote the algebra of analytic functions on which are bounded on bounded subsets of lying at a positive distance from the boundary of We endow with the usual Fréchet topology. denotes the set of continuous homomorphisms . We study the relation between the Arens regularity of the space and the structure of .
 1.
R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839848. MR 13:659f
 2.
R. Aron and P. Berner, A HahnBanach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), 324. MR 80e:46029
 3.
R. Aron, Y. Choi and J. G. Llavona, Estimates by polynomials, Preprint 1993.
 4.
R. Aron, B. Cole and T. Gamelin, Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415 (1991), 5193.MR 92f:46056
 5.
R. Aron, B. Cole and T. Gamelin, Weakstar continuous analytic functions, Canad. J. Math. 47 (1995), 673683.
 6.
R. Aron, J. Diestel, and A.K. Rajappa, Weakly continous functions on Banach spaces containing , Banach Spaces (Proc. of the Missouri Conference), Lecture Notes in Math. 1166, 1985, pp. 13. MR 87g:46022
 7.
R. Aron, C. Herves and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 (1983), 189204. MR 84g:46066
 8.
A. Davie and T. Gamelin, A theorem on polynomialstar approximation, Proc. Amer. Math. Soc. 106 (1989), 351356. MR 89k:46023
 9.
J. Diestel, Sequences and series in Banach spaces, Graduate Texts in Math. 92, SpringerVerlag, New York, 1984. MR 85i:46020
 10.
S. Dineen, Complex Analysis in Locally Convex Spaces, Math. Studies 57, NorthHolland, Amsterdam, 1981. MR 84b:46050
 11.
S. Dineen and R. Timoney, Complex geodesics on convex domains, Progress in Functional Analysis, edited by K. Bierstedt et al., Math. Studies, vol. 170, NorthHolland, Amsterdam, 1992, pp. 333365. MR 92i:46002
 12.
N. Dunford and J. Schwartz, Linear Operators, Part I, Interscience, New York 1958. MR 22:8302
 13.
G. A. Edgar, An ordering for Banach spaces, Pacific J. Math. 108 (1983), 8398. MR 85:85g
 14.
P. Galindo, D. García, M. Maestre and J. Mujica, Extension of multilinear mappings on Banach spaces, Stud. Math. 108 (1994), 5576. MR 95f:46072
 15.
G. Godefroy and B. Iochum, Arens regularity of Banach algebras and the geometry of Banach spaces, J. Funct. Anal. 80 (1988), 4759. MR 89j:46051
 16.
R.C. Gunning and H. Rossi, Analytic functions of several complex variables, PrenticeHall, 1965. MR 31:4927
 17.
P. Harmand, D. Werner and W. Werner, MIdeals in Banach Spaces and Banach Algebras, Lecture Notes in Math. 1547, 1993.MR 94k:46022
 18.
D. Leung, Banach spaces with property (w), Glasgow Math. J. 35 (1993), 207217.MR 94h:46018
 19.
D. Leung, Private communication.
 20.
M. Lindström and R. Ryan, Applications of ultraproducts to infinite dimensional holomorphy, Math. Scand. 71 (1992), 229242. MR 94c:46090
 21.
J. Mujica, Complex Analysis in Banach Spaces, NorthHolland, Amsterdam, 1986. MR 88d:46084
 22.
R. Ryan, Weakly compact holomorphic mappings on Banach spaces, Pacific J. Math. 131 (1988), 179190.MR 89a:46103
 23.
A. Ülger, Weakly compact bilinear forms and Arens regularity,
 1.
 R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839848. MR 13:659f
 2.
 R. Aron and P. Berner, A HahnBanach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), 324. MR 80e:46029
 3.
 R. Aron, Y. Choi and J. G. Llavona, Estimates by polynomials, Preprint 1993.
 4.
 R. Aron, B. Cole and T. Gamelin, Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415 (1991), 5193.MR 92f:46056
 5.
 R. Aron, B. Cole and T. Gamelin, Weakstar continuous analytic functions, Canad. J. Math. 47 (1995), 673683.
 6.
 R. Aron, J. Diestel, and A.K. Rajappa, Weakly continous functions on Banach spaces containing , Banach Spaces (Proc. of the Missouri Conference), Lecture Notes in Math. 1166, 1985, pp. 13. MR 87g:46022
 7.
 R. Aron, C. Herves and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 (1983), 189204. MR 84g:46066
 8.
 A. Davie and T. Gamelin, A theorem on polynomialstar approximation, Proc. Amer. Math. Soc. 106 (1989), 351356. MR 89k:46023
 9.
 J. Diestel, Sequences and series in Banach spaces, Graduate Texts in Math. 92, SpringerVerlag, New York, 1984. MR 85i:46020
 10.
 S. Dineen, Complex Analysis in Locally Convex Spaces, Math. Studies 57, NorthHolland, Amsterdam, 1981. MR 84b:46050
 11.
 S. Dineen and R. Timoney, Complex geodesics on convex domains, Progress in Functional Analysis, edited by K. Bierstedt et al., Math. Studies, vol. 170, NorthHolland, Amsterdam, 1992, pp. 333365. MR 92i:46002
 12.
 N. Dunford and J. Schwartz, Linear Operators, Part I, Interscience, New York 1958. MR 22:8302
 13.
 G. A. Edgar, An ordering for Banach spaces, Pacific J. Math. 108 (1983), 8398. MR 85:85g
 14.
 P. Galindo, D. García, M. Maestre and J. Mujica, Extension of multilinear mappings on Banach spaces, Stud. Math. 108 (1994), 5576. MR 95f:46072
 15.
 G. Godefroy and B. Iochum, Arens regularity of Banach algebras and the geometry of Banach spaces, J. Funct. Anal. 80 (1988), 4759. MR 89j:46051
 16.
 R.C. Gunning and H. Rossi, Analytic functions of several complex variables, PrenticeHall, 1965. MR 31:4927
 17.
 P. Harmand, D. Werner and W. Werner, MIdeals in Banach Spaces and Banach Algebras, Lecture Notes in Math. 1547, 1993.MR 94k:46022
 18.
 D. Leung, Banach spaces with property (w), Glasgow Math. J. 35 (1993), 207217.MR 94h:46018
 19.
 D. Leung, Private communication.
 20.
 M. Lindström and R. Ryan, Applications of ultraproducts to infinite dimensional holomorphy, Math. Scand. 71 (1992), 229242. MR 94c:46090
 21.
 J. Mujica, Complex Analysis in Banach Spaces, NorthHolland, Amsterdam, 1986. MR 88d:46084
 22.
 R. Ryan, Weakly compact holomorphic mappings on Banach spaces, Pacific J. Math. 131 (1988), 179190.MR 89a:46103
 23.
 A. Ülger, Weakly compact bilinear forms and Arens regularity,
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Additional Information
R. M. Aron
Affiliation:
Department of Mathematics, Kent State University, Kent, Ohio 44242
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
DOI:
http://dx.doi.org/10.1090/S000299479601553X
PII:
S 00029947(96)01553X
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 Int9023951
Supported in part by DGICYT pr. 910326 and by grant 93081; the research of the second author was undertaken in part during the academic year 199394 while visiting Kent State University
The third author supported in part by DGICYT pr. 910326
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.910326 and P.B.910538
Article copyright:
© Copyright 1996
American Mathematical Society
