Algebras of real analytic functions: Homomorphisms and bounding sets

Peter Biström; Jesús Jaramillo; Mikael Lindström

Studia Mathematica (1995)

  • Volume: 115, Issue: 1, page 23-37
  • ISSN: 0039-3223

Abstract

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This article deals with bounding sets in real Banach spaces E with respect to the functions in A(E), the algebra of real analytic functions on E, as well as to various subalgebras of A(E). These bounding sets are shown to be relatively weakly compact and the question whether they are always relatively compact in the norm topology is reduced to the study of the action on the set of unit vectors in l ∞ of the corresponding functions in A ( l ∞ ) . These results are achieved by studying the homomorphisms on the function algebras in question, an idea that is also reversed in order to obtain new results for the set of homomorphisms on these algebras.

How to cite

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Biström, Peter, Jaramillo, Jesús, and Lindström, Mikael. "Algebras of real analytic functions: Homomorphisms and bounding sets." Studia Mathematica 115.1 (1995): 23-37. <http://eudml.org/doc/216196>.

@article{Biström1995,
abstract = {This article deals with bounding sets in real Banach spaces E with respect to the functions in A(E), the algebra of real analytic functions on E, as well as to various subalgebras of A(E). These bounding sets are shown to be relatively weakly compact and the question whether they are always relatively compact in the norm topology is reduced to the study of the action on the set of unit vectors in $l_∞$ of the corresponding functions in $A(l_∞)$. These results are achieved by studying the homomorphisms on the function algebras in question, an idea that is also reversed in order to obtain new results for the set of homomorphisms on these algebras.},
author = {Biström, Peter, Jaramillo, Jesús, Lindström, Mikael},
journal = {Studia Mathematica},
keywords = {bounding sets in real Banach spaces; relatively weakly compact; homomorphisms; function algebras},
language = {eng},
number = {1},
pages = {23-37},
title = {Algebras of real analytic functions: Homomorphisms and bounding sets},
url = {http://eudml.org/doc/216196},
volume = {115},
year = {1995},
}

TY - JOUR
AU - Biström, Peter
AU - Jaramillo, Jesús
AU - Lindström, Mikael
TI - Algebras of real analytic functions: Homomorphisms and bounding sets
JO - Studia Mathematica
PY - 1995
VL - 115
IS - 1
SP - 23
EP - 37
AB - This article deals with bounding sets in real Banach spaces E with respect to the functions in A(E), the algebra of real analytic functions on E, as well as to various subalgebras of A(E). These bounding sets are shown to be relatively weakly compact and the question whether they are always relatively compact in the norm topology is reduced to the study of the action on the set of unit vectors in $l_∞$ of the corresponding functions in $A(l_∞)$. These results are achieved by studying the homomorphisms on the function algebras in question, an idea that is also reversed in order to obtain new results for the set of homomorphisms on these algebras.
LA - eng
KW - bounding sets in real Banach spaces; relatively weakly compact; homomorphisms; function algebras
UR - http://eudml.org/doc/216196
ER -

References

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  1. [1] R. Alencar, R. Aron and S. Dineen, A reflexive space of holomorphic functions in infinitely many variables, Proc. Amer. Math. Soc. 90 (1984), 407-411. Zbl0536.46015
  2. [2] P. Biström, S. Bjon and M. Lindström, Function algebras on which homomorphisms are point evaluations on sequences, Manuscripta Math. 73 (1991), 179-185. Zbl0764.46048
  3. [3] P. Biström and J. A. Jaramillo, C ∞ -bounding sets and compactness, Math. Scand. 75 (1994), 82-86. 
  4. [4] J. Bochnak and J. Siciak, Analytic functions in topological vector spaces, Studia Math. 39 (1971), 77-112. Zbl0214.37703
  5. [5] J. Bourgain and J. Diestel, Limited operators and strict cosingularity, Math. Nachr. 119 (1984), 55-58. Zbl0601.47019
  6. [6] T. K. Carne, B. Cole and T. W. Gamelin, A uniform algebra of analytic functions on a Banach space, Trans. Amer. Math. Soc. 314 (1989), 639-659. Zbl0704.46033
  7. [7] J. Castillo and C. Sánchez, Weakly-p-compact, p-Banach-Saks and super-reflexive Banach spaces, J. Math. Anal. Appl., to appear. Zbl0878.46009
  8. [8] K. Ciesielski and R. Pol, A weakly Lindelöf function space C(K) without any continuous injection into c 0 ( Γ ) , Bull. Polish Acad. Sci. Math. 32 (1984), 681-688. Zbl0571.54014
  9. [9] A. M. Davie and T. W. Gamelin, A theorem on polynomial-star approximation, Proc. Amer. Math. Soc. 106 (1989), 351-356. Zbl0683.46037
  10. [10] J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math. 92, Springer, 1984. 
  11. [11] K. Floret, Weakly Compact Sets, Lecture Notes in Math. 801, Springer, 1980. Zbl0437.46006
  12. [12] P. Galindo, D. García, M. Maestre and J. Mujica, Extension of multilinear mappings on Banach spaces, Studia Math. 108 (1994), 55-76. Zbl0852.46004
  13. [13] M. Garrido, J. Gómez and J. A. Jaramillo, Homomorphisms on function algebras, Extracta Math. 7 (1992), 46-52. Zbl0808.46035
  14. [14] T. Husain, Multiplicative Functionals on Topological Algebras, Research Notes in Math. 85, Pitman, 1983. Zbl0514.46030
  15. [15] J. A. Jaramillo and A. Prieto, The weak-polynomial convergence on a Banach space, Proc. Amer. Math. Soc. 118 (1993), 463-468. Zbl0795.46042
  16. [16] H. Jarchow, Locally Convex Spaces, Teubner, 1981. Zbl0466.46001
  17. [17] K. John, H. Toruńczyk and V. Zizler, Uniformly smooth partitions of unity on superreflexive Banach spaces, Studia Math. 70 (1981), 129-137. Zbl0485.46006
  18. [18] B. Josefson, Bounding subsets of l ∞ ( A ) , J. Math. Pures Appl. 57 (1978), 397-421. Zbl0403.46011
  19. [19] A. Kriegl and P. Michor, More smoothly real compact spaces, Proc. Amer. Math. Soc. 117 (1993), 467-471. Zbl0813.46026
  20. [20] J. G. Llavona, Approximation of Continuously Differentiable Functions, North-Holland, 1986. Zbl0642.41001
  21. [21] J. Mujica, Complex homomorphisms of the algebras of holomorphic functions on Fréchet spaces, Math. Ann. 241 (1979), 73-82. Zbl0386.46023
  22. [22] S. Negrepontis, Banach spaces and topology, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, 1984, 1047-1142. 
  23. [23] T. Schlumprecht, A limited set that is not bounding, Proc. Roy. Irish Acad. 90A (1990), 125-129. Zbl0749.46030

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