On the ideal structure of algebras of staralgebra valued functions
Author:
Jorma Arhippainen
Journal:
Proc. Amer. Math. Soc. 123 (1995), 381391
MSC:
Primary 46J20; Secondary 46K05
DOI:
https://doi.org/10.1090/S00029939199512151983
MathSciNet review:
1215198
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
Abstract: The ideal structure of the algebra $C(X,A)$ has been studied in many papers under various topological assumptions on the space X and the algebra A. In this paper we shall study the case where X is a completely regular topological space and A is a locally convex star algebra. In such case the structure of closed (proper) ideals can be described not only by using points of X and some family of closed ideals of A, as usual, but also by using points of the carrier space $\Delta (A)$ of A and some family of closed ideals of $C(X,A)$ depending on those points and also by using different kind of slice ideals of $C(X,A)$.

M. Abel, Description of closed ideals in algebras of continuous vectorvalued functions, Math. Notes, vol. 30, Princeton Univ. Press, Princeton, NJ, 1981, pp. 887892.
 Richard Arens, A generalization of normed rings, Pacific J. Math. 2 (1952), 455–471. MR 51445
 Jorma Arhippainen, On the ideal structure and approximation properties of algebras of continuous $B^*$algebravalued functions, Acta Univ. Oulu. Ser. A Sci. Rerum Natur. 187 (1987), 103. MR 918787
 Jorma Arhippainen, On commutative locally $m$convex algebras, Tartu Ül. Toimetised 928 (1991), 15–28 (English, with Estonian summary). MR 1150230
 Jorma Arhippainen, On the ideal structure of algebras of LMCalgebra valued functions, Studia Math. 101 (1992), no. 3, 311–318. MR 1153787, DOI https://doi.org/10.4064/sm1013311318 , On locally convex square algebras, Preprint series in Math., Univ. of Oulu, 1992.
 Edward Beckenstein, Lawrence Narici, and Charles Suffel, Topological algebras, NorthHolland Publishing Co., AmsterdamNew YorkOxford, 1977. NorthHolland Mathematics Studies, Vol. 24; Notas de Matemática, No. 60. [Mathematical Notes, No. 60]. MR 0473835
 William E. Dietrich Jr., The maximal ideal space of the topological algebra $C(X,\,E)$, Math. Ann. 183 (1969), 201–212. MR 254605, DOI https://doi.org/10.1007/BF01351380 , Function algebras on completely regular spaces, Diss., Northwestern Univ., Evanston, IL, 1971.
 James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606 W. Hery, Rings of continuous Banach algebravalued functions, Doct. Diss. Absttrs 45, Polytech. Instit. of New York, 1974.
 William Hery, Maximal ideals in algebras of continuous $C(S)$ valued functions, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 58 (1975), no. 2, 195–199 (English, with Italian summary). MR 423060
 William J. Hery, Maximal ideals in algebras of topological algebra valued functions, Pacific J. Math. 65 (1976), no. 2, 365–373. MR 435854
 Anastasios Mallios, Topological algebras. Selected topics, NorthHolland Mathematics Studies, vol. 124, NorthHolland Publishing Co., Amsterdam, 1986. Notas de Matemática [Mathematical Notes], 109. MR 857807
 Ernest A. Michael, Locally multiplicativelyconvex topological algebras, Mem. Amer. Math. Soc. 11 (1952), 79. MR 51444
 Peter D. Morris and Daniel E. Wulbert, Functional representation of topological algebras, Pacific J. Math. 22 (1967), 323–337. MR 213876
 Leopoldo Nachbin, Elements of approximation theory, Van Nostrand Mathematical Studies, No. 14, D. Van Nostrand Co., Inc., Princeton, N.J.Toronto, Ont.London, 1967. MR 0217483
 João Bosco Prolla, Approximation of vector valued functions, NorthHolland Publishing Co., AmsterdamNew YorkOxford, 1977. NorthHolland Mathematics Studies, Vol. 25; Notas de Matemática, No. 61. [Notes on Mathematics, No. 61]. MR 0500122
 João Prolla, On the spectra of nonArchimedean function algebras, Functional analysis, holomorphy, and approximation theory (Proc. Sem., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1978) Lecture Notes in Math., vol. 843, Springer, Berlin, 1981, pp. 547–560. MR 610845
 João Prolla, Topological algebras of vectorvalued continuous functions, Mathematical analysis and applications, Part B, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New YorkLondon, 1981, pp. 727–740. MR 634265
 Bertram Yood, Banach algebras of continuous functions, Amer. J. Math. 73 (1951), 30–42. MR 42068, DOI https://doi.org/10.2307/2372157
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46J20, 46K05
Retrieve articles in all journals with MSC: 46J20, 46K05
Additional Information
Article copyright:
© Copyright 1995
American Mathematical Society