On the ideal structure of algebras of star-algebra valued functions

Author:
Jorma Arhippainen

Journal:
Proc. Amer. Math. Soc. **123** (1995), 381-391

MSC:
Primary 46J20; Secondary 46K05

DOI:
https://doi.org/10.1090/S0002-9939-1995-1215198-3

MathSciNet review:
1215198

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Abstract: The ideal structure of the algebra 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 of *A* and some family of closed ideals of depending on those points and also by using different kind of slice ideals of .

**[1]**M. Abel,*Description of closed ideals in algebras of continuous vector-valued functions*, Math. Notes, vol. 30, Princeton Univ. Press, Princeton, NJ, 1981, pp. 887-892.**[2]**Richard Arens,*A generalization of normed rings*, Pacific J. Math.**2**(1952), 455–471. MR**0051445****[3]**Jorma Arhippainen,*On the ideal structure and approximation properties of algebras of continuous 𝐵*-algebra-valued functions*, Acta Univ. Oulu. Ser. A Sci. Rerum Natur.**187**(1987), 103. MR**918787****[4]**Jorma Arhippainen,*On commutative locally 𝑚-convex algebras*, Tartu Ül. Toimetised**928**(1991), 15–28 (English, with Estonian summary). MR**1150230****[5]**Jorma Arhippainen,*On the ideal structure of algebras of LMC-algebra valued functions*, Studia Math.**101**(1992), no. 3, 311–318. MR**1153787**, https://doi.org/10.4064/sm-101-3-311-318**[6]**-,*On locally convex square algebras*, Preprint series in Math., Univ. of Oulu, 1992.**[7]**Edward Beckenstein, Lawrence Narici, and Charles Suffel,*Topological algebras*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. North-Holland Mathematics Studies, Vol. 24; Notas de Matemática, No. 60. [Mathematical Notes, No. 60]. MR**0473835****[8]**William E. Dietrich Jr.,*The maximal ideal space of the topological algebra 𝐶(𝑋,𝐸)*, Math. Ann.**183**(1969), 201–212. MR**0254605**, https://doi.org/10.1007/BF01351380**[9]**-,*Function algebras on completely regular spaces*, Diss., Northwestern Univ., Evanston, IL, 1971.**[10]**James Dugundji,*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606****[11]**W. Hery,*Rings of continuous Banach algebra-valued functions*, Doct. Diss. Absttrs 45, Polytech. Instit. of New York, 1974.**[12]**William Hery,*Maximal ideals in algebras of continuous 𝐶(𝑆) valued functions*, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)**58**(1975), no. 2, 195–199 (English, with Italian summary). MR**0423060****[13]**William J. Hery,*Maximal ideals in algebras of topological algebra valued functions*, Pacific J. Math.**65**(1976), no. 2, 365–373. MR**0435854****[14]**Anastasios Mallios,*Topological algebras. Selected topics*, North-Holland Mathematics Studies, vol. 124, North-Holland Publishing Co., Amsterdam, 1986. Notas de Matemática [Mathematical Notes], 109. MR**857807****[15]**Ernest A. Michael,*Locally multiplicatively-convex topological algebras*, Mem. Amer. Math. Soc.,**No. 11**(1952), 79. MR**0051444****[16]**Peter D. Morris and Daniel E. Wulbert,*Functional representation of topological algebras*, Pacific J. Math.**22**(1967), 323–337. MR**0213876****[17]**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****[18]**João Bosco Prolla,*Approximation of vector valued functions*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. North-Holland Mathematics Studies, Vol. 25; Notas de Matemática, No. 61. [Notes on Mathematics, No. 61]. MR**0500122****[19]**João Prolla,*On the spectra of non-Archimedean 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****[20]**João Prolla,*Topological algebras of vector-valued continuous functions*, Mathematical analysis and applications, Part B, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 727–740. MR**634265****[21]**Bertram Yood,*Banach algebras of continuous functions*, Amer. J. Math.**73**(1951), 30–42. MR**0042068**, https://doi.org/10.2307/2372157

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DOI:
https://doi.org/10.1090/S0002-9939-1995-1215198-3

Article copyright:
© Copyright 1995
American Mathematical Society