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Transactions of the American Mathematical Society

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Singly generated homogeneous $ F$-algebras


Author: Ronn Carpenter
Journal: Trans. Amer. Math. Soc. 150 (1970), 457-468
MSC: Primary 46.50
DOI: https://doi.org/10.1090/S0002-9947-1970-0262829-8
MathSciNet review: 0262829
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Abstract: With each point $ m$ in the spectrum of a singly generated $ F$-algebra we associate an algebra $ {A_m}$ of germs of functions. It is shown that if $ {A_m}$ is isomorphic to the algebra of germs of analytic functions of a single complex variable, then the spectrum of $ A$ contains an analytic disc about $ m$. The algebra $ A$ is called homogeneous if the algebras $ {A_m}$ are all isomorphic. If $ A$ is homogeneous and none of the algebras $ {A_m}$ have zero divisors, we show that $ A$ is the direct sum of its radical and either an algebra of analytic functions or countably many copies of the complex numbers. If $ A$ is a uniform algebra which is homogeneous, then it is shown that $ A$ is either the algebra of analytic functions on an open subset of the complex numbers or the algebra of all continuous functions on its spectrum.


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  • [1] R. Arens, Dense inverse limit rings, Michigan Math. J. 5 (1958), 169-182. MR 21 #3780. MR 0105034 (21:3780)
  • [2] F. T. Birtel, Singly-generated Liouville $ F$-algebras, Michigan Math. J. 11 (1964), 89-94. MR 28 #4376. MR 0161167 (28:4376)
  • [3] R. M. Brooks, Boundaries for locally $ m$-convex algebras, Duke Math. J. 34 (1967), 103-116. MR 37 #762. MR 0225167 (37:762)
  • [4] -, On the spectrum of finitely generated locally $ M$-convex algebras, Studia Math. 29 (1968), 143-150.
  • [5] -, On singly-generated locally $ M$-convex algebras (preprint).
  • [6] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 33 #1824. MR 0193606 (33:1824)
  • [7] N. Dunford and J. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [8] A. Gleason, Finitely generated ideals in Banach algebras, J. Math. Mech. 13 (1964), 125-132. MR 28 #2458. MR 0159241 (28:2458)
  • [9] R. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, N. J., 1965. MR 31 #4927. MR 0180696 (31:4927)
  • [10] R. McKissick, A nontrivial normal sup norm algebra, Bull. Amer. Math. Soc. 69 (1963), 391-395. MR 26 #4166. MR 0146646 (26:4166)
  • [11] S. N. Mergelyan, On the representation of functions by series of polynomials on closed sets, Dokl. Akad. Nauk SSSR 78 (1951), 405-408; English transl., Amer. Math. Soc. Transl. (1) 3 (1962), 287-293. MR 13, 23. MR 0041929 (13:23f)
  • [12] E. A. Michael, Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc. No. 11 (1952). MR 14, 482. MR 0051444 (14:482a)
  • [13] M. Rosenfeld, Commutative $ F$-algebras, Pacific J. Math. 16 (1966), 159-166. MR 32 #8196. MR 0190786 (32:8196)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0262829-8
Keywords: $ F$-algebra, singly generated, homogeneous, analytic disc, uniform algebra
Article copyright: © Copyright 1970 American Mathematical Society

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