Prime ideals in algebras of continuous functions
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- by H. G. Dales and R. J. Loy PDF
- Proc. Amer. Math. Soc. 98 (1986), 426-430 Request permission
Abstract:
Let ${X_0}$ be a compact Hausdorff space, and let ${\mathbf {C}}({X_0})$ be the Banach algebra of all continuous complex-valued functions on ${X_0}$. It is known that, assuming the continuum hypothesis, any nonmaximal, prime ideal ${\mathbf {P}}$ such that $|{\mathbf {C}}({X_0})/{\mathbf {P}}| = {2^{{\aleph _0}}}$ is the kernel of a discontinuous homomorphism from ${\mathbf {C}}({X_0})$ into some Banach algebra. Here we consider the converse question of which ideals can be the kernels of such a homomorphism. Partial results are obtained in the case where ${X_0}$ is metrizable.References
- W. G. Bade and P. C. Curtis Jr., Homomorphisms of commutative Banach algebras, Amer. J. Math. 82 (1960), 589–608. MR 117577, DOI 10.2307/2372972
- H. G. Dales, A discontinuous homomorphism from $C(X)$, Amer. J. Math. 101 (1979), no. 3, 647–734. MR 533196, DOI 10.2307/2373803
- H. G. Dales, Automatic continuity: a survey, Bull. London Math. Soc. 10 (1978), no. 2, 129–183. MR 500923, DOI 10.1112/blms/10.2.129
- H. G. Dales and W. H. Woodin, An introduction to independence for analysts, London Mathematical Society Lecture Note Series, vol. 115, Cambridge University Press, Cambridge, 1987. MR 942216, DOI 10.1017/CBO9780511662256
- Jean Esterle, Semi-normes sur ${\cal C}(K)$, Proc. London Math. Soc. (3) 36 (1978), no. 1, 27–45 (French). MR 482215, DOI 10.1112/plms/s3-36.1.27 —, Solution d’un problème d’Ërdos, Gillman et Hendriksen et application a l’étude des homomorphisme de $C(K)$, Acta Math. Acad. Sci. Hungar. 30 (1977), 113-127.
- Jean Esterle, Sur l’existence d’un homomorphisme discontinu de ${\cal C}(K)$, Proc. London Math. Soc. (3) 36 (1978), no. 1, 46–58 (French). MR 482217, DOI 10.1112/plms/s3-36.1.46
- Jean Esterle, Injection de semi-groupes divisibles dans des algèbres de convolution et construction d’homomorphismes discontinus de ${\cal C}(K)$, Proc. London Math. Soc. (3) 36 (1978), no. 1, 59–85 (French). MR 482218, DOI 10.1112/plms/s3-36.1.59
- J. Esterle, Homomorphismes discontinus des algèbres de Banach commutatives séparables, Studia Math. 66 (1979), no. 2, 119–141 (French, with English summary). MR 565154, DOI 10.4064/sm-66-2-119-141
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, Graduate Texts in Mathematics, No. 43, Springer-Verlag, New York-Heidelberg, 1976. Reprint of the 1960 edition. MR 0407579
- Sandy Grabiner, A formal power series operational calculus for quasi-nilpotent operators. II, J. Math. Anal. Appl. 43 (1973), 170–192. MR 358410, DOI 10.1016/0022-247X(73)90267-9
- B. E. Johnson, Norming $C(U)$ and related algebras, Trans. Amer. Math. Soc. 220 (1976), 37–58. MR 415326, DOI 10.1090/S0002-9947-1976-0415326-6
- Allan M. Sinclair, Homomorphisms from $C_{0}(R)$, J. London Math. Soc. (2) 11 (1975), no. 2, 165–174. MR 377517, DOI 10.1112/jlms/s2-11.2.165
- Russell C. Walker, The Stone-Čech compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83, Springer-Verlag, New York-Berlin, 1974. MR 0380698 W. H. Woodin, Set theory and discontinuous homomorphisms from Banach algebras, Mem. Amer. Math. Soc. (to appear).
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 98 (1986), 426-430
- MSC: Primary 46J10; Secondary 46J20, 54C40
- DOI: https://doi.org/10.1090/S0002-9939-1986-0857934-6
- MathSciNet review: 857934