The kernels of radical homomorphisms and intersections of prime ideals
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Abstract:
We establish a necessary condition for a commutative Banach algebra $A$ so that there exists a homomorphism $\theta$ from $A$ into another Banach algebra such that the prime radical of the continuity ideal of $\theta$ is not a finite intersection of prime ideals in $A$. We prove that the prime radical of the continuity ideal of an epimorphism from $A$ onto another Banach algebra (or of a derivation from $A$ into a Banach $A$-bimodule) is always a finite intersection of prime ideals. Under an additional cardinality condition (and assuming the Continuum Hypothesis), this necessary condition is proved to be sufficient. En route, we give a general result on norming commutative semiprime algebras; extending the class of algebras known to be normable. We characterize those locally compact metrizable spaces $\Omega$ for which there exists a homomorphism from $\mathcal C_0(\Omega )$ into a radical Banach algebra whose kernel is not a finite intersection of prime ideals.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
- Julian Cusack, Automatic continuity and topologically simple radical Banach algebras, J. London Math. Soc. (2) 16 (1977), no. 3, 493–500. MR 461136, DOI 10.1112/jlms/s2-16.3.493
- H. G. Dales, The uniqueness of the functional calculus, Proc. London Math. Soc. (3) 27 (1973), 638–648. MR 333738, DOI 10.1112/plms/s3-27.4.638
- 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, A discontinuous homomorphism from $C(X)$, Amer. J. Math. 101 (1979), no. 3, 647–734. MR 533196, DOI 10.2307/2373803
- H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs. New Series, vol. 24, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. MR 1816726
- H. G. Dales and R. J. Loy, Prime ideals in algebras of continuous functions, Proc. Amer. Math. Soc. 98 (1986), no. 3, 426–430. MR 857934, DOI 10.1090/S0002-9939-1986-0857934-6
- 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
- H. Garth Dales and W. Hugh Woodin, Super-real fields, London Mathematical Society Monographs. New Series, vol. 14, The Clarendon Press, Oxford University Press, New York, 1996. Totally ordered fields with additional structure; Oxford Science Publications. MR 1420859
- 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
- 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
- Jean Esterle, Universal properties of some commutative radical Banach algebras, J. Reine Angew. Math. 321 (1981), 1–24. MR 597976, DOI 10.1515/crll.1981.321.1
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Band 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156915
- Karl H. Hofmann and Sidney A. Morris, The structure of compact groups, De Gruyter Studies in Mathematics, vol. 25, Walter de Gruyter & Co., Berlin, 1998. A primer for the student—a handbook for the expert. MR 1646190
- 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
- Irving Kaplansky, Normed algebras, Duke Math. J. 16 (1949), 399–418. MR 31193
- Keith Kendig, Elementary algebraic geometry, Graduate Texts in Mathematics, No. 44, Springer-Verlag, New York-Berlin, 1977. MR 0447222, DOI 10.1007/978-1-4615-6899-5
- Sidney A. Morris, Pontryagin duality and the structure of locally compact abelian groups, London Mathematical Society Lecture Note Series, No. 29, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR 0442141, DOI 10.1017/CBO9780511600722
- Walter Rudin, Averages of continuous functions on compact spaces, Duke Math. J. 25 (1958), 197–204. MR 98313
- 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
- Allan M. Sinclair, Automatic continuity of linear operators, London Mathematical Society Lecture Note Series, No. 21, Cambridge University Press, Cambridge-New York-Melbourne, 1976. MR 0487371, DOI 10.1017/CBO9780511662355
- Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, Graduate Studies in Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2002. MR 1900941, DOI 10.1090/gsm/046
Additional Information
- Hung Le Pham
- Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England
- Address at time of publication: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2E1
- Email: hung@maths.leeds.ac.uk, hlpham@math.ualberta.ca
- Received by editor(s): June 1, 2005
- Received by editor(s) in revised form: April 7, 2006
- Published electronically: July 23, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 1057-1088
- MSC (2000): Primary 46H40, 46J10; Secondary 46J05, 13C05, 43A20
- DOI: https://doi.org/10.1090/S0002-9947-07-04325-5
- MathSciNet review: 2346483