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Banach algebras with uncomplemented radical


Authors: Gregory F. Bachelis and Sadahiro Saeki
Journal: Proc. Amer. Math. Soc. 100 (1987), 271-274
MSC: Primary 46J20; Secondary 43A45
DOI: https://doi.org/10.1090/S0002-9939-1987-0884465-0
MathSciNet review: 884465
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Abstract: A condition is given on the invertible elements of a commutative Banach algebra $ \mathbf{B}$ which implies that the radical of $ \mathbf{B}$ is not complemented by any closed subalgebra. This condition is then applied to show that certain quotient algebras of group algebras have an uncomplemented radical.


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  • [1] G. F. Bachelis, Some radical quotients in harmonic analysis, Radical Banach Algebras and Automatic Continuity (Long Beach, Calif., 1981), Lecture Notes in Math., vol. 975, Springer-Verlag, Berlin and New York, 1983, pp. 301-308. MR 697593 (84k:43001)
  • [2] G. F. Bachelisd and S. Saeki, Power bounded elements in a Banach algebra (in preparation).
  • [3] W. G. Bade and P. C. Curtis, Jr., Homomorphisms of commutative Banach algebras, Amer. J. Math. 82 (1960), 589-608. MR 0117577 (22:8354)
  • [4] -, The Wedderburn decomposition of commutative Banach algebras, Amer. J. Math. 82 (1960), 851-866. MR 0123200 (23:A529)
  • [5] H. G. Dales and J. Esterle, Discontinuous homomorphisms from $ C(X)$, Bull. Amer. Math. Soc. 83 (1977), 257-258. MR 0430786 (55:3791)
  • [6] C. Feldman, The Wedderburn principal theorem in Banach algebras, Proc. Amer. Math. Soc. 2 (1951), 771-777. MR 0044042 (13:361b)
  • [7] I. Gelfand, Zur Theorie der Charaktere der Abelschen topologischen Gruppen, Rec. Math. (N.S.) 9 [51] (1941), 49-50. MR 0004635 (3:36d)
  • [8] E. A. Gorin and V. Y. Lin, On a condition on the radical of a Banach algebra ensuring strong decomposability, Mat. Zametki 2 (1967), 589-592. MR 0221288 (36:4340)
  • [9] A. Y. Helemskii, Example of a compact space which has the uniqueness property without being an $ F$-space, Mat. Zametki 1 (1967), 735-740. MR 0212574 (35:3445)
  • [10] H. Helson, On the ideal structure of group algebras, Ark. Mat. 2 (1952), 83-86. MR 0049912 (14:246d)
  • [11] M. Henrikson, Some remarks on a paper of Aronszajn and Panitchpakdi, Pacific J. Math. 7 (1957), 1619-1621. MR 0092954 (19:1186f)
  • [12] E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R.I., 1957. MR 0089373 (19:664d)
  • [13] J.-P. Kahane, Séries de Fourier absolument convergentes, Springer-Verlag, Berlin, 1970. MR 0275043 (43:801)
  • [14] Y. Katznelson and W. Rudin, The Stone-Weierstrass property in Banach algebras, Pacific J. Math. 11 (1961), 253-265. MR 0126738 (23:A4032)
  • [15] W. Rudin, Fourier analysis on groups, Interscience, New York, 1962. MR 0152834 (27:2808)
  • [16] S. Saeki, A characterization of SH-sets, Proc. Amer. Math. Soc. 30 (1971), 497-503. MR 0283500 (44:731)
  • [17] -, An elementary proof of a theorem of Henry Helson, Tôhoku Math. J. 20 (1968), 244-247. MR 0231139 (37:6694)
  • [18] -, Extremally disconnected sets in groups, Proc. Amer. Math. Soc. 52 (1975), 317-318. MR 0372541 (51:8748)
  • [19] -, Helson sets which disobey spectral synthesis, Proc. Amer. Math. Soc. 47 (1975), 371-377. MR 0361625 (50:14070)
  • [20] -, Tensor products of Banach algebras and harmonic analysis, Tôhoku Math. J. 24 (1972), 281-299. MR 0324320 (48:2672)
  • [21] J. D. Stegeman, Extension of a theorem of Henry Helson, Proc. Internat. Congr. Math., Abstracts, Section 5 (1966), p. 28.

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DOI: https://doi.org/10.1090/S0002-9939-1987-0884465-0
Article copyright: © Copyright 1987 American Mathematical Society

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