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Continuity of epimorphisms and derivations on vector-valued group algebras

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Abstract

This paper deals with the automatic continuity theory for the convolution algebra of all Bochner integrable functions from a locally compact abelian group G into an arbitrary unital complex Banach algebra A. For non-compact G, it is shown that all epimorphisms and all derivations on this vector-valued group algebra are necessarily continuous while for compact G, such results depend heavily on the automatic continuity properties of the range algebra a.

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References

  1. E. Albrecht and M. M. Neumann, Automatic continuity for operators of local type. In: 3 Radical Banach Algebras and Automatic Continuity. LNM 975, 342–355 (1983).

    Article  MathSciNet  Google Scholar 

  2. F. F. Bonsall and 1 Duncan, Complete normed algebras. New York-Heidelberg-Berlin 1973.

  3. H. G. Dales, Automatic continuity: a survey. Bull. London Math. Soc. 10, 129–183 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  4. H. G. Dales, Convolution algebras on the real line. In: Radical Banach Algebras and Automatic Continuity. LNM 975, 180–209 (1983).

    Article  MathSciNet  Google Scholar 

  5. J. Diestel and J. J. Uhl, Vector measures. Math. Surveys 15, Providence, RI 1977.

  6. I. M. Gelfand, D. A. Raikov and G. E. Shilov, Commutative normed rings. New York 1964.

  7. E. Hille and R. S. Phillips, Functional analysis and semi-groups. Amer. Math. Soc. Colloq. Publ. 31, Providence, RI 1957.

  8. N. P. Jewell and A. M. Sinclair, Epimorphisms and derivations on L 1 (0,1) are continuous. 3 Bull. London. Math. Soc. 8, 135–139 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  9. B. E. Johnson, The uniqueness of the (complete) norm topology. Bull. Amer. Math. Soc. 73, 3 537–539 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  10. B. E. Johnson, Continuity of derivations on commutative algebras. Amer. J. Math. 91, 1–10 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  11. B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky. Amer. J. Math. 90, 1067–1073 (1968).

    Article  MATH  MathSciNet  Google Scholar 

  12. G. P. Johnson, Spaces of functions with values in a Banach algebra. Trans. Amer. Math. Soc. 92, 411–429(1959).

    Article  MATH  MathSciNet  Google Scholar 

  13. R. Kantrowitz and M. M. Neumann, Automatic continuity of homomorphisms and derivations on algebras of continuous vector-valued functions. Czechoslovak Math. J. 45, 747–756 (1995).

    MATH  MathSciNet  Google Scholar 

  14. M. M. Neumann, Automatic continuity of linear operators. In: Functional Analysis: Surveys and Recent Results II. North-Holland Math. Stud. 38, 269–296 (1980).

    Article  Google Scholar 

  15. W. Rudin, Fourier analysis on groups. New York-London 1962.

  16. A. M. Sinclair, Automatic continuity of linear operators. London Math. Soc. Lecture Note Ser. 21, Cambridge 1976.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, Mississippi State University, 39762, Mississippi State, MS, USA

    Michael M. Neumann

  2. Departamento de Análisis Matemático, Universidad de Granada, 18071, Granada, Spain

    M. Victoria Velasco

Authors
  1. Michael M. Neumann
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  2. M. Victoria Velasco
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Dedicated to Heinz Konig on the occasion of his 65th birthday

Research supported by Grant SNF 11-1015 from the Danish Science Research Council.

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Neumann, M.M., Velasco, M.V. Continuity of epimorphisms and derivations on vector-valued group algebras. Arch. Math. 68, 151–158 (1997). https://doi.org/10.1007/s000130050044

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  • DOI: https://doi.org/10.1007/s000130050044

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