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manuscripta mathematica

, Volume 45, Issue 2, pp 127–146 | Cite as

The norm-strict bidual of a Banach algebra and the dual of Cu(G)

  • Michael Grosser
  • Viktor Losert
Article

Abstract

To each Banach algebra A we associate a (generally) larger Banach algebra A+ which is a quotient of its bidual A″. It can be constructed using the strict topology on A and the Arens product on A″. A+ has certain more pleasant properties than A″, e.g. if A has a bounded right approximate identity, then A+ has a two-sided unit. In the special case A=L1(G) (G a locally compact abelian group) one gets A+=Cu(G)′, the dual of the space of bounded, uniformly continuous functions on G, and we show that the center of the convolution algebra Cu(G)′ is precisely the space M(G) of finite measures on G.

Keywords

Continuous Function Abelian Group Number Theory Algebraic Geometry Topological Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [A1]
    ARENS, R.F.: Operations induced in functions classes, Monatsh. Math.55, 1–19 (1951)Google Scholar
  2. [A2]
    ARENS, R.F.: The adjoint of a bilinear operation, Proc. Amer. Math. Soc.2, 839–848 (1951)Google Scholar
  3. [CLM]
    CIGLER, J., LOSERT, V., MICHOR, P.: Banach modules and functors on categories of Banach spaces, Lecture Notes in Pure and Applied Mathematics, vol. 46, New York: Marcel Dekker Inc. 1979Google Scholar
  4. [CY]
    CIVIN, P., YOOD, B.: The second conjugate space of a Banach algebra as an algebra, Pacific J. Math.11, 847–870 (1961)Google Scholar
  5. [D]
    DAVENPORT, J.W.: Multipliers on a Banach algebra with a bounded approximate identity, Pacific J. Math.63, 131–135 (1976)Google Scholar
  6. [G1]
    GROSSER, M.: Bidualräume und Vervollständigungen von Banachmoduln, Universität Wien, Dissertation 1976; Lecture Notes in Mathematics, vol. 717, Berlin-Heidelberg-New York: Springer 1979Google Scholar
  7. [G2]
    GROSSER, M.: L1(G) as an ideal in its second dual space, Proc. Amer. Math. Soc.73, 363–364 (1979)Google Scholar
  8. [G3]
    GROSSER, M.: Module tensor products of Ko (X,X) with its dual, to appear in: Colloquia Mathematica Societatis Janos Bolyai, vol. 35, Functions, Series, Operators, Budapest (Hungary), 1980, Amsterdam-Oxford-New York: North HollandGoogle Scholar
  9. [GLR]
    GULICK, S.L., LIU, T.S., ROOIJ, A.C.M. van,: Group algebra modules I, Canad. J. Math.19, 133–150 (1967)Google Scholar
  10. [HR]
    HEWITT, E., ROSS, K.A.: Abstract harmonic analysis, Part I, Berlin-Heidelberg-New York: Springer 1963Google Scholar
  11. [ST]
    SENTILLES, F.D., TAYLOR, D.C.: Factorization in Banach algebras and the general strict topology, Trans. Amer. Math. Soc.142, 141–152 (1969)Google Scholar
  12. [Tak]
    TAKAHASI, SIN-EI: Dixmier's representation theorem of central double centraliziers on Banach algebras, Trans. Amer. Math. Soc.253, 229–236 (1979)Google Scholar
  13. [Tay]
    TAYLOR, D.C.: The strict topology for double centralizer algebras. Trans. Amer. Math. Soc.150, 633–643 (1970)Google Scholar
  14. [To]
    TOMIUK, B.J.: Multipliers on Banach algebras, Studia Math.54, 267–283 (1975)Google Scholar
  15. [W]
    WENDEL, J.G.: Left centralizers and isomorphisms of group algebras, Pacific J. Math.2, 251–261 (1952)Google Scholar
  16. [Y]
    YOUNG, N.J.: The irregularity of multiplication in group algebras, Quart. J. Math. Oxford Ser. 224, 59–62 (1973)Google Scholar
  17. [Z1]
    ZAPPA, A.; The center of the convolution algebra Cu(G)*, Rend. Sem. Mat. Univ. Padova,52, 71–83 (1974)Google Scholar
  18. [Z2]
    ZAPPA, A.: The center of an algebra of operators, Coll. Math.39, 343–349 (1978)Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Michael Grosser
    • 1
  • Viktor Losert
    • 1
  1. 1.Institut für MathematikUniversität WienWienAustria

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