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Preduals of semigroup algebras

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Abstract

For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C 0(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak*-continuous. Given a discrete semigroup S, the convolution algebra ℓ 1(S) also carries a coproduct. In this paper we examine preduals for ℓ 1(S) making both the product and the coproduct weak*-continuous. Under certain conditions on S, we show that ℓ 1(S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ 1(S) when S is either ℤ+×ℤ or (ℕ,⋅).

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References

  1. Chou, C.: Weakly almost periodic functions and almost convergent, functions on a group. Trans. Am. Math. Soc. 206, 175–200 (1975)

    Article  MATH  Google Scholar 

  2. Dales, H.G.: Banach Algebras and Automatic Continuity. Clarendon, Oxford (2000)

    MATH  Google Scholar 

  3. Dales, H.G., Lau, A.T.-M., Strauss, D.: Banach algebras on semigroups and on their compactifications. Mem. Am. Math. Soc. (to appear)

  4. Daws, M.: Dual Banach algebras: representations and injectivity. Stud. Math. 178, 231–275 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Daws, M., Le Pham, H., White, S.: Conditions implying the uniqueness of the, weak*-topology on certain group, algebras. Houst. J. Math. 35, 253–276 (2009)

    MATH  Google Scholar 

  6. Godefroy, G.: Parties admissibles d’un espace de Banach. Applications. Ann. Sci. École Norm. Sup. 16, 109–122 (1983)

    MATH  MathSciNet  Google Scholar 

  7. Godefroy, G.: Existence and uniqueness of isometric preduals: a survey. In: Banach Space Theory, Iowa City, IA, 1987. Contemp. Math., vol. 85, pp. 131–193. Am. Math. Soc., Providence (1989)

    Google Scholar 

  8. Godefroy, G., Saphar, P.D.: Duality in spaces of operators and smooth norms on Banach spaces. Ill. J. Math. 32, 672–695 (1988)

    MATH  MathSciNet  Google Scholar 

  9. Palmer, T.W.: Banach Algebras and the General Theory of *-Algebras. Vol. I. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  10. Runde, V.: Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule. Math. Scand. 95, 124–144 (2004)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, University of Leeds, Leeds, LS20 8DW, UK

    Matthew Daws

  2. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2E1, Canada

    Hung Le Pham

  3. Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, UK

    Stuart White

Authors
  1. Matthew Daws
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  2. Hung Le Pham
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  3. Stuart White
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Corresponding author

Correspondence to Matthew Daws.

Additional information

Communicated by Joachim Hilbert.

The second name author was supported by a Killam Postdoctoral Fellowship and a Honorary PIMS PDF.

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Daws, M., Pham, H.L. & White, S. Preduals of semigroup algebras. Semigroup Forum 80, 61–78 (2010). https://doi.org/10.1007/s00233-009-9186-5

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  • DOI: https://doi.org/10.1007/s00233-009-9186-5

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