Abstract
We define module operator amenability for completely contractive Banach algebras which are Banach modules over another Banach algebra with compatible actions and study module operator amenability of the Fourier algebra \(A(S)\) of an inverse semigroup \(S\) with the set of idempotents \(E\), as a module over the semigroup algebra \(\ell ^{1}(E)\). We show that when \(E\) has a minimum element, module operator amenability of \(A(S)\) is equivalent to amenability of \(S\).
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References
Amini, M.: Module amenability for semigroup algebras. Semigroup Forum 69, 243–254 (2004)
Amini, M.: Corrigendum: module amenability for semigroup algebras. Semigroup Forum 72, 493–494 (2006)
Amini, M., Bodaghi, A., Ebrahimi Bagha, D.: Module amenability of the second dual and module topological center of semigroup algebras. Semigroup Forum 80, 302–312 (2010)
Barnes, B.A.: Representations of the \(l^{1}\)-algebra of an inverse semigroup. Trans. Am. Math. Soc. 218, 361–396 (1976)
Bohm, G., Szlachanyi, K.: Weak \(C^{*}\)-Hopf algebras and multiplicative isometries. J. Operator Theory 45, 357–376 (2001)
Conway, J.B.: A Course in Operator Theory. American Mathematical Society, Providence (1991)
Dales, H.G., Lau, A.T.-M., Strauss, D.: Banach algebras on semigroups and their compactification. Memoirs of the American Mathematical Society, 205, No. 966 (2010)
Duncan, J., Namioka, I.: Amenability of inverse semigroups and their semigroup algebras. Proc. R. Soc. Edinb. 80A, 309–321 (1978)
Doran, R.S., Wichmann, J.: Approximate Identities and Factorization in Banach Modules. Lecture Notes in Math., vol. 768. Springer-Verlag, Berlin (1979)
Effros, E.G., Ruan, Z.J.: Operator Spaces. Oxford University Press, New York (2000)
Eymard, P.: L’alg\(\grave{e}\)bre de Fourier d’un groupe localement compact. Bull. Soc. Math. France 92, 181–236 (1964)
Forrest, B.E., Runde, V.: Amenability and weak amenability of the Fourier algebra. Math. Z. 250, 731–744 (2005)
Johnson, B.E.: Cohomology in Banach Algebras. Memoirs of the American Mathematical Society, No. 127. American Mathematical Society, Providence (1972)
Johnson, B.E.: Non-amenability of the Fourier algebra of a compact group. J. Lond. Math. Soc. 50(2), 361–374 (1994)
Leptin, H.: Sur l’alg\(\grave{e}\)bre de Fourier d’un groupe localement compact. C. R. Acad. Sci. Paris Ser. A 266, 1180–1182 (1968)
Munn, W.D.: A class of irreducible matrix representations of an arbitrary inverse semigroup. Proc. Glasgow Math. Assoc. 5, 41–48 (1961)
Ruan, Z.J.: The operator amenability of A(G). Am. J. Math. 117, 1449–1474 (1995)
Wordingham, J.R.: The left regular \(*\)-representation of an inverse semigroup. Proc. Am. Math. Soc. 86, 55–58 (1982)
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The first author was partly supported by a grant from IPM (No. 90430215).
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Communicated by Jerome A. Goldstein.
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Amini, M., Rezavand, R. Module operator amenability of the Fourier algebra of an inverse semigroup. Semigroup Forum 92, 45–70 (2016). https://doi.org/10.1007/s00233-014-9678-9
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DOI: https://doi.org/10.1007/s00233-014-9678-9