Regularity and amenability of the second dual of weighted group algebras
DOI:
https://doi.org/10.4067/S0716-09172007000300004Keywords:
Arens product, Weighted group algebra, Amenability.Abstract
For a wide variety of Banach algebras A (containing the group algebras L1(G), M(G) and A(G)) the Arens regularity of A** is equivalent to that A, and the amenability of A** is equivalent to the amenability and regularity of A. In this paper, among other things, we show that this variety contains the weighted group algebras L1(G, w) and M(G, w).References
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[B-R] Baker J.W. and Rejali A.,“ On the Arens regularity of the weighted convolution algebras, J. London Math. Soc. (2) 40, pp. 535-546, (1987).
[D-G-H] Dales H.G., Ghahramani F. and Helemskii A.Y.A., “ The amenability of measure algebras”, J. London Math. Soc. (2) 66, pp. 213-226, (2002).
[D-H] Duncan J. and Houssainun S.A.R.,“ The second dual of a Banach algebra”, Proc. Roy. Soc. Edinburgh A 84, pp. 309-325, (1979).
[D-L] Dales H.G. and Lau A.T.-M.,“ The second duals of Beurling algebras”, Mem. Amer. Math. Soc. 177, No. 386, (2005).
[F-R] Forrest B. and Runde V.,“ Amenability and weak amenability of the Fourier algebra”, Math. Z. 250 4, pp. 731-744, (2005).
[G-L] Ghahramani F. and Laali J.“ Amenability and topological centers of the second duals of Banach algebras”, Bull. Austral. Math. Soc. 65, pp. 191-197, (2002).
[G-L-L] Ghahramani F., Lau A.T.M. and Losert V.,“ Isometric isomorphisms between Banach algebras related to locally compact groups”, Trans, Amer. Math. Soc. 321, pp. 273-383, (1990).
[G-L-W] Ghahramani F., Loy R.J. and Willis G.,“ Amenability and weak amenability of second conjugate Banach algebras”, Proc. Amer. Math. Soc. 124, pp. 1489-1497, (1996).
[Go] Gourdeau F.,“Amenability and the second dual of a Banach algebra“, Studia Math. 125, pp. 45-80, (1997).
[Gro] Gronback N.,“ Amenability of weighted convolution algebras on locally compact groups”, Trans. Amer. Math. Soc. 319, pp. 765-775, (1990).
[Gra] Granirer E.,“ Amenability and semisimplicity for second duals of quotients of the Fourier algebra A(G),”J. Austral. Math. Soc. (Series A) 63, pp. 289-296, (1997).
[L-L] Lau A.T.M and Loy R.J.,“ Weak amenability of Banach algebras on locally compact groups”, J. Funct. Anal. 145, pp. 175-204, (1997).
[P] Pym J.S.,“ Remarks on the second duals of Banach algebras”, Nigerian Math. Soc. 2, pp. 31-33, (1983).
[R1] Rejali A.,“ The analogue of weighted group algebra for semitopological semigroups”, J. Sci. I.R. Iran 6, pp. 113-120, (1995).
[R2] Rejali A.,“ Weighted function spaces on topological groups”, Bull. Iranian Math. Soc. 22, pp. 43-63, (1996).
[S] Sherman S.,“ The second adjoint of a C*-algebra,” Proc. Intern. Congr. Math. Cambridge, I, 470, (1950).
[U1] Ulger A.,“ Arens regularity of weakly sequentially complete Banach algebras”, Proc. Amer. Math. Soc. 127, pp. 3221-3227, (1999).
[U2] Ulger A.,“ Centeral elements of A** for certain Banach algebras A without bounded approximate identities,”Glasgow Math. J. 41, pp. 369-377, (1999).
[W] White M.,“ Characters on weighted amenable groups”, Bull. London Math. Soc. 23, pp. 375-380, (1991).
[Y1] Young N.,“ The irregularity of multiplication in group algebras,” Quart. J. Math. Oxford (2) 24, pp. 59-62, (1973).
[Y2] Young N.,“ Separate continuity and multilinear operations”, Proc. London Math. Soc. (3) 26, pp. 289-319, (1973).
[Y3] Young N.,“ Periodicity of functionals and representations of normed algebras on reflexive spaces,” Proc. Edinburgh Math. Soc. 20, pp. 100-120, (1976).
[B-R] Baker J.W. and Rejali A.,“ On the Arens regularity of the weighted convolution algebras, J. London Math. Soc. (2) 40, pp. 535-546, (1987).
[D-G-H] Dales H.G., Ghahramani F. and Helemskii A.Y.A., “ The amenability of measure algebras”, J. London Math. Soc. (2) 66, pp. 213-226, (2002).
[D-H] Duncan J. and Houssainun S.A.R.,“ The second dual of a Banach algebra”, Proc. Roy. Soc. Edinburgh A 84, pp. 309-325, (1979).
[D-L] Dales H.G. and Lau A.T.-M.,“ The second duals of Beurling algebras”, Mem. Amer. Math. Soc. 177, No. 386, (2005).
[F-R] Forrest B. and Runde V.,“ Amenability and weak amenability of the Fourier algebra”, Math. Z. 250 4, pp. 731-744, (2005).
[G-L] Ghahramani F. and Laali J.“ Amenability and topological centers of the second duals of Banach algebras”, Bull. Austral. Math. Soc. 65, pp. 191-197, (2002).
[G-L-L] Ghahramani F., Lau A.T.M. and Losert V.,“ Isometric isomorphisms between Banach algebras related to locally compact groups”, Trans, Amer. Math. Soc. 321, pp. 273-383, (1990).
[G-L-W] Ghahramani F., Loy R.J. and Willis G.,“ Amenability and weak amenability of second conjugate Banach algebras”, Proc. Amer. Math. Soc. 124, pp. 1489-1497, (1996).
[Go] Gourdeau F.,“Amenability and the second dual of a Banach algebra“, Studia Math. 125, pp. 45-80, (1997).
[Gro] Gronback N.,“ Amenability of weighted convolution algebras on locally compact groups”, Trans. Amer. Math. Soc. 319, pp. 765-775, (1990).
[Gra] Granirer E.,“ Amenability and semisimplicity for second duals of quotients of the Fourier algebra A(G),”J. Austral. Math. Soc. (Series A) 63, pp. 289-296, (1997).
[L-L] Lau A.T.M and Loy R.J.,“ Weak amenability of Banach algebras on locally compact groups”, J. Funct. Anal. 145, pp. 175-204, (1997).
[P] Pym J.S.,“ Remarks on the second duals of Banach algebras”, Nigerian Math. Soc. 2, pp. 31-33, (1983).
[R1] Rejali A.,“ The analogue of weighted group algebra for semitopological semigroups”, J. Sci. I.R. Iran 6, pp. 113-120, (1995).
[R2] Rejali A.,“ Weighted function spaces on topological groups”, Bull. Iranian Math. Soc. 22, pp. 43-63, (1996).
[S] Sherman S.,“ The second adjoint of a C*-algebra,” Proc. Intern. Congr. Math. Cambridge, I, 470, (1950).
[U1] Ulger A.,“ Arens regularity of weakly sequentially complete Banach algebras”, Proc. Amer. Math. Soc. 127, pp. 3221-3227, (1999).
[U2] Ulger A.,“ Centeral elements of A** for certain Banach algebras A without bounded approximate identities,”Glasgow Math. J. 41, pp. 369-377, (1999).
[W] White M.,“ Characters on weighted amenable groups”, Bull. London Math. Soc. 23, pp. 375-380, (1991).
[Y1] Young N.,“ The irregularity of multiplication in group algebras,” Quart. J. Math. Oxford (2) 24, pp. 59-62, (1973).
[Y2] Young N.,“ Separate continuity and multilinear operations”, Proc. London Math. Soc. (3) 26, pp. 289-319, (1973).
[Y3] Young N.,“ Periodicity of functionals and representations of normed algebras on reflexive spaces,” Proc. Edinburgh Math. Soc. 20, pp. 100-120, (1976).
Published
2017-04-12
How to Cite
[1]
A. Rejali and H. R. E. Vishki, “Regularity and amenability of the second dual of weighted group algebras”, Proyecciones (Antofagasta, On line), vol. 26, no. 3, pp. 259-267, Apr. 2017.
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