Quotient Arens regularity of $L^1(G)$

Subject Areas : History and biography

A. Zivari-Kazempour ¹

1 - Department of Mathematics, University of Ayatollah Borujerdi, Borujerd, Iran

Received: 2015-12-09 Accepted : 2016-02-28 Published : 2015-12-01

Keywords: Arens product, Quotient Arens regular, Introverted subspace, Weakly almost periodic,

Abstract :

Let $\mathcal{A}$ be a Banach algebra with BAI and $E$ be an introverted subspace of $\mathcal{A}^\prime$.In this paper we study the quotient Arens regularity of $\mathcal{A}$ with respect to $E$ and prove that the group algebra $L^1(G)$ for a locally compact group $G$, is quotient Arens regular with respect to certain introverted subspace $E$ of $L^\infty(G)$.Some related result are given as well.

References:

[1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839-848.

[2] J. F. Berglund, H. D. Junghenn and P. Milnes, Analysis on Semigroups, Wiley-Interscience, New York, 1989.

[3] P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847-870.

[4] H. G. Dales, Banach algebras and automatic continuity, London Math. Soc. Monographs 24, Clarenden Press, Oxford, 2000.

[5] H. G. Dales and A. T. M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005), 1-199.

[6] H. G. Dales, A. T.-M. Lau and D. Strauss, Banach algebras on semigroups and on their compactification, Mem. Amer. Math. Soc. 205 (2010), 1-165.

[7] E. Hewitt and K. Ross, Abstract Harmonic Analysis, Volume II, Springer-Verlag, Berlin, 1970.

[8] N. Isik, J. Pym and A. Ulger, The second dual of the group algebra of a compact group, J. London Math. Soc. 35 (1987), 135-158.

[9] A. T. M. Lau and A. Ulger, Topological centres of certain dual algebras, Trans. Amer. Math. Soc. 348 (1996), 1191-1212.

[10] A. Ulger, Arens regularity of the algebra A⊗Bb , Trans. Amer. Math. Soc, 305 (1988), 623-639.

[11] A. Ulger, Arens regularity sometimes implies RNP, Pacific J. Math. 143 (1990), 377-399.

[12] P. K. Wong, Arens product and the algebra of duble multipliers, Proc. Amem. Math. Soc. 94 (1985), 441-444.

[13] N. J. Young, The irregularity of multiplication in group algebras, Quart. J. Math. 24 (1973), 59-62.