## Banach Journal of Mathematical Analysis

### Quasi-multipliers of the dual of a Banach algebra

M. Adib, J. Bracic, and A. Riazi

#### Abstract

In this paper we extend the notion of quasi-multipliers to the dual of a Banach algebra $A$ whose second dual has a mixed identity. We consider algebras satisfying weaker condition than Arens regularity. Among others we prove that for an Arens regular Banach algebra which has a bounded approximate identity the space $QM_{r}(A^{*})$ of all bilinear and separately continuous right quasi-multipliers of $A^{*}$ is isometrically isomorphic to $A^{**}.$ We discuss the strict topology on $QM_{r}(A^{*})$ and apply our results to $C^{*}-$algebras and to the group algebra of a compact group.

#### Article information

Source
Banach J. Math. Anal., Volume 5, Number 2 (2011), 6-14.

Dates
First available in Project Euclid: 14 August 2011

https://projecteuclid.org/euclid.bjma/1313362997

Digital Object Identifier
doi:10.15352/bjma/1313362997

Mathematical Reviews number (MathSciNet)
MR2780864

Zentralblatt MATH identifier
1222.47058

#### Citation

Adib, M.; Riazi, A.; Bracic, J. Quasi-multipliers of the dual of a Banach algebra. Banach J. Math. Anal. 5 (2011), no. 2, 6--14. doi:10.15352/bjma/1313362997. https://projecteuclid.org/euclid.bjma/1313362997

#### References

• C.A. Akemann and G.K. Pedersen, Complications of semicontinuity in $C^*-$algebra theory, Duke Math. J. 40 (1973), 785–795.
• Z. Argün and K. Rowlands, On quasi-multipliers, Studia Math. 108 (1994), 217–245.
• P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847–870.
• H.G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monographs, Clarendon press, 2000.
• B. Dearden, Quasi-multipliers of Pedersen's ideal, Rocky Mountain J. Math. 22 (1992), 157–163.
• R.E. Edwards, Functional Analysis, Theory and Application, Holt, Rinehart and Winston, 1965.
• M. Grosser, Quasi-multipliers of the algebra of approximable operators and its duals, Studia Math. 124 (1997), 291–300.
• M.S. Kassem and K. Rowlands, The quasi-strict topology on the space of quasi-multipliers of a $B^*-$algebra, Math. Proc. Cambridge Philos. Soc. 101 (1987), 555–566.
• M. Kaneda, Quasi-multipliers and algebrizations of an operator space, J. Funct. Anal. 251 (2007), 346–359.
• M. Kaneda and V.I. Paulsen, Quasi-multipliers of operator spaces, J. Funct. Anal. 217 (2004), 347–365.
• G. Köthe, Topological Vector Space I, I. New York-Heidelberg-Berlin: Springer, (1969).
• H. Lin, The structure of quasi-multipliers of $C^*-$algebras, Trans. Amer. Math. Soc. 315 (1987), 147–172.
• H. Lin, Fundamental approximate identities and quasi-multipliers of simple AFC*-algebras, J. Func. Anal. 79 (1988), 32–43.
• H. Lin, Support algebras of $\sigma-$unital $C^*-$algebras and their quasi-multipliers, Trans. Amer. math. Soc. 325, (1991), 829–854.
• M. McKennon, Quasi-multipliers, Trans. Amer. Math. Soc. 233 (1977), 105–123.
• A. Ülger, Arens regularity sometimes implies the RNP, Pacific. J. Math 143 (1990), 377–399.
• R. Vasudevan and S. Goel, Embedding of quasi-multipliers of a Banach algebra into its second dual, Math. Proc. Cambridge Philos. Soc. 95 (1984), 457–466.
• R. Vasudevan, S. Goel and S. Takahasi, The Arens product and quasi-multipliers, Yokohama. Math. J. 33, (1985), 49–66.
• S. Watanabe, A Banach algebra which is an ideal in the second dual space, Sci. Rep. Niigata Univ. Ser. A 11 (1974), 95–101.
• R. Yilmaz and K. Rowlands, On orthomorphisms, quasi-orthomorphisms and quasi-multipliers, J. Math. Anal. Appl. 313 (2006), 120–131.
• N. Young, The irregularity of multiplication in group algebras, Quart. J. Math. Oxford 24 (1973), 59–62.