On a question related to bounded approximate identities of ideals in Banach algebras.

In this paper we give an example of a Banach algebra A and a closed ideal I of A such that the multiplier algebra of I is equal to A but I does not have any bounded approximate identity. In the case that I has an approximate identity, we give a necessary condition on I for which A = M (I) , where M (I) denotes the multiplier algebra of I. Finally, as a corollary of our results, we show that the Fourier algebra of an amenable group is strictly dense in the Fourier–Stieltjes algebra. [ABSTRACT FROM AUTHOR]