Let $\cal A$ be a Banach algebra with a bounded approximate identity, when a closed two sided ideal on it has a bounded approximate identity?


**In particular**, let $\cal B$ be a Banach algebra with a bounded approximate identity and $T:\cal A\to \cal B$ be an algebra homomorphism which is norm decreasing and surjective. **When $\ker (T)$ has a bounded approximate identity?**


***Thank you so much!***



Bounded approximate identity and kernel of algebra homomorphism
**Let $\cal A$ be a Banach algebra with a bounded approximate identity, when a closed two sided ideal on it has a bounded approximate identity?**


This is a very nice problem that has been studied for special Banach algebras. I'm not sure if there is a "nice" characterization for a general Banach-algebra. Here is a list of results that I'm aware of for certain Banach algebras:


(i) This is always true for $C^\*$-algebras. Since any closed two-sided of a $C^\*$-algebra is a $C^\*$-algebra and so it has a bounded approximate identity. 


(ii) B. Forrest, E. Kaniuth and A.T. Lau has studied this problem for Fourier algebras in "Ideals with bounded approximate identities in Fourier algebras, J. Funct. Anal. 203 (2003), no. 1, 286–304." 


(iii) If $\cal A$ is an amenable Banach algebra and $\cal I$ is a closed ideal then $\cal I$ has a bounded approximate identity if and only if $\cal I$ is amenable if and only if $\cal I$ is weakly complemented. For definitions and a proof you can see theorem 2.3.7. "Volker Runde, Lectures on Amenability, Lecture Notes in Mathematics, 1774. Springer-Verlag, Berlin, 2002."


