# 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?

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!

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Are you interested in negative results? – Bill Johnson Nov 6 '13 at 0:26
No, I want to ker(T) has a bounded approximate identity. In particular, when A is weakly amenable and has a quasi-central bounded approximate identity, I want to see ker(T) has a bounded approximate identity! – Albert harold Nov 6 '13 at 5:12
If $\cal A$ is amenable then it has a bounded approximate identity. Out of curiosity, is there any connection between weak-amenability and quasi-central bounded approximate identity? – Alvin Nov 6 '13 at 15:14
Every ideal with quasi-central bounded approximate identity in weakly amenable Banach algebras, has trace extension property, so it is weakly amenable! – Albert harold Nov 7 '13 at 5:40
In my opinion the question is far too broad. "Most" ideals in "most" unital Banach algebras fail to have approximate identities, unless one is looking at very special classes such as Cstar-algebras – Yemon Choi Feb 27 '14 at 5:06

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?
(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.
(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."
Thanks ALVIN. $\cal A$ is not amenable, but $\cal A$ is weakly amenable and has a quasi-central bounded approximate identity. What do you think about it? – Albert harold Nov 5 '13 at 15:49