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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 at 5:06

1 Answer 1

up vote 1 down vote accepted

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."

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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

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