In trying to follow the proof of Proposition 4.11 in


M. C. White, [Injective modules for uniform algebras](http://www.mas.ncl.ac.uk/~nmcw/papers/ua21.pdf), Proc. London Math. Soc. 73 (1996) 155--184


there is a part which seems unclear.


Let $I$ be a left ideal in a unital Banach algebra $A$.
Assume $I$ is weakly complemented as a Banach $A$-module in $A$. That is, the s.e.s
$$
0\xleftarrow{} I^\*\xleftarrow{i^\*} A^\*\xleftarrow{\pi^\*} (A/I)^\*\xleftarrow{} 0
$$
splits in the category mod-$A$. Let $\sigma$ be a right inverse $A$-morphism of $\pi^\*$.


In Proposition 4.11 of the paper it is stated that $I$ has a bounded right approximate identity $(e\_\alpha)$ such that 
$$
\sup\_\alpha\Vert 1-e\_\alpha\Vert\leq C
$$ 
for $C=\Vert\sigma\Vert$. I understand the proof until the moment the author claims (see the end of the page 12 )



> 
> "...so $e\in I^{\perp\perp}$. As $I^{\perp\perp}$ is the weak-star closure of $I$ in $A ' '$, we may choose a net in $I$ which tends to $1-e$ in the $\sigma(A ' ', A ')$-topology and which is bounded by $\Vert 1- e\Vert$. This net is a weak bounded approximate identity..."
> 
> 
> 


(The author uses $X'$ for the dual of a Banach space $X$.)


As far as I can see, he constructs a net of the form $(1-e\_\alpha)\_{\alpha}$ such that


1) $\sup\_{\alpha}\Vert 1-e\_\alpha \Vert \leq \Vert 1-e\Vert$


2) the net $(1-e\_\alpha)\_{\alpha}$ converges to $1-e$ in the $\sigma(A ' ', A ')$ topology


3) $(e\_\alpha)\_{\alpha}$ is contained in $I$


4) $(e\_\alpha)\_{\alpha}$ is weak bounded approximate identity


Paragraphs 1) and 2) are just the Goldstine theorem. Paragraph 4) is a simple computation. The main problem is paragrpah 3), I can't show that $(e\_\alpha)\_{\alpha}$ is contained in $I$.


But even if we prove somehow that $(e\_\alpha)\subset I$. There is one more step in the proof 



> 
> "...a weak bounded approximate identity has a norm bounded approximate identity, with the same bound..."
> 
> 
> 


I agree that the statement is true, but if we look carefully it says that we would have genuine approximate identity $(f\_\beta)$ with $\sup\_{\beta}\Vert f\_\beta\Vert\leq\sup\_\alpha\Vert e\_\alpha\Vert$, though we want $\sup\_\beta\Vert 1-f\_\beta\Vert\leq \sup\_\alpha\Vert 1-e\_\alpha\Vert$


That are steps I do not understand.



Existence of certain bounded approximate identity
Thanks to the hints of Narutaka Ozawa and Yemon Choi I can answer this question by myself.


Recall that $I^{\perp\perp}$ is the weak${}^\*$ closure of $I$ in $A^{\*\*}$, so by Goldstine theorem we can choose a net $(e\_\nu'')\_{\nu\in N''}\subset I$ such that it weak${}^\*$ converges to $e$. Clearly $(1-e\_\nu'')\_{\nu\in N''}$ converges to $1-e$ in the same topology. By lemma 1.1 from [this paper](http://arxiv.org/abs/1405.5551) there exists a net in $\operatorname{conv}(1-e\_\nu'')\_{\nu\in N''}=1-\operatorname{conv}(e\_\nu'')\_{\nu\in N''}$ that weak${}^\*$ converges to $1-e$ with norm bound $\Vert 1-e\Vert$. Denote this net as $(1-e\_\nu')\_{\nu\in N'}$, then it is easy to check that $(e\_\nu')\_{\nu\in N'}$ weak${}^\*$ converges to $e$ and a weak right approximate identity for $I$. By proposition 33.2 in Approximate identities and factorization in Banach algebras by Doran R. S., Wichmann J. there is a net $(e\_\nu)\_{\nu\in N}\subset\operatorname{conv}(e\_\nu')\_{\nu\in N'}$ which is a right bounded approximate identity for $I$. For any $\nu\in N$ the vector $1-e\_\nu$ is in $\operatorname{conv}(1-e\_\nu')\_{\nu\in N'}$, then taking into account the norm bound on $(1-e\_\nu')\_{\nu\in N'}$ we get 
$$
\sup\_{\nu\in N}\Vert 1-e\_\nu\Vert
\leq\Vert 1-e\Vert
$$


