Exercise 30, § 6

Let $A$ be a Banach algebra having an approximate unit. Denote by $\widetilde{A}$ the Banach algebra obtained from $A$ by adjunction of a unit element, denoted by 1.

a) Let $x\in A$. Then $x$ belongs to the closed left ideal generated by $x$. b) For every $\varepsilon >0$ and every finite family $(x_i)_{i\in I}$ of elements of A, there exists $e\in A$ such that $\|e\|\leqslant 1$ and $\|ex_i-x_i\|< \varepsilon$ for all $i\in I$.

c) Let $x\in A,\varepsilon >0$ and $0< \gamma \leqslant 1/8$. There exists $\eta >0$ such that for every $e\in A$ satisfying $\|e\|\leqslant 1$ and $\|ex-x\|< \eta$, then $\alpha = (1-\gamma )\cdot 1+\gamma e$ is invertible in $\widetilde{A}$ and $\|x-\alpha^{-1}x\|< \varepsilon$.

d) Let $x\in A,\varepsilon >0,\delta >0$ and $0< \gamma \leqslant 1/8$. There exists a sequence $(e_n)_{n\geqslant 1}$ in A having the following properties:

(i) We have $\|e_n\|<2$ for all $n\geqslant 1$ ;

(ii) The element

is invertible in $\widetilde{A}$ for all $n\geqslant 1$ and

(Assuming that $(e_1, . . . , e_n)$ have been defined, consider an arbitrary element $e_{n+1}$ of norm $<2$ ; observe that

where $e'_k= ((1-\gamma )\cdot 1 +e_{n+1})^{-1}e_k$, and determine $e_{n+1}$ satisfying the required conditions by exploiting the fact that the set of invertible elements of $\widetilde{A}$ is open, and that the mapping $a\mapsto a^{-1}$ is continuous.)

e) For every $x\in A$, there exist $y$ and $z$ in A such that $x=yz$ (“Cohen factorization theorem”). Let $\delta >0$. We may suppose that $z$ belongs to the closed left ideal generated by $x$ and that $\|y-z\|< \delta$. (Consider a sequence $(e_n)_{n\geqslant 1}$ as above and define $y_n$ as previously; the sequence $z_n=y_n^{-1}x$ converges to an element $z\in A$, and $z_n$ belongs to the closed left ideal generated by $x$; the sequence $(y_n)$ converges to $y\in A$ and $x=yz.$)