Exercise 29, § 6

a) Let $G$ be a locally compact topological group. The Banach algebra $L^1(G)$ has an approximate identity. (Use INT, VIII, § 2, n$^o7$, Lemma 4). b) Let A be a Banach algebra admitting an approximate identity. For every character $\chi \in \mathsf{X}(A)$, one has $\|\chi \|= 1$.

c) The Banach subalgebra of $L^1(\mathbf{Z})$ formed by the sequences $(x_n)$ such that $x_n= 0$ if $n <0$ has no approximate identity. (Construct a nonzero character of norm $<1$.)

d) Let G be the free group generated by two elements $a$ and $b$ and let $\mu$ be a Haar measure on G. Let I be the closed ideal of $L^1(G)$ formed by the $f\in L^1(G)$ such that $\mu(f) = 0$. Let $\xi =e^{2i\pi /3}$. Let $x=\varepsilon_e+\xi \varepsilon_a+\xi^{-1}\varepsilon_b\in L^1(G)$. For every $f\in L^1(G)$, one has $\|x*f-x\|\geqslant 1$. The Banach algebra I has no approximate identity.