Exercise 3, § 3

Let $x \in X$. Show that the centralizer of $x$ in $A(X)$ is the subalgebra generated by $x$. Deduce that the only elements of $L(X)$ which commute with $x$ are the multiples of $x$. In particular, the centre of $L(X)$ is $\{0\}$ if $\operatorname{Card}(X) \geq 2$ and the centre of $L(X)/\left( \sum_{n > p} L^n(X) \right)$ reduces to the canonical image of $L^p(X)$.