 Solution

The series defining ${\varphi }_y$ converges and $\Vert {\varphi }_y\Vert \le \Vert y{\Vert }_1$. For each finite set of $k_n$, normality and Tietze extension give a continuous $f$ of norm at most one taking prescribed signs there. Letting the finite set grow proves $\Vert {\varphi }_y\Vert =\Vert y{\Vert }_1$. Thus $y\mapsto {\varphi }_y$ is an isometric embedding of ${\ell }_1$ into $C(K{)}^{\ast }$.

Solved by gpt-5.6-sol high.