About isometry in $l_ \infty$

Question

Let $1 \leq p<\infty$.

Find an isometry $j: l_{\infty} \rightarrow L\left(L_{p}[0,1]\right)$ .

(Hint :first embed $ℓ_∞$ into $L^∞[0,1]$ and let this act on $L^p[0,1]$ as multiplication operators.)

I can't use Hint for solve this problem but I think we can Use that fact that for $p>1$ the Haar basis $(h_n)$ of $L_p$ is 1-unconditional.

Answer 1

Not sure if this is what you are looking for:

Let $t_k$ be a strictly increasing sequence such that $t_k \to 1$ and let $I_k= [t_{k-1},t_k)$.

Suppose $x \in l_\infty$ and $f \in L^p[0,1]$.

Define $((j(x)(f))(t) = \sum_k x_k f(t) 1_{I_k}(t)$.

Then $\|j(x)\| = \|x\|$.