Is the Lorentz space $L_{W,1}(0,1)$ isomorphic to $L_1(0,1)$?

Question

Let $W$ be a positive non-increasing continuous function on $(0,1]$ so that $\lim_{t \rightarrow 0} W(t)=\infty$, $W(1)=1$ and $\int_0^1 W(t) dt =1$. For $1 \leq p <\infty$, the Lorentz function space $L_{W,p}(0,1)$ is the space of all measurable functions $f$ on $(0,1)$ such that $$ \|f\|_{W,p} = \left(\int_0^1 f^*(t)^p \; W(t) \; dt \right)^{1/p} < \infty \; , $$ where $f^*$ is the decreasing rearrangement of $|f|$.

The space $L_{W,1} (0,1)$ is a r.i. function space on $(0,1)$ [Lindenstrauss and Tzafriri, Classical Banach Spaces II, pages 120-121] and $L_{W,1} (0,1)\subsetneq L_1(0,1)$.

Questions. Are the spaces $L_{W,1} (0,1)$ and $L_1 (0,1)$ isomorphic as Banach spaces?

Are they isomorphic as Banach lattices?

Answer 1

The answer, as expected, is no. There must be several ways to prove it. Corollary 2.e.8 in Lindenstrauss-Tzafriri's book (vol. 2) provides one.