Let $(\Omega, \Sigma, \mu)$ be a $\sigma$-finite measure space, $1 \leq p<\infty$, and $(X, |\cdot|)$ be a Banach space. Assume $\varphi$ is an isometric automorphism of $L_p := L_p (\Omega, \mu, E)$. I would like to ask if $$ [f \in L_p \text{ and } B \in \Sigma] \implies\varphi (f 1_B) = \varphi(f) 1_B. $$
Informally, does $\varphi$ preserve the support of $f$. If yes, then my second question in this thread has a positive answer.
Could you elaborate on my question? Thank you so much!
Consider $X=\mathbb{R}$ with Lebesgue measure, and for some fixed $x\in \mathbb{R}$ set $\varphi(f)(t):= f(x+t)$. Then $\varphi$ induces an isometric isomorphism on $L^p(\mathbb{R})$ for any $1\leq p\leq \infty$. However, the desired property $\varphi(f1_B)= \varphi(f)1_B$ is not true for all $f$ and $B$.