Why is the output of the LPF half that of the real/imaginary part of the complex envelope? The passband signal $\ u_p(t) =
I_{[−1,1]}(t) \cos 400πt $ is passed through an LPF as $\ u_p(t) \cos(401\pi t).$ The question is to find the output. The example does the following:

If we let $\ u_1 = u_{c1} +ju_{s1}  $ denote the complex envelope with respect to the reference $\ e^{j401t} $,  the output of the LPF is $\ u_{c1}/2.$ Why is it half?

The passband signal can be rewritten as,
$\ u_p(t) = \Re(I_{[−1,1]}(t)e^{−jt}e^{j401t})$ to find $\ u_{c1}$
but how is this justified if we want to find the output of $\ u_p(t) \cos(401\pi t) $ and not just $\ u_p $ that is just rearranged.  Shouldn't the translation induced by the cosine be taken into account. I'd appreciate any help.
The idea of this exercise is to show what happens if you have a frequency offset at the receiver, i.e. if the actual carrier frequency is different from what the receiver thinks it should be.

You can write the passband signal as

$$u_p(t)=\text{Re}\left(I_{[-1,1]}e^{j400\pi t}\right)\tag{1}$$

Since you're demodulating with a frequency of $\omega_0=401\pi$, it is advantageous to rewrite the passband signal with that reference frequency:

$$u_p(t)=\text{Re}\left(I_{[-1,1]}e^{-j\pi t}e^{j401\pi t}\right)\tag{2}$$

From $(2)$ you can read off the new complex envelop with respect to the actual demodulation frequency $\omega_0=401\pi$:

$$u(t)=u_c(t)+ju_s(t)=I_{[-1,1]}e^{-j\pi t}=I_{[-1,1]}\cos(\pi t)-jI_{[-1,1]}\sin(\pi t)$$

Demodulation with $\cos(401\pi t)$ will give you the I component $u_c(t)$. The factor $1/2$ simply occurs because by multiplying with a cosine you get $u_c(t)\cos^2(\omega_0t)=u_c(t)(\frac12+\frac12\cos(2\omega_0t))$, and the LPF removes the component at $2\omega_0$, which leaves you with $\frac12 u_c(t)$.