I'm reading A. Deitmar and S. Echterhoff's book "Principles of Harmonic Analysis", and have a confusion of the proof in the snapshot: Why do we need to prove $\|f\|\equiv\|f\eta\|$ in order to show $m$ is a continuous map? I think that the proof closes as soon as we have $|m(f)|\le\|f\chi\bar{\chi_0}\|$
Note that $L$ is an inclusion from $L^1(A)$ to $B(L^2(A))$ given by $\langle L(f)\phi,\psi\rangle=\int_A f(y)\langle L_y\phi,\psi\rangle dy$
