Failure of weak-star convergence to commute with squaring

In $L^{\infty }(\mathbb{R})=(L^1(\mathbb{R}){)}^{\ast }$, the sequence $f_n(x)=\sin (nx)$ converges weak-star to zero by the Riemann-Lebesgue lemma, while $f_n^2$ converges weak-star to the constant $1/2$. Nonlinear pointwise operations therefore need not preserve weak-star limits.