Examples of semigroups of contractive Fourier multipliers but not positive?

Question

Can you show me a concrete an example of semigroup $(T_t)_{t\geq 0}$ of Fourier multipliers such that each operator $T_t$ induces a contractive Fourier multiplier $T_t\colon L^p(\mathbb{T}) \to L^p(\mathbb{T})$ for any $1 \leq p \leq \infty$ and such that $(T_t)_{t\geq 0}$ is a $C_0$-semigroup of self-adjoint contractive operators on $L^2(\mathbb{T})$ but such that $(T_t)_{t\geq 0}$ is not a semigroup of positive operators (in the sense that it maps positive functions into positive functions)?