Absence of local unconditional structure in spaces of smooth functions on two-dimensional torus

Consider a finite collection $\{T_1,\dots ,T_J\}$ of differential operators with constant coefficients on ${\mathbb{T}}^2$ and the space of smooth functions generated by this collection, namely, the space of functions $f$ such that $T_j f \in C($\mathbb{T}$^2)$. We prove that under a certain natural condition this space is not isomorphic to a quotient of a $C(S)$-space and does not have a local unconditional structure. This fact generalizes the previously known result that such spaces are not isomorphic to a complemented subspace of $C(S)$.