Absence of local unconditional structure in spaces of smooth functions on two-dimensional torus (Q2146194)

It is well known that the space \(C^k(\mathbb{T})\) of \(k\) times continuously differentiable functions on the unit circle is isomorphic to \(C(\mathbb{T})\). A quite different situation happens for the space \(C^k(\mathbb{T}^2)\). Under a certain natural condition, it is well known that \(C^k(\mathbb{T}^2)\) is not isomorphic to a complemented subspace of any \(C(S)\) space. In this paper something stronger is proved: \(C^k(\mathbb{T}^2)\) does not have local unconditional structure. Under suitable conditions, it is also proved that \(C^k(\mathbb{T}^2)\) is not isomorphic to any quotient space of any \(C(S)\).