Finite-codimensional Chebyshev subspaces in the complex space $C(Q)$

We consider finite-codimensional Chebyshev subspaces in the complex space $C(Q)$, where $Q$ is a compact Hausdorff space, and prove analogs of some theorems established earlier for the real case by Garkavi and Brown (in particular, we characterize such subspaces). It is shown that if the real space $C(Q)$ contains finite-codimensional Chebyshev subspaces, then the same is true of the complex space $C(Q)$ (with the same $Q$).