The $C$-convexity of Banach spaces with unconditional bases

A Banach space is called $C$-convex if the space $c_0$ cannot be represented finitely in it. Necessary and sufficient conditions for the $C$-convexity of a space with an unconditional basis and of the product of a space $Y$ with respect to the unconditional basis of a space $X$ are obtained. These conditions are rendered concrete for two classes of spaces: The Orlich space of sequences is $C$-convex if and only if its normalizing function satisfies the $\Delta_2$-condition; the Lorentz space of sequences is $C$-convex if and only if its normalizing sequence satisfies the condition $\varliminf\limits_{n\to\infty}\sum_{i=1}^{2n}c_i\bigl/\sum_{i=1}^nc_i=1$.