We explore \( C^* \) -convexity as a framework for constructing operator norms on matrix algebras, generalizing classical convexity to non-commutative settings. We construct a one-parameter family of norms on \({\mathbb {M}}_n\) whose unit balls are \(C^*\) -convex and admit a characterization by \(2\times 2\) block matrix positivity. This family forms a log-convex interpolation path from the spectral norm ( \(\alpha = 0\) ) to the numerical radius ( \(\alpha = 1\) ), with dual L-norms connecting the trace norm to the dual numerical radius. We derive explicit dual norm formulas and demonstrate that these norms are computable via semidefinite programming, achieving exact linear matrix inequalities for dyadic \(\alpha \) . Furthermore, we prove complete contractivity of these norms under unital completely positive maps, enhancing their relevance to quantum channels.
