Tomiyama-type maps with a diagonal perturbation

We investigate a two-parameter family of linear maps on matrix algebras, constructed as diagonal perturbations of classical Tomiyama maps. Employing the Choi matrix method alongside block-positivity techniques, we derive explicit necessary and sufficient conditions for positivity, complete positivity, and k-positivity across arbitrary dimensions. These conditions provide a transparent geometric characterization of the positivity regions within the parameter space.