  We give the Fock representation of a noncommutative $\mathbb{C}P^N$ and gauge
theories on it. The Fock representation is constructed based on star products
given by deformation quantization with separation of variables and operators
which act on states in the Fock space are explicitly described by functions of
inhomogeneous coordinates on ${\mathbb C}P^N$. Using the Fock representation,
we are able to discuss the positivity of Yang-Mills type actions and the
minimal action principle. Other types of actions including the Chern-Simons
term are also investigated. BPS-like equations on noncommutative
$\mathbb{C}P^1$ and $\mathbb{C}P^2$ are derived from these actions. There are
analogies between BPS-like equations on $\mathbb{C}P^1$ and monopole equations
on ${\mathbb R}^3$, and BPS-like equations on $\mathbb{C}P^2$ and instanton
equations on ${\mathbb R}^8$. We discuss solutions of these BPS-like equations.
