Given a completely bounded map $u:Z\to M$ from an operator space $Z$ into a von Neumann algebra (or merely a unital dual algebra) $M$, we define $u$ to be $C$-semidiscrete if for any operator algebra $A$, the tensor operator $I_A\otimes u$ is bounded from $A{\otimes }_{\min }Z$ into $A{\otimes }_{\mathrm{nor}}M$, with norm less than $C$. We investigate this property and characterize it by suitable approximation properties, thus generalizing the Choi-Effros characterization of semidiscrete von Neumann algebras. Our work is an extension of some recent work of Pisier on an analogous...