In uniform circular motion, a particle moves with a constant speed along a circular path of constant radius. The angular velocity vector $\vec{\omega}$ points along the axis of rotation (perpendicular to the plane of the circle). Since the speed and the plane of motion are fixed, both the magnitude and direction of the angular velocity remain constant. The angular momentum of the particle about the center is given by $\vec{L} = I\vec{\omega}$ (or $\vec{L} = m\vec{r} \times \vec{v}$). Since the moment of inertia $I$ (or mass $m$ and radius $r$) and the angular velocity $\vec{\omega}$ are constant, the angular momentum $\vec{L}$ also stays constant.