  In this paper, we investigate Lefschetz properties of Veronese subalgebras.
We show that, for a sufficiently large $r$, the $r$\textsuperscript{th}
Veronese subalgebra of a Cohen-Macaulay standard graded $K$-algebra has
properties similar to the weak and strong Lefschetz properties, which we call
the `almost weak' and `almost strong' Lefschetz properties. By using this
result, we obtain new results on $h$- and $g$-polynomials of Veronese
subalgebras.
