Lefschetz properties and the Veronese construction

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.