Nonself-adjoint semicrossed products by abelian semigroups

Let $\mathcal{S}$ be the semigroup $$\mathcal{S}$=\sum^{\oplus k}_{i=1}\Sc{S}_i$,whereforeach$i\in I$,$$\mathcal{S}$_i$ is a countable subsemigroup of the additive semigroup $\text{B}{R}_{+}$ containing 0. We consider representations of $\mathcal{S}$ as contractions $\{T_s{\}}_{s\in \mathcal{S}}$ on a Hilbert space with the Nica-covariance property: $T_s^{\ast }{T}_t=T_t{T}_s^{\ast }$ whenever $t\wedge s=0$. We show that all such representations have a unique minimal isometric Nica-covariant dilation. This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant representations of the action of $\mathcal{S}$ on an operator algebra $\mathcal{A}$ by completely contractive endomorphisms. We conclude by calculating the $C^{\ast }$-envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis).