Let $\mathcal{ S}$ be the semigroup $\mathcal{ S}=\sum^{ \oplus k}_{ i=1}\Sc{ S}_i$, where for each $i\in I$, $\mathcal{ S}_i$ is a countable subsemigroup of the additive semigroup $\B{ R}_+$ 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: $T_s^*T_t=T_tT_s^*$ 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 $C^*$-envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis).
Categories: math.OA