$C^{\ast }$-envelopes of universal free products and semicrossed products for multivariable dynamics

We show that for a class of operator algebras satisfying a natural condition the $C^{\ast }$-envelope of the universal free product of operator algebras $A_i$ is given by the free product of the $C^{\ast }$-envelopes of the $A_i$. We apply this theorem to, in special cases, the $C^{\ast }$-envelope of the semicrossed products for multivariable dynamics in terms of the single variable semicrossed products of Peters.