$C^*$-envelopes of universal free products and semicrossed products for multivariable dynamics Benton Duncan 47L3046L09$C^*$-envelopesfree productssemicrossed productsuniversal operator algebrasmultivariable dynamics We show that, for a class of operator algebras satisfying a natural condition, the $C^{*}$-envelope of the universal free product of operator algebras $A_i$ is given by the free product of the $C^{*}$-envelopes of the $A_i$. We apply this theorem to, in special cases, the $C^{*}$-envelope of the semicrossed products for multivariable dynamics in terms of the single variable semicrossed products of Peters. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3273 10.1512/iumj.2008.57.3273 en Indiana Univ. Math. J. 57 (2008) 1781 - 1788 state-of-the-art mathematics http://iumj.org/access/