$C^*$-envelopes of universal free products and semicrossed products for multivariable dynamics

TL;DR

This paper establishes that for certain operator algebras, their $C^*$-envelope of a universal free product equals the free product of their individual $C^*$-envelopes, with applications to multivariable dynamics.

Contribution

It proves a general theorem relating $C^*$-envelopes of free products of operator algebras to their individual $C^*$-envelopes, and applies it to multivariable dynamical systems.

Findings

$C^*$-envelope of free product equals free product of $C^*$-envelopes under natural conditions

Application to semicrossed products for multivariable dynamics

Explicit description of $C^*$-envelopes in special cases

Abstract

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.