C-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∗-envelope of the universal free product of operator algebras Ai is given by the free product of the C ∗-envelopes of the Ai. 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. An important invariant for non-selfdajoint operator algebras is the C-envelope. This is a minimal C-algebra containing the operator algebra in a completely isometric manner. The utility of such a C-algebra was laid out in [1] and [2] and its existence was proved by Hamana in [10] using injective envelopes. Unfortunately as with most universal objects identifying the requisite C-algebra is often difficult, and is often carried out on a case by case basis. There have been several important classes of operator algebras which have received intensive study: the semigroupoid algebras of [13] as special cases of the tensor algebras over Ccorrespondences of [15], and the semicrossed products of [18]. Both algebras try to encode some sort of dynamics on an underlying C-algebra. In both of these cases however the dynamics are constrained significantly by either avoiding interactions between morphisms as in the first algebras, or by constraining the dynamics to a single variable in the case of the second algebra. Recently a new attempt at multivariable dynamics has been initiated in [8]. There, two possible universal objects related to a multivariable system of dynamics are defined and studied. In particular they let τ = (τ1, τ2, · · · , τn) be a tuple of continuous self maps of a locally compact Hausdorff space. They then study universal operator algebras which encode these dynamics. To do this they look at the universal operator algebra generated by C0(X) and contractions Si encoding the dynamics of τi via a covariance relation Sif(x) = f(τi(x))Si for all f ∈ C0(X). There are two universal operator algebras they study the first they call the semicrossed product, and the second the tensor algebra. The only difference being an additional constraint on the tensor algebra, that the contractions Si are a family of row contractions. This additional constraint allows a cleaner analysis and more concrete theorems. In particular, building on the groundbreaking work in [12] and [15], the C-envelopes of the tensor algebras are identified in Theorem 5.1 of [8]. The semicrossed products however are less tractable since they lack this constraint. While some results can be proved in these examples they are often less satisfying. In particular a good understanding of the C-envelopes is lacking. In this paper we begin to address this issue by recognizing the semicrossed products 2000 Mathematics Subject Classification. 47L30, 46L09.