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\(C^*\)-envelopes of tensor algebras for multivariable dynamics. (English) Zbl 1219.47143

Proc. Edinb. Math. Soc., II. Ser. 53, No. 2, 333-351 (2010); corrigendum ibid. 54, No. 3, 643-644 (2011).
A multivariable dynamical system is a pair \((X, \sigma)\), where \(X\) is a locally compact Hausdorff space and \(\sigma=(\sigma_1,\dots, \sigma_n)\) is a family of proper continuous maps from \(X\) into itself. The tensor algebra \(\mathcal{A}(X, \sigma)\) of the system \((X, \sigma)\) was introduced and studied in [K. R. Davidson and E. G. Katsoulis, “Operator algebras for multivariable dynamics”, Mem. Am. Math. Soc. 982 (2011; Zbl 1236.47001)]. It is the universal operator algebra generated by \(C_0(X)\) and \(n\) isometries \(\mathfrak{s}_1,\dots, \mathfrak{s}_n\) with pairwise orthogonal ranges satisfying the covariance relations
\[ f\mathfrak{s}_i=\mathfrak{s}_i(f\circ \sigma_i) \]
for all \(f \in C_0(x)\) and \(1\leq i\leq n\).
It has been shown in the aforementioned paper that the \(C^*\)-envelope of \(\mathcal{A}(X, \sigma)\) is a Cuntz-Pimsner algebra. In the paper under review, a new description of the \(C^*\)-envelope of \(\mathcal{A}(X, \sigma)\) is given.
The authors show that in the surjective case (that is, if \(X=\cup_{i=1}^n\sigma_i(X)\)), the \(C^*\)-envelope is a crossed product by an endomorphism and also a groupoid \(C^*\)-algebra. In the non-surjective case, they show that it is a full corner of such an algebra.
A condition for the simplicity of the \(C^*\)-envelope of \(\mathcal{A}(X, \sigma)\) is also obtained. It is shown that, if the space \(X\) is compact, the \(C^*\)-envelope is simple if and only if the system is minimal.

MSC:

47L55 Representations of (nonselfadjoint) operator algebras
47L40 Limit algebras, subalgebras of \(C^*\)-algebras
46L05 General theory of \(C^*\)-algebras
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37B99 Topological dynamics
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