Given an inverse semigroup \(S\) acting (by partial homomorphisms) on a topological space \(X\), we can construct the transformation groupoid \(X\rtimes S\). Then, the full crossed product \(C^*\)-algebra \(C_0(X)\rtimes S\) turns out to be canonically isomorphic to the full groupoid \(C^*\)-algebra \(C^*(X\rtimes S)\), and the same result holds for the reduced versions of both algebras.
When \(S\) acts on a locally compact groupoid \(G\) by partial equivalences (see [A. Buss and R. Meyer, Rocky Mt. J. Math. 47, No. 1, 53–159 (2017; Zbl 1404.46058)]), we can construct a transformation groupoid \(G\rtimes S\), inducing an action of \(S\) on the full groupoid \(C^*\)-algebra \(C^*(G)\) by Hilbert bimodules. Actions of \(S\) on \(C^*\)-algebras by Hilbert bimodules are equivalent to saturated Fell bundles \((A_t)_{t\in S}\) over \(S\), the unit fiber \(A_1:=A\) being the \(C^*\)-algebra on which the action takes place. Then, the full section \(C^*\)-algebra \(C^*((A_t)_{t\in S})\) of the Fell bundle plays the role of the full crossed product for the action, and it is denoted by \(A\rtimes S\). In the particular case of the action of \(S\) on a locally compact groupoid \(G\), the full section \(C^*\)-algebra describing the action of \(S\) on \(C^*(G)\) is identified with \(C^*(G\rtimes S)\) (see [loc. cit.]), so that we have an isomorphism \[ C^*(G)\rtimes S\cong C^*(G\rtimes S). \eqno(\ast) \]
In the paper under review, the authors deal with the problem of determining whether there exists a version of \((\ast)\) for the reduced crossed product \(C^*\)-algebra versions of the above \(C^*\)-algebras. The definition of the basic ingredient for this job – the reduced algebra \(A\rtimes_r S\) – is given in [R. Exel, New York J. Math. 17, 331–382 (2011; Zbl 1228.46061)], in analogy with the construction of the reduced crossed product for an action of a group or groupoid; this is done by using induced representations of \(A\) on \(A\rtimes S\). But when we move to the reduced crossed product on inverse semigroups, the problem is that the “transformation groupoid” need not be Hausdorff, and then the target algebra of the conditional expectation from \(A\rtimes S\) to \(A\) takes values in a larger algebra than \(A\). The authors deal with this problem by defining a weak conditional expectation \(E: A\rtimes S\rightarrow A''\), which induces a \(C^*\)-correspondence from \(A\rtimes S\) to \(A''\). They use such an expectation to show that any representation of \(A\) extends (in a unique way) to a normal representation of \(A''\). Thus, by tensoring each extended representation with the \(C^*\)-correspondence, we obtain induced representations of \(A\rtimes S\). Hence, we can define \(A\rtimes _r S\) to be the quotient of \(A\rtimes S\) defined by the \(C^*\)-seminorm coming from these induced representations.
The authors prove that the new definition of \(A\rtimes_r S\) obtained with the previous construction coincides with that given in [loc. cit.]. In the particular case of the action of \(S\) on a locally compact groupoid, they obtain an onto map \[ C_r^*(G)\rtimes_r S\rightarrow C_r^*(G\rtimes S) \] that need not be faithful, and they prove faithfulness of that map when either \(G\) is a closed subgroupoid of \(G\rtimes S\), or \(G\) is “inner exact” in the sense of C. Anantharaman-Delaroche [“Some remarks about the weak containment property for groupoids and semigroups”, arXiv:1604.01724].