A note on crossed products
An action of a locally compact group \(G\) on a C*-algebra \(\mathfrak{A}\) is concretely represented on a Hilbert space \(\mathfrak{H}\) by covariant representations of \(G\) and \(\mathfrak{A}\), in the sense that the group action is implemented by a unitary group representation on the image of \(\mathfrak{A}\). Covariant representations on \(L^2(G, \mathfrak{H})\) can be induced from representations of \(\mathfrak{A}\) on \(\mathfrak{H}\) by considering an orbit representation of \(\mathfrak{A}\) and the left regular representation of \(G\). There are two important C*-algebras related to the C*-dynamics. On one hand the full C*-crossed product \(\mathfrak{A} \rtimes G\) is the universal C*-algebra with respect to the covariant representations of the action. On the other hand the reduced C*-crossed product \(\mathfrak{A} \rtimes_r G\) is given by the covariant representation on \(L^2(G, \mathfrak{A})\). The paper under review gives an easy approach to two key results in the theory. First that the reduced C*-crossed product is independent of the faithful representation of \(\mathfrak{A}\), in the sense that it is canonically \(*\)-isomorphic to an induced representation of any faithful representation of \(\mathfrak{A}\). Secondly that \(\mathfrak{A} \rtimes G = \mathfrak{A} \rtimes_r G\) when \(G\) is amenable. Both proofs are well-presented, while the paper assumes only basic knowledge of the C*-algebras theory.
scientific article; zbMATH DE number 7363315
scientific article