Simplicity and tracial weights on non-unital reduced crossed products (Q6543115)

The reduced \(C^*\)-algebra \(C^*_r (G)\) of a discrete group \(G\) is the norm closure of the algebra of operators on \(\ell^2(G)\) generated by the left regular representation \(\lambda_G\) of \(G\). The group \(G\) is said to be \(C^*\)-simple if \(C^*_r (G)\) is simple.\N\NLet \(G\) be a discrete group, \(H\) be a Hilbert space and \(A\) be a unital \(C^*\)-algebra endowed with an action of \(G\) by automorphisms. Then the reduced crossed product \(C^*\)-algebra \(A\rtimes_r G\) is a closed subalgebra of \(B(\ell^2(G,H))\) which is defined uniquely, up to isomorphism, through a \(*\)-representation \(\pi:A\rightarrow B(H)\). It is proved in [\textit{E. Breuillard} et al., Publ. Math. Inst. Hautes Etud. Sci. 126, 35--71 (2017; Zbl 1391.46071)] that if \(G\) is a discrete \(C^*\)-simple group and \(A\) does not have any proper \(G\)-invariant ideal, then \(A\rtimes_r G\) is simple.\N\NIn the present paper, replacing the role of the unit element of \(A\) by a completely full element, the author proves for a \(C^*\)-simple group \(G\) that if $A$ is a \(C^*\)-algebra with a completely full element and \((\alpha, u): G\curvearrowright A\) is a twisted action then the reduced twisted crossed product \(C^*\)-algebra \(A\rtimes _{r, \alpha, u} G\) is simple if and only if \(A\) has no proper \(G\)-invariant ideal.\N\NHe also proves that the uniqueness of proper tracial weights (up to scalar multiple) is stable under taking the reduced twisted crossed product \(C^*\)-algebra of a trace preserving twisted action over a discrete group with trivial amenable radical.\N\NThere are applications concerning locally compact groups. For instance, it is proved for a locally compact group \(G\) with an open normal subgroup \(N\) that, if \(N\) and \(G/N\) are \(C^*\)-simple, then so is \(G\).