Non-Abelian factors for actions of Z and other non-C*-simple groups

Let Gamma be a countable group and (X, Gamma) a compact topological dynamical system. We study the question of the existence of an intermediate C*-subalgebra A C-r*(Gamma) < A < C(X) (sic)(r) Gamma, which is not of the form A = C(Y) (sic)(r) Gamma, corresponding to a factor map (X, Gamma) -> (Y, Gamma). Here C-r*(Gamma) is the reduced C*-algebra of Gamma and C(X) (sic)(r) Gamma is the reduced C*-crossed-pro duct of (X, Gamma). Our main results are: (1) For Gamma which is not C*-simple, when (X, Gamma) admits a Gamma-invariant probability measure, then such a sub-algebra always exists. (2) For Gamma = Z and (X, Gamma) an irrational rotation of the circle X = R/Z, we give a full description of all these non-crossed-product subalgebras. (c) 2024 Elsevier Inc. All rights reserved.