We consider an extendible endomorphism α of a C*-algebra A. We associate to it a canonical C*-dynamical system (B, β) that extends (A, α) and is "reversible" in the sense that the endomorphism β admits a unique regular transfer operator β⁎. The theory for (B, β) is analogous to the theory of classic crossed products by automorphisms, and the key idea is to describe the counterparts of classic notions for (B, β) in terms of the initial system (A, α). We apply this idea to study the ideal structure of a non-unital version of the crossed product C*(A, α, J) introduced recently by the author and A. V. Lebedev. This crossed product depends on the choice of an ideal J in (ker α)⊥, and if J = ( ker α)⊥ it is a modification of Stacey's crossed product that works well with non-injective α's. We provide descriptions of the lattices of ideals in C*(A, α, J) consisting of gauge-invariant ideals and ideals generated by their intersection with A. We investigate conditions under which these lattices coincide with the set of all ideals in C*(A, α, J). In particular, we obtain simplicity criteria that besides minimality of the action require either outerness of powers of α or pointwise quasinilpotence of α.
The ideal structure of the Hecke $C^*$ -algebra of Bost and Connes
