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Ideal structure of crossed products by endomorphisms via reversible extensions of $C^*$-dynamical systems

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 $(\kerα)^\bot$, and if $J=(\kerα)^\bot$ 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 $α$.