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

We consider an extendible endomorphism $\alpha$ of a $C^*$-algebra $A$. We associate to it a canonical $C^*$-dynamical system $(B,\beta)$ that extends $(A,\alpha)$ and is `reversible' in the sense that the endomorphism $\beta$ admits a unique regular transfer operator $\beta_*$. The theory for $(B,\beta)$ is analogous to the theory of classic crossed products by automorphisms, and the key idea is to describe the counterparts of classic notions for $(B,\beta)$ in terms of the initial system $(A,\alpha)$. We apply this idea to study the ideal structure of a non-unital version of the crossed product $C^*(A,\alpha,J)$ introduced recently by the author and A. V. Lebedev. This crossed product depends on the choice of an ideal $J$ in $(\ker\alpha)^\bot$, and if $J=(\ker\alpha)^\bot$ it is a modification of Stacey's crossed product that works well with non-injective $\alpha$'s. We provide descriptions of the lattices of ideals in $C^*(A,\alpha,J)$ 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 $C^*(A,\alpha,J)$. In particular, we obtain simplicity criteria that besides minimality of the action require either outerness of powers of $\alpha$ or pointwise quasinilpotence of $\alpha$.