Ideal structure of crossed products by endomorphisms via reversible extensions of $C^*$dynamical systems
Abstract
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 nonunital 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 noninjective $\alpha$'s. We provide descriptions of the lattices of ideals in $C^*(A,\alpha,J)$ consisting of gaugeinvariant 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 $\alpha$.
 Publication:

arXiv eprints
 Pub Date:
 April 2014
 DOI:
 10.48550/arXiv.1404.4928
 arXiv:
 arXiv:1404.4928
 Bibcode:
 2014arXiv1404.4928K
 Keywords:

 Mathematics  Operator Algebras;
 46L05
 EPrint:
 35 pages, Appendix on $C^*(A,\alpha,J)$ viewed as relative CuntzPimsner algebras is added, this version is accepted to Internat. J. Math