Crossed products by endomorphisms of $C_0(X)$-algebras

In the first part of the paper, we develop a theory of crossed products of a $C^{\ast }$-algebra $A$ by an arbitrary (not necessarily extendible) endomorphism $\alpha :A\to A$. We consider relative crossed products $C^{\ast }(A,\alpha ;J)$ where $J$ is an ideal in $A$, and describe up to Morita-Rieffel equivalence all gauge invariant ideals in $C^{\ast }(A,\alpha ;J)$ and give six term exact sequences determining their $K$-theory. We also obtain certain criteria implying that all ideals in $C^{\ast }(A,\alpha ;J)$ are gauge invariant, and that $C^{\ast }(A,\alpha ;J)$ is purely infinite. In the second part, we consider a situation where $A$ is a $C_0(X)$-algebra and $\alpha$ is such that $\alpha (fa)=\mathrm{\Phi }(f)\alpha (a)$, $a\in A$, $f\in C_0(X)$where$\Phi$isanendomorphismof$C_0(X)$\mathrm{.}Pictoriallyspeaking,$\alpha$isamixtureofatopologicaldynamicalsystem$(X,\varphi)$dualto$(C_0(X),\Phi)$andacontinuousfieldofhomomorphisms$\alpha_x$ between the fibers $A(x)$, $x\in X$, of the corresponding $C^{\ast }$-bundle. For systems described above, we establish efficient conditions for the uniqueness property, gauge-invariance of all ideals, and pure infiniteness of $C^{\ast }(A,\alpha ;J)$. We apply these results to the case when $X=$Prim$(A)$ is a Hausdorff space. In particular, if the associated $C^{\ast }$-bundle is trivial, we obtain formulas for $K$-groups of all ideals in $C^{\ast }(A,\alpha ;J)$. In this way, we constitute a large class of crossed products whose ideal structure and $K$-theory is completely described in terms of $(X,\varphi,{\alpha_{x}}_{x\in X};Y)$where$Y$isaclosedsubsetof$X$.