Crossed products by endomorphisms of C0(X)-algebras

In the first part of the paper, we develop a theory of crossed products of a C⁎-algebra A by an arbitrary (not necessarily extendible) endomorphism α:A→A. We consider relative crossed products C⁎(A,α;J) where J is an ideal in A, and describe up to Morita-Rieffel equivalence all gauge-invariant ideals in C⁎(A,α;J) and give six term exact sequences determining their K-theory. We also obtain certain criteria implying that all ideals in C⁎(A,α;J) are gauge-invariant, and that C⁎(A,α;J) is purely infinite.In the second part, we consider a situation where A is a C0(X)-algebra and α is such that α(fa)=Φ(f)α(a), a∈A, f∈C0(X) where Φ is an endomorphism of C0(X). Pictorially speaking, α is a mixture of a topological dynamical system (X,φ) dual to (C0(X),Φ) and a continuous field of homomorphisms αx between the fibers A(x), x∈X, of the corresponding C⁎-bundle.For systems described above, we establish efficient conditions for the uniqueness property, gauge-invariance of all ideals, and pure infiniteness of C⁎(A,α;J). We apply these results to the case when X=Prim(A) is a Hausdorff space. In particular, if the associated C⁎-bundle is trivial, we obtain formulas for K-groups of all ideals in C⁎(A,α;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,φ,{αx}x∈X;Y) where Y is a closed subset of X.