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 f459e5e9a7" title="Click to view the MathML source">(X,φ,{αx}x∈X;Y) where Y is a closed subset of X.
