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

Abstract

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 $\alpha (fa)=\Phi (f)\alpha (a)$, $a\in A$, $f\in C_0(X)$ where $\Phi$ is an endomorphism of $C_0(X)$. Pictorially speaking, $\alpha$ is a mixture of a topological dynamical system $(X,\phi )$ dual to $(C_0(X),\Phi )$ and a continuous field of homomorphisms ${\alpha }_x$ 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 $(X,\phi ,\{{\alpha }_x{\}}_{x\in X};Y)$ where $Y$ is a closed subset of $X$.

This is a version to appear in J. Funct. Anal

References in corpus (7)

• A construction of $C^{\ast }$-algebras from $C^{\ast }$-correspondences
• Partial-isometric crossed products by semigroups of endomorphisms
• Ideal structure of $C^{\ast }$-algebras associated with $C^{\ast }$-correspondences
• Pure infiniteness and ideal structure of $C^{\ast }$-algebras associated to Fell bundles
• Crossed product of a C*-algebra by an endomorphism, coefficient algebras and transfer operators
• Filling families and strong pure infiniteness
• Ideal structure of crossed products by endomorphisms via reversible extensions of $C^{\ast }$-dynamical systems

Cited by in corpus (5)

• Aperiodicity: the almost extension property and uniqueness of pseudo-expectations
• Pure infiniteness and ideal structure of $C^{\ast }$-algebras associated to Fell bundles
• Classification of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras
• Topological freeness for $C^{\ast }$-correspondences
• Nijenhuis operators on homogeneous spaces related to $C^{\ast }$-algebras