Crossed products by endomorphisms of C<sub>0</sub>(X)-algebras

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Crossed products by endomorphisms of C₀(X)-algebras

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作者：B.K. Kwaśniewski^a ; ^b ; ^{bartoszk@math.uwb.edu.pl" class="auth_mail" title="E-mail the corresponding author}
关键词：primary ; 46L05 ; secondary ; 46L35
刊名：Journal of Functional Analysis
出版年：2016
出版时间：15 March 2016
年：2016
卷：270
期：6
页码：2268-2335
全文大小：1065 K

文摘

In the first part of the paper, we develop a theory of crossed products of a C^⁎

overflow="scroll"> C^{⁎}

-algebra A by an arbitrary (not necessarily extendible) endomorphism α:A→A

overflow="scroll"> α : A \to A

. We consider relative crossed products C^⁎(A,α;J)

overflow="scroll"> 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)

overflow="scroll"> 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)

overflow="scroll"> C^{⁎} (A, α; J)

are gauge-invariant, and that C^⁎(A,α;J)

overflow="scroll"> C^{⁎} (A, α; J)

is purely infinite.

In the second part, we consider a situation where A is a C₀(X) $overflow="scroll"> C_{0} (X)$ -algebra and α is such that α(fa)=Φ(f)α(a) $overflow="scroll"> α (f a) = Φ (f) α (a)$ , a∈A $overflow="scroll"> a \in A$ , f∈C₀(X) $overflow="scroll"> f \in C_{0} (X)$ where Φ is an endomorphism of C₀(X) $overflow="scroll"> C_{0} (X)$ . Pictorially speaking, α is a mixture of a topological dynamical system (X,φ) $overflow="scroll"> (X, φ)$ dual to (C₀(X),Φ) $overflow="scroll"> (C_{0} (X), Φ)$ and a continuous field of homomorphisms α_x $overflow="scroll"> α_{x}$ between the fibers A(x) $overflow="scroll"> A (x)$ , x∈X $overflow="scroll"> x \in X$ , of the corresponding C^⁎ $overflow="scroll"> C^{⁎}$ -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) $overflow="scroll"> C^{⁎} (A, α; J)$ . We apply these results to the case when X=Prim(A) $overflow="scroll"> X = Prim (A)$ is a Hausdorff space. In particular, if the associated C^⁎ $overflow="scroll"> C^{⁎}$ -bundle is trivial, we obtain formulas for K -groups of all ideals in C^⁎(A,α;J) $overflow="scroll"> C^{⁎} (A, α; J)$ . 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,{α_x}_x∈X;Y) $overflow="scroll"> (X, φ, {α_{x}}_{x \in X}; Y)$ where Y is a closed subset of X.

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