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Going-Down functors and the K\"unneth formula for crossed products by \'etale groupoids. (arXiv:1810.04415v1 [math.OA])
来源于:arXiv
We study the connection between the Baum-Connes conjecture for an ample
groupoid $G$ with coefficient $A$ and the K\"unneth formula for the K-theory of
tensor products by the crossed product $A\rtimes_r G$. To do so we develop the
machinery of Going-Down functors for ample groupoids. As an application we
prove that both the uniform Roe algebra of a coarse space which uniformly
embeds into a Hilbert space and the maximal Roe algebra of a space admitting a
fibred coarse embedding into a Hilbert space satisfy the K\"unneth formula. We
also provide a stability result for the K\"unneth formula using controlled
K-theory, and apply it to give an example of a space that does not admit a
coarse embedding into a Hilbert space, but whose uniform Roe algebra satisfies
the K\"unneth formula. As a by-product of our methods, we also prove a
permanence property for the Baum-Connes conjecture with respect to equivariant
inductive limits of the coefficient algebra. 查看全文>>