The coarse Baum–Connes conjecture along a factor with filtered coefficients

Let AA be a filtered CC^{\ast}-algebra, let XX and YY be proper metric spaces, and let NXN_X be a locally finite net in XX. For each k0k\geq 0, let Pk(NX)P_k(N_X) be the kk-Rips complex, and let CL,Pk(NX),f(Pk(NX)×Y,A)C^{\ast}_{L,P_k(N_X),f}(P_k(N_X)\times Y,A) denote the localization algebra along the first factor, while Cf(Pk(NX)×Y,A)C_f^{\ast}(P_k(N_X)\times Y,A) denotes the Roe algebra with filtered coefficients. The coarse Baum–Connes conjecture along XX with filtered coefficients. Evaluation at zero induces an isomorphism

e:limkK(CL,Pk(NX),f(Pk(NX)×Y,A))limkK(Cf(Pk(NX)×Y,A)).e_{\ast}: \lim_{k\rightarrow \infty} K_{\ast}(C^{\ast}_{L,P_k(N_X),f}(P_k(N_X)\times Y, A))\rightarrow \lim_{k\rightarrow \infty} K_{\ast}(C_f^{\ast}(P_k(N_X) \times Y, A)).

This is a product-space version of the filtered-coefficient coarse Baum–Connes assertion, but the supplied text does not state whether it has been proved or disproved.

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Primary source

Jianguo Zhang, “The coarse Baum-Connes conjecture with filtered coefficients and product metric spaces”, arXiv:2410.11662 (2025).

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