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Coarse Baum-Connes conjecture along one direction with filtered coefficients for products

Establish that for proper metric spaces X and Y and a locally finite net N_X in X, the evaluation-at-zero map e_* induces an isomorphism between the direct limit over k of K_*(C^*_{L,P_k(N_X), f}(P_k(N_X) × Y, A)) and the direct limit over k of K_*(C^*_f(P_k(N_X) × Y, A)).

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Background

To bridge the conjecture for product spaces with that for their factors, the authors introduce localization algebras along a specified factor (here, X) and formulate a corresponding conjecture that the evaluation-at-zero map remains an isomorphism in this context.

This conjecture is instrumental in proving closure under products for the filtered-coefficients version and serves as a key technical component connecting factor-wise conjectures for X and Y to the product X × Y.

References

Now we introduce the coarse Baum-Connes conjecture along X with filtered coefficients for X\times Y. Let X, Y be two proper metric spaces and N_X be a locally finite net in X. Then the evaluation at zero map e induces the following isomorphism.

— The coarse Baum-Connes conjecture with filtered coefficients and product metric spaces (2410.11662 - Zhang, 15 Oct 2024) in Conjecture Conj-CBC-along-X, Section 4.1 (The coarse Baum-Connes conjecture along one direction with filtered coefficients)