Maximal coarse Baum-Connes and maximal coarse Novikov conjectures

Establish that for every metric space X with bounded geometry, the maximal coarse assembly map μ_max: lim_{d→∞} K_*(P_d(X)) → K_*(C^*_{max}(X)) is an isomorphism; equivalently, show that the maximal coarse Novikov conjecture holds by proving μ_max is injective.

Background

In addition to the reduced Roe algebra C^*(X), the paper considers the maximal Roe algebra C^*_{max}(X) and the associated maximal assembly map μ_max. The maximal conjectures parallel the reduced ones but use the universal (maximal) C*-norm.

These maximal versions are important in settings where reduced and maximal completions may behave differently; later sections develop maximal relative analogues using geometric ideals to define quotient algebras at infinity.