Uniform \(K\)-homology theory
The author modifies the construction of analytic \(K\)-homology for metric spaces by introducing uniformity of approximation for \(K\)-homology cycles. A bounded operator \(T\) on a Hilbert space \(H\) is said to be \((\varepsilon,M)\)-approximable if there is a rank \(M\) operator \(k\) such that \(\|T-k\|<\varepsilon\). Let \(X\) be a metric space, \(C_R(X)=\{f\in C_c(X):\text{diam}(\text{supp}(f))\leq R; \|f\|\leq 1\}\), and let \(E:C_R(X)\to\mathbb B(H)\) be a linear map. \(E\) is \((\varepsilon,R,M)\)-approximable if \(E(f)\) is \((\varepsilon,M)\)-approximable for any \(f\in C_R(X)\). \(E\) is uniformly approximable if for every \(R\geq 0\), \(\varepsilon>0\) there exists \(M>0\) such that \(E\) is \((\varepsilon,R,M)\)-approximable. Let \(\varphi:C_0(X)\to\mathbb B(H)\) be a \(*\)-homomorphism. A triple \((H,\varphi,T)\) defines a \(K\)-homology cycle if several linear maps from \(C_0(X)\) to \(\mathbb B(H)\) (like \(f\mapsto [T,\varphi(f)]\), \((1-T^*T)\varphi(f)\), etc.), take values in the space of compact operators. The author defines uniform cycles by requiring that these linear maps should also be uniformly approximable. These uniform cycles are used to define uniform analytic \(K\)-homology theory in the standard way. Then the author defines an index map from this theory to the \(K\)-theory of uniform Roe \(C^*\)-algebras analogous to the coarse assembly map, shows that, for a torsion-free countable group \(\Gamma\), the direct limit \(\lim_{d\to\infty}K^u_*(P_d\Gamma)\) of the uniform \(K\)-homology of the Rips complexes \(P_d(\Gamma)\) of \(\Gamma\) is isomorphic to \(K^{\text{top}}_*(\Gamma,l^\infty\Gamma)\), and proves a criterion for amenability in terms of vanishing of a `fundamental class' similar to that in the uniformly finite homology.
scientific article; zbMATH DE number 5572188
scientific article