Asymptotic coarsening of proper metric spaces and index problems
First of all let us recall the notion of coarse structure. A metric space is called proper if every closed ball \(B(x,r)\) is compact. Let \((M,d)\) be a proper metric space and \(H_M\) be a separable Hilbert space equipped with a faithful and non-degenerate *-representation of \(C_0(M)\) whose range contains no non-zero compact operator. A bounded operator \(T:H_M\to H_M\) has finite propagation if \(\sup \{d(m_1,m_2):(m_1,m_2)\in \text{supp}(T)\} <\infty \) where \(m_i \in M\) and \(\text{supp}(T)\) is the complement of the set of points \((m_1,m_2)\in M\times M\) for which there exist \(g_1,g_2 \in C_0(M)\) such that \(g_1Tg_2=0, g_1(m_1)\neq 0 g_2(m_2)\neq 0.\) A bounded operator \(T:H_M\to H_M\) is called locally compact if the operators \(gT\) and \(Tg\) are compact for all \(g\in C_0(M).\) The Roe algebra \(C^*(M)\) is the operator norm closure of the *-algebra of all locally compact, finite propagation operators acting on \(H_M.\) This algebra can be considered as the noncommutative quotient of coarsening a metric space by identifying all pairs of points that are finite distance apart. The main purpose of this paper is to introduce an asymptotic coarse structure on proper metric spaces and study the associated Roe type \(C^{\ast}\)-algebra \(C^{\ast}_A(X)\) and the assembly map from \(K\)-homology \(K_{\ast}(X)\) to the \(K\)-theory of \(C^{\ast}_A(X).\) The \(C^{\ast}\)-algebra \(C^{\ast}_A(X),\) which is generated by locally compact operators over \(X\) with asymptotically controlled propagation, can be regarded as the noncommutative quotient of asymptotic coarsening of the space \(X.\) Asymptotically coarsening a space may be thought of as identifying all pairs of points which live in an entourage in the asymptotic coarse structure. An asymptotic Lipschitz homotopy invariance theorem for the \(K\)-theory of \(C^{\ast}_A(X)\) and the \(K\)-homology of \(X\) is established. It is shown that the assembly map is an isomorphism for asymptotically scaleable metric spaces.
scientific article; zbMATH DE number 1945287