The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. (English) Zbl 0956.19004

First of all let us recall the coarse Baum-Connes conjecture. A metric space is called proper if every closed ball \(B(x,r)\) is compact. Let \((M,d)\) be a proper metric space and \(H_M\) 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 \(H_M.\) A metric space is called locally finite if every ball contains finitely many elements. Let \(\Gamma\) be a locally finite discrete metric space. For each \(d\geq 0,\) the Rips complex \(P_d(\Gamma)\) is the simplicial polyhedron where the set of all vertices is \(\Gamma ,\) and a finite subset \(\{\gamma _0,\dots,\gamma _n\}\subset \Gamma\) spans a simplex iff \(d(\gamma _i ,\gamma _j) \leq d\) for all \(0\leq i,j \leq n.\) Endow \(P_d(\Gamma)\) with the spherical metric. Recall that a discrete metric space \(X\) is said to have bounded geometry if for all \(r>0\), there exists \(N(r)>0 \) such that the number of elements in \(B(x,r)\) is at most \(N(r)\) for all \(x\in X.\) Every finitely generated group, as a metric space with a word-length metric, has bounded geometry.
The coarse Baum-Connes conjecture: if \(\Gamma\) is a discrete metric space with bounded geometry, then the index map from \(\lim _{d\to \infty} K_*(P_d(\Gamma))\) to \(\lim _{d\to \infty} K_*(C^*(P_d(\Gamma)))\) is an isomorphism, where \(K_*(P_d(\Gamma))=KK_*(C_0(P_d(\Gamma)), {\mathbb C})\) is the locally finite K-homology group of \(P_d(\Gamma).\)
A map \(f\) from a metric space \((X,d)\) to a separable infinite-dimensional Hilbert space \(H\) is said to be a uniform embedding if there exist non-decreasing functions \(r_1\) and \(r_2\) from \([0,\infty)\) to \({\mathbb R}\) such that (i) \(\forall x,y\in X r_1 (d(x,y)) \leq \|f(x)-f(y)\|\leq r_2 (d(x,y)),\) (ii) \(\lim _{t\to \infty} r_i(t)= \infty.\)
The main purpose of this remarkable paper is to prove the following result. Theorem 1.1: Let \(\Gamma\) be a discrete metric space with bounded geometry. If \(\Gamma\) admits a uniform embedding into Hilbert space, then the coarse Baum-Connes conjecture holds for \(\Gamma .\)


19K56 Index theory
46L80 \(K\)-theory and operator algebras (including cyclic theory)
58B34 Noncommutative geometry (à la Connes)
19K35 Kasparov theory (\(KK\)-theory)
Full Text: DOI