## 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 $$\Gamma .$$

### MSC:

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