Twisted Roe algebras and their $K$-theory

Let $X$ be a metric space with bounded geometry. The coarse Baum-Connes conjecture claims that a certain coarse assembly map

is an isomorphism, where $KX_{*}(X)$ is the coarse $K$-homology of the space $X$ and $C^{*}(X)$ is the associated Roe algebras. This conjecture provides algorithm to compute the higher indices for elliptic operators on non-compact Riemannian manifolds. It has many applications in topology and geometry. In particular, it implies the Novikov conjecture on homotopy invariance of higher signatures, and the Gromov-Lawson-Rosenberg conjecture regarding nonexistence of positive scalar curvature metrics on closed aspherical manifolds.

The coarse Baum-Connes conjecture has been verified for a large class of metric spaces with bounded geometry, including all coarsely embeddable spaces [16], and certain relative expanders [5]. The purpose of the paper is to enlarge the class of metric spaces satisfying the coarse Baum-Connes conjecture.

In order to study the $K$-theory of Roe algebras, we introduce a twisted version of Roe algebras for a metric space with bounded geometry using a coarse algebra. Let $Z$ be a metric space with bounded geometry and $\mathcal{A}$ a stable $C^{*}$-algebra. The coarse $Z$-algebra $\Gamma(Z,\mathcal{A})$ to be some certain $C^{*}$-subalgebra of $\ell^{\infty}(Z,\mathcal{A})$. Typical examples of a coarse $Z$-algebra include $\ell^{\infty}(Z,\mathcal{K})$ and $C_{0}(Z,\mathcal{K})$, where $\mathcal{K}$ is the algebra of compact operators on an infinite-dimensional separable Hilbert space, and $C_{0}(Z,\mathcal{K})$ is the algebra of all functions from $Z$ to $\mathcal{K}$ vanishing at infinity. Using coarse algebras, we define the twisted Roe algebra $C^{*}_{\Gamma(Z,\mathcal{A})}(M,\mathcal{A})$ for a metric space $M$ with a coarse equivalence $f:M\to Z$. The twisted Roe algebras can be viewed as a coarse analogue of crossed product of a group $G$ and a $G$-$C^{*}$-algebra. Furthermore, we have the coarse imprimitivity theorem.

As an application, we have $C^{*}_{C_{0}(X,\mathcal{K})}(X,\mathcal{K})\sim_{\text{Morita}}\mathcal{K}.$ This can be seen as a coarse analogue of $C_{0}(G,\mathcal{K})\rtimes_{r}G\sim_{\text{Morita}}\mathcal{K}$. We also have $C^{*}_{\ell^{\infty}(X,\mathcal{K})}(X,\mathcal{K})\cong C^{*}(X)$, where $C^{*}(X)$ is the Roe algebra of $X$.

For metric space $X$ with bounded geometry and coarse $X$-algebra $\Gamma(X,\mathcal{A})$, we introduce a coarse assembly map

The twisted coarse Baum-Connes conjecture claims that the coarse assembly map is an isomorphism. In the case when $\Gamma(X,\mathcal{A})=\ell^{\infty}(X,\mathcal{K})$, the twisted coarse Baum-Connes conjecture is the usual coarse Baum-Connes conjecture, while it is the coarse Baum-Connes conjecture with coefficients in $\mathcal{A}$ when $\Gamma(X,\mathcal{A})=\ell^{\infty}(X,\mathcal{A})$.

In [16], Yu showed that the coarse Baum-Connes conjecture holds for a metric space which admits a coarse embedding into Hilbert space. We strengthen the result to the twisted Baum-Connes conjecture with coefficients in any coarse algebra.

Let $Z\to X\to Y$ be a coarse fibration. It is possible that $X$ does not admit a coarse embedding into Hilbert space, even though $Z$ and $Y$ are coarsely embeddable into Hilbert space. The following is the main result of this paper.

We take a different approach to prove the twisted coarse Baum-Connes conjecture for the coarse fibration, other than the Dirac-dual-Dirac method.

Group extensions provide examples of coarse fibrations. In [2], Arzhantseva and Tessera showed that admitting a coarse embedding into Hilbert space is not preserved by group extensions. They constructed in [2] an extension of groups $1\to N\to G\to G/N\to 1$ such that $N$ and $G/N$ are coarsely embeddable into Hilbert space, but $G$ is not coarsely embeddable. It is natural to ask if the coarse Baum-Connes conjecture holds for such a group $G$. In [4], the first author showed that the Novikov conjecture with coefficients holds for $G$, and it follows that the coarse assembly map is injective. In this paper, we strengthen this result to the following.

The paper is organized as follows. In Section 2, we introduce the concept of twisted Roe algebras and study their properties, especially the coarse imprimitivity theorem. In Section 3, we introduce the twisted coarse Baum-Connes conjecture and show how our framework of twisted coarse Baum-Connes can be used to prove the usual coarse Baum-Connes conjecture. In particular, we formulate a coarse algebra for a metric space admitting a coarse embedding into Hilbert space. In Section 4, we show the coarse Baum-Connes conjecture for a metric space $X$ with a coarse fibration structure $Z\to X\to Y$ when the fiber $Z$ and the baser space $Y$ are coarse embeddable into Hilbert space.

In this section, we will first introduce a coarse algebra for a metric space and use it to define twisted Roe algebra. We will then consider the properties of the twisted Roe algebras.

Let $Z$ be a metric space with bounded geometry in the sense that for any $R>0$, there exists $N_{R}\in\mathbb{N}$ such that

A partial translation on $X$ is a bijection $v:D\to R$ from a subset $D\subseteq Z$ onto another subset $R\subseteq Z$ such that $\sup_{x\in D}d(x,v(x))\leq R$ for some constant $R>0$. Recall that a $C^{*}$-algebra $\mathcal{A}$ is stable if $\mathcal{A}\otimes\mathcal{K}\cong\mathcal{A}$, where $\mathcal{K}$ is the algebra of compact operators on an infinite-dimensional Hilbert space.

Let $\ell^{\infty}(Z,\mathcal{A})$ be the $C^{*}$-algebra of all bounded map from $Z$ to $\mathcal{A}$. For any map $\xi\in\ell^{\infty}(X,\mathcal{A})$ and any partial translation $v:D\to R$ on $Z$, we define a map from $Z$ $\mathcal{A}$ by

In the case when $R\cap{\rm Supp}(f)=\emptyset$, we set $v^{*}f\equiv 0$ on $Z$.

Let $C_{0}(Z,\mathcal{A})$ be the $C^{*}$-algebra of all maps from $Z$ to $\mathcal{A}$ vanishing at infinity. It is obvious that $\ell^{\infty}(Z,\mathcal{A})$ is also a coarse $Z$-algebra for any stable algebra $\mathcal{A}$. Moreover, every coarse $Z$-algebra with fiber $\mathcal{A}$ contains $C_{0}(Z,\mathcal{A})$ as an ideal.

Let $M$ and $Z$ be metric spaces. We say that $M$ and $Z$ are coarsely equivalent if there exists a proper map $f:M\to Z$ and non-decreasing functions $\rho_{+},\rho_{-}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ with $\rho_{\pm}(t)\to\infty$ as $t$ tends to $\infty$, such that

• (1)

  $\rho_{-}(d_{M}(x,x^{\prime}))\leq d_{X}(f(x),f(x^{\prime}))\leq\rho_{+}(d_{M}(x,x^{\prime}))$ for all $x,x^{\prime}\in M$;

• (2)

  there exists $r>0$ such that $Z=N_{r}(f(M))$, where $N_{r}(f(M))=\{x\in Z:\exists~{}m\in M,~{}\text{such~{}that}~{}d(x,f(m))\leq r\}$.

The map $f$ is called a coarse equivalence and $\rho_{\pm}$ is called the controlling functions associated with $f$. We say that a family of metric spaces $(M_{n})$ is uniformly coarsely equivalent to $Z$ if there exists a coarse equivalence $f_{n}:M_{n}\to Z$ for each $n$ such that the family of coarse equivalences $(f_{n})_{n\in\mathbb{N}}$ shares the same controlling functions $\rho_{\pm}$ and the constant $r$ such that $N_{r}(f_{n}(M_{n}))=Z$ for all $n\in\mathbb{N}$.

Let $M$ be a proper metric space in the sense that the closure of a bounded subset of $M$ is compact. In order to define the twisted Roe algebra for the space $M$, we fix a countable, dense subset $\widehat{M}\subseteq M$.

Consider

The space $\mathcal{H}_{X,\mathcal{A}}$ is a right Hilbert $\mathcal{A}$-module as follows. For any $\xi,\eta\in\ell^{2}(\widehat{M},\mathcal{A})$ and $a\in\mathcal{A}$, we define

It is called a geometric $M$-$\mathcal{A}$-module. We say the map $T:\widehat{M}\times\widehat{M}\to\mathcal{A}$ is norm-bounded if it is bounded when it is viewed as $\mathcal{A}$-linear map on $\ell^{2}(\widehat{M},\mathcal{A})$ by

for all $\xi\in\ell^{2}(\widehat{M},\mathcal{A})$.

Let $Z$ be a metric space with bounded geometry, and $f:M\to Z$ a coarse equivalence. Without loss of generality, we assume that the map $f$ is a Borel map (cf. [14, Lemma A.3.12]). Then, we obtain a uniformly bounded Borel cover $\left\{B_{x}=f^{-1}(x)\right\}_{x\in Z}$. By the stability of the algebra $\mathcal{A}$, we consider a $*$-isomorphism

If $T:\widehat{M}\times\widehat{M}\to\mathcal{A}$ is norm-bounded, locally compact operator with finite propagation, then for any $x,y\in\widehat{M}$, the operator $\chi_{B_{x}}T\chi_{B_{y}}$ lies in the algebra $\mathcal{A}\otimes\mathcal{K}(\ell^{2}(\widehat{M}))$. We obtain a map $\widetilde{T}:X\times X\to\mathcal{A}$ by

for all $x,y\in X$. Since each $B_{x}$ is bounded and $T$ is locally compact, the map $\widetilde{T}$ is well-defined.

Let $v:D\to R$ be a partial translation on $X$ and let $T:\widehat{M}\times\widehat{M}\to\mathcal{A}$ be a norm-bounded, locally compact map with finite propagation. Then, we can consider a map $T^{v}:Z\to\mathcal{A}$ by

Now, we are ready to define the twisted Roe algebra for a proper metric space $M$ with a coarse equivalence $f:M\to Z$ using a coarse $X$-algebra.

Example 2.5 can be viewed as a coarse analogue of the identity $C_{0}(\Gamma,\mathcal{K})\rtimes\Gamma\cong\mathcal{K}(\ell^{2}(\Gamma))\otimes\mathcal{K}\cong\mathcal{K}$ for a countable discrete group $\Gamma$.

The twisted Roe algebras are invariant under coarse equivalence in the following sense:

The proof is similar to the classic case (cf. [14, Lemma 5.1.12, Remark 5.1.13]). The key point is that the coefficient algebra $\mathcal{A}$ is assumed to be stable.

The following definition is introduced in [9].

Let $X$, $Y$ and $Z$ be metric spaces with bounded geometry, and let $Z\to X\to Y$ be a coarse $Z$-fibration. By Definition 2.8, there exists $\phi_{y}:Z_{y}\to Z$ a coarse equivalence for each $y\in Y$. Let $\Gamma(Z,\mathcal{A})$ be a coarse $Z$-algebra. This twisted coefficient algebra induces a twisted coefficient algebra on $X$, which can be seen as a coarse analogue of the induction representation for groups, see [10]. For each $y\in Y$, we define

where for any $\xi\in\ell^{\infty}(Z_{y},\mathcal{A})$, the function $(\phi_{y})_{*}(\xi)\in\ell^{\infty}(Z,\mathcal{A})$ is defined by

where $\psi:\mathcal{K}(\ell^{2}(Z_{y}))\otimes\mathcal{A}\to\mathcal{A}$ is an isomorphism. Since $X$ has bounded geometry and $\phi_{y}$ is a coarse equivalence, this direct sum $\bigoplus_{x\in\phi^{-1}_{y}(z)}\xi(x)$ is finite which defines an element in $\mathcal{K}(\ell^{2}(Z_{y}))\otimes\mathcal{A}$, thus $(\phi_{y})_{*}(\xi)$ is well-defined. Define the induced coarse $X$-algebra associated with $\Theta(Z,\mathcal{A})$ to be

Since $X=\bigsqcup_{y\in Y}Z_{y}$ as a set, we can view an element in $\Theta_{ind}^{Z}(X,\mathcal{A})$ as an $\ell^{\infty}$-function from $X$ to $\mathcal{A}$.

We now prove that $\Theta_{ind}^{Z}(X,\mathcal{A})$ is a coarse $X$-algebra. For any partial translation $v:D\to R$ on $X$ and any element $\xi\in\Theta_{ind}^{Z}(X,\mathcal{A})$ with $\textup{supp}(\xi)\subseteq D$, by definition, one has that

Since $v$ is a partial translation, there exists $R>0$ such that $d(x,v(x))\leq R$ for any $x\in D$. As $p:X\to Y$ is bornologous, there exists $S$ such that $d_{Y}(p(x),p(v(x)))\leq S$. Thus if we view $\xi$ as a $C_{0}$-function from $Y$ to $\bigsqcup_{y\in Y}\Theta(Z_{y},\mathcal{A})$, we define the $\varepsilon$-support of $\xi$ to be

Then the $\varepsilon$-support of $v_{*}(\xi)$ is totally contained in $B_{Y}(\textup{supp}_{\varepsilon}(\xi),S)$, as a result, $\|v_{*}(\xi)|_{Z_{y}}\|_{\infty}\to 0$ as $y\to\infty$.

On the other hand, for such a partial translation $v:D\to R$, we denoted by $R_{y}=Z_{y}\cap R$ for each $y\in Y$. We denote

Define $v_{y^{\prime}y}$ to be the restriction of $v$ on $D_{y^{\prime}y}$. For each $v_{y^{\prime}y}$ and $\xi\in\Theta_{ind}^{Z}(X,\mathcal{A})$, one can check that $(v_{y^{\prime}y})^{*}(\xi|_{D_{y^{\prime}y}})$ indeed defines an element in $\Theta(Z_{y},\mathcal{A})$ using condition (3) in Definition 2.8. Since $v$ is a partial translation, there are only finitely many $y^{\prime}$ such that $D_{y^{\prime}y}$ is nonempty. This proves that $\Theta_{ind}^{Z}(X,\mathcal{A})$ is a coarse $X$-algebra, and its completion is denoted by $\Gamma_{ind}^{Z}(X,\mathcal{A})$. We can then define the twisted Roe algebra of $X$.

In this section, we shall introduce the twisted version of the coarse Baum-Connes conjecture. In [8], a notion of coarsely proper algebra was introduced to give a conceptual framework for the geometric Dirac-dual-Dirac method. In this section, we will understand the coarsely proper algebra from the perspective of coarse $X$-algebras.

Let $X$ be a metric space with bounded geometry, $\mathcal{A}$ a stable $C^{*}$-algebra, and $\Gamma(X,\mathcal{A})$ be a coarse $X$-algebra. Let $M$ be a metric space with coarse equivalence $f:M\to X$.