Homotopy type of the unitary group of the uniform Roe algebra on $\mathbb{Z}^{n}$

For a $C^{\ast}$-algebra $A$, let $\operatorname{GL}_{d}(A)$ and $\operatorname{U}_{d}(A)$ denote the space of the invertible and unitary matrices with entries in $A$, respectively. It it well-known that they always have the same homotopy type. We will often refer only to $\operatorname{U}_{d}(A)$ but most statements are valid for $\operatorname{GL}_{d}(A)$ as well. There have been a lot of works on the homotopy theory of $\operatorname{U}_{d}(A)$ and some of them have important applications. For finite-dimensional case, the complex-valued unitary matrices $\operatorname{U}_{d}(\mathbb{C})$ is just the usual unitary group acting linearly on $\mathbb{C}^{d}$. For inifinite-dimensional case, Kuiper [Kui65] proved that the space of all unitary operators on an inifinite-dimensional Hilbert space is contractible. This result is basic in the Atiyah–Singer index theory. This kind of contractibility result has been extended to $\operatorname{U}_{d}(A)$ of some other algebras $A$ while $A$ is all the bounded operators on a infinite dimensional Hilbert space in the original result. Of course it is not always the case for $\operatorname{U}_{d}(A)$ of other infinite-dimensional $C^{\ast}$-algebras $A$. In general, it is hard to determine the homotopy type of $\operatorname{U}_{d}(A)$.

Let us use the notation

It is well-known that $\operatorname{U}_{\infty}(A)$ has the same homotopy type as $\operatorname{U}_{1}(A\otimes\mathcal{K})$ where $\mathcal{K}$ is the space of compact operators. The $K$-theory $K_{i}(A)$ ($i=0,1$) is a basic homotopy invariant of $A$, which is characterized as

Since $\operatorname{U}_{d}(A)$ is not necessarily homotopy equivalent to $\operatorname{U}_{\infty}(A)$, $K_{i}(A)$ is not a so strong invariant in general. But sometimes the natural map $\operatorname{U}_{d}(A)\to\operatorname{U}_{\infty}(A)$, which we will call the stabilizing map, becomes a homotopy equivalence. Study on such stability seems to trace back to the work of Bass [Bas64]. There have been a various works on this kind of stability. Rieffel introduced the topological stable rank in [Rie83] and applied it to show the stability of the non-commutative torus in [Rie87], which is a key tool in the present work. It is difficult in general to determine how stable a given $C^{\ast}$-algebra is.

In the present paper, we study the stability of the uniform Roe algebra $C^{\ast}_{u}(|\mathbb{Z}^{n}|)$ on $\mathbb{Z}^{n}$ and investigate its homotopy type. The uniform Roe algebra $C^{\ast}_{u}(X)$ of a metric space $X$ introduced by Roe in [Roe88] to establish an index theory on open manifolds, where the index lives in the $K$-theory $K_{\ast}(C^{\ast}_{u}(X))$. The algebra $C^{\ast}_{u}(X)$ itself is also important since it encodes a kind of “large scale geometry” of $X$. Studying the homotopy type of $\operatorname{U}_{d}(C^{\ast}_{u}(X))$ will provide more insights from a homotopy theoretic viewpoint, which cannot be obtained only from its $K$-theory. But there are only a few works on the homotopy type of $\operatorname{U}_{d}(C^{\ast}_{u}(X))$ yet. For example, Manuilov and Troitsky [MT21] studied some condition for $\operatorname{U}_{d}(C^{\ast}_{u}(X))$ being contractible. In the present work, we observe the other extreme, that is, $\operatorname{U}_{d}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))$ has a highly nontrivial homotopy type.

We give some comment on the relation with our previous work [KKT] on the space $\mathcal{U}$ of finite propagation unitary operators on $\mathbb{Z}$. Note that $\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}|))$ can be viewed as a kind of completion of $\mathcal{U}$. We determined the homotopy type of $\mathcal{U}$ there. But it is not clear whether $\mathcal{U}$ has the same homotopy type as $\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}|))$. Actually, they turn out to have the same homotopy type (Theorem 1.2). Also, the method there does not seem to be extended to $\mathbb{Z}^{n}$ when $n\geq 2$. We employ rather operator algebraic technique in the present paper. Our method here reduces the problem to determine the homotopy type of $\operatorname{U}_{d}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))$ to the one to show the surjectivity of the homomorphism on $K$-theory $K_{\ast}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\to K_{\ast}(C^{\ast}(|\mathbb{Z}^{n}|))$ induced from the inclusion (Proposition 7.4), where $C^{\ast}(|\mathbb{Z}^{n}|)$ denotes the Roe algebra of $\mathbb{Z}^{n}$.

For stability, we show the following theorem in Section 3.

Let $K(V,i)$ denote the Eilenberg–MacLane space of type $(V,i)$ and $B\operatorname{U}_{\infty}(\mathbb{C})$ denote the classifying space of the unitary group $\operatorname{U}_{\infty}(\mathbb{C})$. Also, for based spaces $X_{i}$ ($i=1,2,\ldots$), define

For the homotopy type of $\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))$, we show the following results when $n=1,2$.

More detailed descriptions of the vector spaces $V_{0}$ and $V_{1}$ appear in the proof of Lemma 7.7.

We will see the existence of a homotopy section of the inclusion $\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}|))\to\operatorname{U}_{1}(B^{\mathrm{SW}})$ in Section 6, where $\operatorname{U}_{1}(B^{\mathrm{SW}})$ is the Segal–Wilson restricted unitary group [SW85] having the homotopy type of $\mathbb{Z}\times B\operatorname{U}_{\infty}(\mathbb{C})$. This implies Theorem 1.2. We also show in Section 7 that, for any integer $n\geq 1$, the inclusion $\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\to\operatorname{U}_{1}(C^{\ast}(|\mathbb{Z}^{n}|))$ admits a homotopy section if and only if the homomorphism $K_{\ast}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\to K_{\ast}(C^{\ast}(|\mathbb{Z}^{n}|))$ is surjective. Since we can see it is surjective when $n=1,2$, Theorems 1.2 again and 1.3 follows. If one could show the surjectivity for $n\geq 3$, then a similar homotopy decomposition will immediately follow.

This paper is organized as follows. We fix our notation in Section 2. In Section 3, we recall Rieffel’s results on stability and show Theorem 1.1. In Section 4, we recall the Bott periodicity realized as a $\ast$-homomorphism. In Section 5, we recall the Segal–Wilson restricted unitary group and show its stability. In Section 6, we show Theorem 1.2 using the Segal–Wilson restricted unitary group. In Section 7, we discuss the homotopy type of $\operatorname{U}_{d}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))$ for general $n\geq 1$ and show Theorems 1.2 again and 1.3.

The $C^{\ast}$-algebra of bounded operators on a Hilbert space $V$ is denoted by $\mathcal{B}(V)$ and the subalgebra of compact operators by $\mathcal{K}(V)$. We write the operator norm of $T\in\mathcal{B}(V)$ as $\|T\|$. The Hilbert space of square summable sequences indexed by a discrete group $\Gamma$ will be written as

We also consider the tensor product Hilbert space $\ell^{2}(\Gamma)\otimes\mathcal{H}$ with an infinite dimensional separable Hilbert space $\mathcal{H}$.

A bounded operator $T\in\mathcal{B}(\ell^{2}(\Gamma))$ can be expressed in the matrix form as

For $T\in\mathcal{B}(\ell^{2}(\Gamma)\otimes\mathcal{H})$, we also have a similar expression $T=(T_{g,h})_{g,h}$ with $T_{g,h}\in\mathcal{B}(\mathcal{H})$.

It is easy to see that the definition of having finite propagation is independent of the choice of generators while the value of $\operatorname{prop}(T)$ depends on the word metric. Since we have

for any finite propagation operators $S,T\in\mathcal{B}(\ell^{2}(\Gamma))$, the subset of finite propagation operators becomes a unital $\ast$-subalgebra of $\mathcal{B}(\ell^{2}(\Gamma))$. Similar properties hold for finite propagation operators $S,T\in\mathcal{B}(\ell^{2}(\Gamma)\otimes\mathcal{H})$ such that the components $T_{g,h}$ and $S_{g,h}$ are compact operators.

We will consider the uniform Roe algebra $C^{\ast}_{u}(|\Gamma|)$ is a subalgebra of the Roe algebra $C^{\ast}(|\Gamma|)$ with respect to some inclusion $\mathbb{C}\subset\mathcal{H}$.

We use the symbol $\ell^{\infty}(\Gamma,\mathbb{C})$ to express the Banach algebra of $\mathbb{C}$-valued bounded sequences indexed by $\Gamma$ rather than the simpler symbol $\ell^{\infty}(\Gamma)$ since we also consider the abelian group of $\mathbb{Z}$-valued bounded sequences $\ell^{\infty}(\Gamma,\mathbb{Z})$.

The group $\Gamma$ acts on the algebras $\ell^{\infty}(\Gamma,\mathbb{C})$ and $\ell^{\infty}(\Gamma,\mathcal{K}(\mathcal{H}))$ by right translation. The action by $x\in\Gamma$ is compatible with the conjugation by $S_{x}$ through the diagonal inclusion $\ell^{\infty}(\Gamma,\mathbb{C})\to C^{\ast}_{u}(|\Gamma|)$ or $\ell^{\infty}(\Gamma,\mathcal{K}(\mathcal{H}))\to C^{\ast}(|\Gamma|)$ given by

Moreover, this inclusion extends to the well-known isomorphisms

from the reduced crossed products of $C^{\ast}$-algebras. For example, see [Roe03, Theorem 4.28].

The $d\times d$-matrix algebra $M_{d}(A)$ of a $C^{\ast}$-algebra $A$ is again a $C^{\ast}$-algebra. The spaces of invertible elements and unitary elements in $M_{d}(A)$ will be denoted as $\operatorname{GL}_{d}(A)$ and $\operatorname{U}_{d}(A)$. The stabilizing maps are given as

where the inductive limits are taken along the inclusions $\operatorname{GL}_{d}(A)\subset\operatorname{GL}_{d+1}(A)$ and $\operatorname{U}_{d}(A)\subset\operatorname{U}_{d+1}(A)$. The inductive limit spaces $\operatorname{GL}_{\infty}(A)$ and $\operatorname{U}_{\infty}(A)$ are well-known to be homotopy equivalent to the spaces $\operatorname{GL}_{1}(A\otimes\mathcal{K}(\mathcal{H}))$ and $\operatorname{U}_{1}(A\otimes\mathcal{K}(\mathcal{H}))$.

The aim of this section is to prove Theorem 1.1. Once the assumption of the following result by Rieffel [Rie87] is verified, the theorem will immediately follow.

For the definitions of the topological stable rank $\operatorname{tsr}(A)\in\mathbb{Z}_{\geq 1}$, see [Rie83]. A $C^{\ast}$-algebra $A$ is said to be tsr-boundedly divisible [Rie87] if there is a constant $K$ such that for any integer $m$, there exists an integer $d\geq m$ such that $A$ is isomorphic to $M_{d}(B)$ for some $C^{\ast}$-algebra $B$ with $\operatorname{tsr}(B)\leq K$. To verify the assumption, we need the following two lemmas.

Let us recall the Bott periodicity of $C^{\ast}$-algebras here. Let $A$ be a $C^{\ast}$-algebra, which might be non-unital. The direct sum $\mathbb{C}\oplus A$ is considered as the unitization with unit $(1,0)\in\mathbb{C}\oplus A$. Define the unitary group $\operatorname{U}^{\prime}_{d}(A)$ by

If $A$ is already unital, we have a canonical isomorphism $\operatorname{U}^{\prime}_{d}(A)\cong\operatorname{U}_{d}(A)$. So we use the same symbol $\operatorname{U}_{d}(A)$ for $\operatorname{U}^{\prime}_{d}(A)$ even if $A$ is not unital.

Consider the following space of continuous functions:

This is a $C^{\ast}$-algebra without unit. Notice that $C_{0}(\mathbb{R}^{m},A)$ is isomorphic to the space $\Omega^{m}A$ of based maps from the $m$-sphere $S^{m}$ to $A$ where the basepoint $\ast\in S^{m}$ is mapped to $0\in A$.

Set the element

where we identify $\mathbb{R}^{2}\cong\mathbb{C}$ in the matrix entries. The Bott map $\beta\colon A\to M_{2}(A\oplus C_{0}(\mathbb{R}^{2},A))$ is a $\ast$-homomorphism defined by

Then we have the commutative square of unital $C^{\ast}$-algebras

where $\epsilon\colon\mathbb{C}\oplus A\to\mathbb{C}$ and $\epsilon\colon M_{2}(\mathbb{C}\oplus A\oplus C_{0}(\mathbb{R}^{2},A))\to M_{2}(\mathbb{C}\oplus A)$ are the projections and $\eta\colon\mathbb{C}\to M_{2}(\mathbb{C}\oplus A)$ is the unit map. This square induces the $\ast$-homomorphism between the kernels of $\epsilon$:

We call this $\beta$ the Bott map as well. It is natural in the following sense: if $f\colon A\to B$ is a $\ast$-homomorphism between $C^{\ast}$-algebras, then the following square commutes:

The Bott periodicity provides the natural homotopy equivalence

which is a group homomorphism. Thus we obtain the following proposition on infinite loop structure.

To study the homotopy type of $\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}|))$, we will relate it with other spaces. One is the Segal–Wilson restricted unitary group $\operatorname{U}_{1}(B^{\mathrm{SW}})$ and the other is the unitary group of the Roe algebra $\operatorname{U}_{1}(C^{\ast}(|\mathbb{Z}|))$. We recall the former in this section.

We have another matrix expression for $T\in\mathcal{B}(\ell^{2}(\mathbb{Z}))$ as

where

The symbol “SW” stands for Segal–Wilson. The unitary group $\operatorname{U}_{1}(B^{\mathrm{SW}})$ is called the restricted unitary group in the work of Segal and Wilson [SW85]. They used it as a model of the infinite Grassmannian.

Let $S=S_{+1}\in B^{\mathrm{SW}}$ the shift operator as in Example 2.2. We have $\operatorname{ind}S^{n}=n$.

The goal of this section is to see the following.

To show this, we do not use a kind of stability as in Section 3.