K-theory for generalized Lamplighter groups

TL;DR

This paper calculates the K-theory of reduced group C*-algebras for generalized Lamplighter groups, providing insights into their algebraic structure.

Contribution

It introduces a method to compute K-theory for a broad class of generalized Lamplighter groups' C*-algebras.

Findings

K-theory computed explicitly for these groups

New techniques for analyzing group C*-algebras

Enhanced understanding of algebraic invariants

Abstract

We compute K-theory for the reduced group C*-algebras of generalized Lamplighter groups.

Equations

$$K_{\ast }(C_{\lambda }^{\ast }((\underset{\Gamma }{⨁}\Sigma )⋊\Gamma ))\cong K_{\ast }(C_{\lambda }^{\ast }(\Gamma ))\oplus \left(\underset{[C]\in \mathcal{C}}{⨁}\underset{[X]\in N_C\backslash F(C)}{⨁}\underset{[\phi ]\in C\backslash \left(({\mathrm{con}}^{\times }\Sigma {)}^{C\cdot X}\right)}{⨁}{K}_{\ast }(C_{\lambda }^{\ast }({\mathrm{Stab}}_C(\phi )))\right).$$

$$K_1(C_{\lambda }^{\ast }(\Gamma ))\cong K_1(C_{\lambda }^{\ast }((\underset{\Gamma }{⨁}\Sigma )⋊\Gamma )).$$

$$K_{\ast }(C_{\lambda }^{\ast }((\underset{\Gamma }{⨁}\Sigma )⋊\Gamma ))\cong K_{\ast }(C_{\lambda }^{\ast }(\Gamma ))\oplus \left(\underset{[X]\in \Gamma \backslash {\mathrm{FIN}}^{\times }}{⨁}\underset{({\mathrm{con}}^{\times }\Sigma {)}^X}{⨁}{K}_{\ast }(\mathbb{C})\right).$$

$$\left\{e_{\phi }\text{: }\phi \in {\left\{1,\dots ,n\right\}}^F,F\subseteq \Gamma \mathrm{finite}\right\}$$

$$D_F=C^{\ast }(\left\{e_{\phi }\text{: }\phi \in {\left\{1,\dots ,n\right\}}^{F^{\prime }}\mathrm{for}{F}^{\prime }\subseteq F\right\}).$$

$$K_{\ast }(D_f)$$

$$M_f=\left(\begin{array}{cccc}1& 0& \dots & 0\\ k_1& 1& & 0\\ ⋮& & \ddots & \\ k_n& 0& & 1\end{array}\right).$$

$$\underset{f\in F}{⨂}{K}_{\ast }(D_f)$$

$$K_{\ast }(C({\left\{0,\dots ,n\right\}}^{\Gamma }){⋊}_r\Gamma )\cong K_{\ast }(C_{\lambda }^{\ast }(\Gamma ))\oplus \left(\underset{[C]\in \mathcal{C}}{⨁}\underset{[X]\in N_C\backslash F(C)}{⨁}\underset{[\phi ]\in C\backslash \left({\left\{1,\dots ,n\right\}}^{C\cdot X}\right)}{⨁}{K}_{\ast }(C_{\lambda }^{\ast }({\mathrm{Stab}}_C(\phi )))\right).$$

$$\left\{{\pi }_F^{-1}(\phi )\text{: }\phi \in {\left\{1,\dots ,n\right\}}^F,F\subseteq \Gamma \mathrm{finite}\right\}$$

$$\underset{[F]\in \Gamma \backslash \mathrm{FIN}}{⨆}{\mathrm{Stab}}_{\Gamma }(F)\backslash \left({\left\{1,\dots ,n\right\}}^F\right)\cong \Gamma \backslash \left\{{\pi }_F^{-1}(\phi )\text{: }\phi \in {\left\{1,\dots ,n\right\}}^F,F\subseteq \Gamma \mathrm{finite}\right\},[\phi ]\mapsto [\phi ],$$

$$K_{\ast }\left(C\left({\left\{0,\dots ,n\right\}}^{\Gamma }\right){⋊}_r\Gamma \right)\cong \underset{[F]\in \Gamma \backslash \mathrm{FIN}}{⨁}\underset{[\phi ]\in {\mathrm{Stab}}_{\Gamma }(F)\backslash \left({\left\{1,\dots ,n\right\}}^F\right)}{⨁}{K}_{\ast }(C_{\lambda }^{\ast }({\mathrm{Stab}}_{\Gamma }(\phi ))).$$

$$\underset{[C]\in \mathcal{C}}{⨆}{N}_C\backslash F(C)\to \Gamma \backslash {\mathrm{FIN}}^{\times },[X]\mapsto [C\cdot X]$$

$${\mathrm{Stab}}_{\Gamma }(C\cdot X)=C.$$

$$K_{\ast }((\underset{\Gamma }{⨂}A){⋊}_r\Gamma )\cong K_{\ast }(C_{\lambda }^{\ast }(\Gamma ))\oplus \left(\underset{[C]\in \mathcal{C}}{⨁}\underset{[X]\in N_C\backslash F(C)}{⨁}\underset{[\phi ]\in C\backslash \left({\left\{1,\dots ,n\right\}}^{C\cdot X}\right)}{⨁}{K}_{\ast }(C_{\lambda }^{\ast }({\mathrm{Stab}}_C(\phi )))\right).$$

$$K_1(C_{\lambda }^{\ast }(\Gamma ))\cong K_1((\underset{\Gamma }{⨂}A){⋊}_r\Gamma ).$$

$$K_{\ast }((\underset{\Gamma }{⨂}A){⋊}_r\Gamma )\cong K_{\ast }(C_{\lambda }^{\ast }(\Gamma ))\oplus \left(\underset{[X]\in \Gamma \backslash {\mathrm{FIN}}^{\times }}{⨁}\underset{{\left\{1,\dots ,n\right\}}^X}{⨁}{K}_{\ast }(\mathbb{C})\right).$$

$$K_{\ast }(C_{\lambda }^{\ast }((\underset{\Gamma }{⨁}\Sigma )⋊\Gamma ))\cong K_{\ast }(C_{\lambda }^{\ast }(\Gamma ))\oplus \left(\underset{[C]\in \mathcal{C}}{⨁}\underset{[X]\in N_C\backslash F(C)}{⨁}\underset{[\phi ]\in C\backslash \left(({\mathrm{con}}^{\times }\Sigma {)}^{C\cdot X}\right)}{⨁}{K}_{\ast }(C_{\lambda }^{\ast }({\mathrm{Stab}}_C(\phi )))\right).$$

$$K_1(C_{\lambda }^{\ast }(\Gamma ))\cong K_1(C_{\lambda }^{\ast }((\underset{\Gamma }{⨁}\Sigma )⋊\Gamma )).$$

$$K_{\ast }(C_{\lambda }^{\ast }((\underset{\Gamma }{⨁}\Sigma )⋊\Gamma ))\cong K_{\ast }(C_{\lambda }^{\ast }(\Gamma ))\oplus \left(\underset{[X]\in \Gamma \backslash {\mathrm{FIN}}^{\times }}{⨁}\underset{({\mathrm{con}}^{\times }\Sigma {)}^X}{⨁}{K}_{\ast }(\mathbb{C})\right).$$

Full text

1. Introduction

The classical Lamplighter group is given by the semidirect product $\left(\bigoplus_{\mathbb{Z}}(\mathbb{Z}/2\mathbb{Z})\right)\rtimes\mathbb{Z}$, where the $\mathbb{Z}$-action on $\bigoplus_{\mathbb{Z}}(\mathbb{Z}/2\mathbb{Z})$ is induced by the canonical translation action of $\mathbb{Z}$ on itself. This construction can be generalized by replacing $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}$ by other groups. The classical Lamplighter group and its generalizations are important examples in group theory which led to solutions of several open problems (see for instance [7, 6, 10]).

The goal of these notes is to derive a $K$-theory formula for group C*-algebras of generalized Lamplighter groups of the form $\left(\bigoplus_{\Gamma}\Sigma\right)\rtimes\Gamma$, where $\Sigma$ is an arbitrary finite group and $\Gamma$ is an arbitrary countable group. As in the classical setting, the $\Gamma$-action on $\bigoplus_{\Gamma}\Sigma$ is induced by the canonical left translation action of $\Gamma$ on itself. Our computations are inspired by [9, 13], which treat the special case of free groups $\Gamma$ ([9] deals with the case $\Gamma=\mathbb{Z}$). Our method, however, is completely different from the ones adopted in [9, 13].

Our main result reads as follows. Let $\Sigma$ be a finite group and $\Gamma$ a countable group. Let ${\rm con}\,\Sigma$ be the set of conjugacy classes in $\Sigma$, and ${\rm con}^{\times}\,\Sigma\mathrel{:=}{\rm con}\,\Sigma\setminus\left\{\left\{1\right\}\right\}$ the set of non-trivial conjugacy classes. Let $\mathcal{C}$ be the set of conjugacy classes of finite subgroups of $\Gamma$. For a finite subgroup $C$ of $\Gamma$, let $F(C)$ be the set of non-empty finite subsets of the right coset space $C\backslash\Gamma$ which are not of the form $\pi^{-1}(Y)$ for a finite subgroup $D\subseteq\Gamma$ with $C\subsetneq D$ and $Y\subseteq D\backslash\Gamma$, where $\pi:\>C\backslash\Gamma\twoheadrightarrow D\backslash\Gamma$ is the canonical projection. The normalizer $N_{C}\mathrel{:=}\left\{\gamma\in\Gamma\text{: }\gamma C\gamma^{-1}=C\right\}$ acts on $F(C)$ by left multiplication, and we denote the set of orbits by $N_{C}\backslash F(C)$. Given $X\in F(C)$, we write $C\cdot X\mathrel{:=}\bigsqcup_{x\in X}C\cdot x$ and let $({\rm con}^{\times}\,\Sigma)^{C\cdot X}$ be the set of functions $C\cdot X\to{\rm con}^{\times}\,\Sigma$. $\gamma\in C$ acts on $\varphi\in({\rm con}^{\times}\,\Sigma)^{C\cdot X}$ via $(\gamma.\varphi)(x)=\varphi(\gamma^{-1}x)$, and we set ${\rm Stab}_{C}(\varphi)=\left\{\gamma\in C\text{: }\gamma.\varphi=\varphi\right\}$ for $\varphi\in({\rm con}^{\times}\,\Sigma)^{C\cdot X}$.

Theorem 1.1**.**

If $\Gamma$ satisfies the Baum-Connes conjecture with coefficients, then the K-theory of $C^{*}_{\lambda}((\bigoplus_{\Gamma}\Sigma)\rtimes\Gamma)$ is given by

[TABLE]

Here we take one representative $C$ out of each class in $\mathcal{C}$, one representative $X$ out of each class in $N_{C}\backslash F(C)$, and one representative $\varphi$ out of each class in $C\backslash\left(({\rm con}^{\times}\,\Sigma)^{C\cdot X}\right)$.

We refer the reader to [1, 14, 5] and the references therein for more information about the Baum-Connes conjecture. For instance, Theorem 1.1 applies to all groups with the Haagerup property [11] and all hyperbolic groups [12].

Note that $\Sigma$ enters our formula only in the form of ${\rm con}^{\times}\,\Sigma$. What is more, if $\Gamma$ is infinite, then for each $[C]\in\mathcal{C}$, we simply get a free abelian group of countably infinite rank, so that $K_{*}(C^{*}_{\lambda}((\bigoplus_{\Gamma}\Sigma)\rtimes\Gamma))$ does not depend on $\Sigma$ at all. This becomes particularly evident in $K_{1}$, where Theorem 1.1 yields the following

Corollary 1.2**.**

Let $\Sigma$ be a finite group and $\Gamma$ a countable group. If $\Gamma$ satisfies the Baum-Connes conjecture with coefficients, then the canonical inclusion $\Gamma\hookrightarrow\Sigma\rtimes\Gamma$ induces an isomorphism

[TABLE]

Moreover, for torsion-free $\Gamma$, our formula becomes particularly simple.

Corollary 1.3**.**

Let $\Sigma$ and $\Gamma$ be as in Theorem 1.1. Assume that $\Gamma$ is torsion-free. Write ${\rm FIN}^{\times}$ for the set of non-empty finite subsets of $\Gamma$. Then, under the same assumptions as in Theorem 1.1, we have

[TABLE]

The proof of our main theorem proceeds in two steps. First, using the Going-Down principle from [2, 8] (see also [5, § 3]), we show that $C^{*}_{\lambda}((\bigoplus_{\Gamma}\Sigma)\rtimes\Gamma)$ has the same K-theory as the crossed product $C(({\rm con}\,\Sigma)^{\Gamma})\rtimes_{r}\Gamma$ for the topological full shift $\Gamma\curvearrowright({\rm con}\,\Sigma)^{\Gamma}$. Here we view ${\rm con}\,\Sigma$ as a finite alphabet. Secondly, we compute K-theory for $C(({\rm con}\,\Sigma)^{\Gamma})\rtimes_{r}\Gamma$ using [3, 4]. As a by-product, we obtain a general K-theory formula for crossed products of topological full shifts (see Proposition 2.4). Both steps require our assumption that $\Gamma$ satisfies the Baum-Connes conjecture with coefficients.

We point out that it is not possible to apply the results in [3, 4] directly because [3, 4] only deal with crossed products attached to actions on commutative C*-algebras.

I am indebted to Alain Valette for inviting me to Neuchatel, and to Sanaz Pooya and Alain Valette for interesting discussions which led to these notes. Moreover, I thank the referee for helpful comments and careful proofreading.

2. K-theory for certain crossed products and generalized Lamplighter groups

We first discuss the following abstract situation: Let $A=\bigoplus_{i=0}^{n}M_{k_{i}}$ be a finite dimensional C*-algebra, where $M_{k}$ is the algebra of $k\times k$-matrices. We assume that $k_{0}=1$, i.e., $A=\mathbb{C}\oplus M_{k_{1}}\oplus\dotso\oplus M_{k_{n}}$. Let $\Gamma$ be a countable group. We form the tensor product $\bigotimes_{\Gamma}A$ as follows: For every finite subset $F\subseteq\Gamma$, we form the ordinary tensor product $\bigotimes_{F}A$, and for $F_{1}\subseteq F_{2}$, we have the canonical embedding $\bigotimes_{F_{1}}A\hookrightarrow\bigotimes_{F_{2}}A,\,x\mapsto x\otimes 1$ (here $1$ denotes the unit of $\bigotimes_{F_{2}\setminus F_{1}}A$, and we used the canonical isomorphism $\bigotimes_{F_{2}}A\cong\left(\bigotimes_{F_{1}}A\right)\otimes\left(\bigotimes_{F_{2}\setminus F_{1}}A\right)$). Then set $\bigotimes_{\Gamma}A\mathrel{:=}\varinjlim_{F}\bigotimes_{F}A$. The left $\Gamma$-action on itself by translations induces an action $\Gamma\curvearrowright\bigotimes_{\Gamma}A$. Our goal is to compute the $K$-theory of $\left(\bigotimes_{\Gamma}A\right)\rtimes_{r}\Gamma$. The special case $A=C^{*}_{\lambda}(\Sigma)$ will lead to Theorem 1.1.

Let $e_{i}$ be a minimal projection in $M_{k_{i}}\subseteq A$. In particular, $e_{0}=1\in\mathbb{C}\subseteq A$. For $F\subseteq\Gamma$ finite, let $\varphi\in\left\{1,\dotsc,n\right\}^{F}$, i.e., $\varphi$ is a function $\varphi:\>F\to\left\{1,\dotsc,n\right\}$. Define $e_{\varphi}\mathrel{:=}\bigotimes_{f\in F}e_{\varphi(f)}\in\bigotimes_{F}A\subseteq\bigotimes_{\Gamma}A$. If $F=\emptyset$, then for $\varphi:\>\emptyset\to\left\{1,\dotsc,n\right\}$, we set $e_{\varphi}\mathrel{:=}1$ (where $1$ denotes the unit of $\bigotimes_{\Gamma}A$). The set

[TABLE]

is a $\Gamma$-invariant family of commuting non-zero projections, which is closed under multiplication up to zero (i.e., the product of two projections in the family is either zero or again a projection in the family). We do not need it now, but the family is also linearly independent (see Lemma 2.3 and the proof of (2)). Let $D$ be the C*-subalgebra of $\bigotimes_{\Gamma}A$ generated by the projections in (1). Let $\iota:\>D\hookrightarrow\bigotimes_{\Gamma}A$ be the canonical embedding. Note that $\iota$ is $\Gamma$-equivariant.

Proposition 2.1**.**

If $\Gamma$ satisfies the Baum-Connes conjecture with coefficients, then $\iota\rtimes_{r}\Gamma$ induces an isomorphism $K_{*}(D\rtimes_{r}\Gamma)\cong K_{*}(\left(\bigotimes_{\Gamma}A\right)\rtimes_{r}\Gamma)$.

Proof.

By the Going-Down principle (see [5, § 3]), it suffices to show that for every finite subgroup $H\subseteq\Gamma$, $\iota\rtimes_{r}H$ induces an isomorphism $K_{*}(D\rtimes_{r}H)\cong K_{*}(\left(\bigotimes_{\Gamma}A\right)\rtimes_{r}H)$.

Let us first treat the case of the trivial subgroup, $H=\left\{1\right\}$. For a fixed finite subset $F\subseteq\Gamma$, let

[TABLE]

Then $D=\varinjlim_{F}D_{F}$. We also have $\bigotimes_{\Gamma}A=\varinjlim_{F}\bigotimes_{F}A$. As $K$-theory is continuous, i.e., preserves direct limits, it suffices to show that $\iota_{F}\mathrel{:=}\iota|_{D_{F}}:\>D_{F}\to\bigotimes_{F}A$ induces an isomorphism in $K_{*}$. Let $[\iota_{F}]\in KK(D_{F},\bigotimes_{F}A)$ be the $KK$-element determined by $\iota_{F}$. Consider the projection $e=\sum_{i=0}^{n}e_{i}$ in $A$. $e$ is a full projection in $A$, and we have $eAe=\bigoplus_{i=0}^{n}\mathbb{C}e_{i}$. The $\bigotimes_{F}A$–$\bigotimes_{F}eAe$-imprimitivity bimodule $\bigotimes_{F}Ae$ gives rise to a $KK$-element $\mathbf{j}_{F}\in KK(\bigotimes_{F}A,\bigotimes_{F}eAe)$. $\mathbf{j}_{F}$ is invertible, and its inverse is the $KK$-element induced by the inclusion $\bigotimes_{F}eAe\hookrightarrow\bigotimes_{F}A$. Hence it suffices to show that the Kasparov product $[\iota_{F}]\cdot\mathbf{j}_{F}\in KK(D_{F},\bigotimes_{F}eAe)$ induces an isomorphism $K_{*}(D_{F})\to K_{*}(\bigotimes_{F}eAe)$.

First, consider the special case of a single element subset, $F=\left\{f\right\}$ for some $f\in\Gamma$. Let us write $D_{f}\mathrel{:=}D_{\left\{f\right\}}$, $\iota_{f}\mathrel{:=}\iota_{\left\{f\right\}}$ and $\mathbf{j}_{f}\mathrel{:=}\mathbf{j}_{\left\{f\right\}}$. Since $D_{f}=\mathbb{C}\cdot 1+\mathbb{C}e_{1}+\dotso+\mathbb{C}e_{n}$ (where $1$ denotes the unit of $\bigotimes_{\Gamma}A$) and $eAe=\mathbb{C}e_{0}\oplus\mathbb{C}e_{1}\oplus\dotso\oplus\mathbb{C}e_{n}$, we can describe the map $K_{*}(D_{F})\to K_{*}(\bigotimes_{F}eAe)$ induced by $[\iota_{f}]\cdot\mathbf{j}_{f}$ by the commutative diagram

[TABLE]

where the upper horizontal map is the map we want to describe, and $M_{f}$ is the $(n+1)\times(n+1)$-matrix

[TABLE]

Obviously, $M_{f}$ is invertible. Note that everything is independent of $f$.

Now consider the case of a general finite subset $F\subseteq\Gamma$. Since $D_{F}=\bigotimes_{f\in F}D_{f}$, we have $K_{*}(D_{F})\cong\bigotimes_{f\in F}K_{*}(D_{f})$, and we also have $K_{*}(\bigotimes_{F}eAe)\cong\bigotimes_{f\in F}K_{*}(eAe)$. The homomorphism $K_{*}(D_{F})\to K_{*}(\bigotimes_{F}eAe)$ induced by $[\iota_{F}]\cdot\mathbf{j}_{F}$ respects this tensor product decomposition, in the sense that we have a commutative diagram

[TABLE]

Again, we see that $M_{F}$ is invertible because all the $M_{f}$, $f\in F$, are.

Now let us deal with the case of an arbitrary finite subgroup $H\subseteq\Gamma$. If we choose an increasing sequence of $H$-invariant finite subsets $F\subseteq\Gamma$ whose union is $\Gamma$, we obtain $H$-equivariant inductive limit decompositions $D=\varinjlim_{F}D_{F}$ and $\bigotimes_{\Gamma}A=\varinjlim_{F}\bigotimes_{F}A$. Hence, again by continuity of $K$-theory, it suffices to show that, for every $F$, $\iota_{F}\rtimes_{r}H:\>D_{F}\rtimes_{r}H\to\left(\bigotimes_{F}A\right)\rtimes_{r}H$ induces an isomorphism in $K_{*}$. Let $\mathbf{j}_{F}\in KK(\bigotimes_{F}A,\bigotimes_{F}eAe)$ be as before. Since the full projection $\bigotimes_{F}e\in\bigotimes_{F}A$ giving rise to $\mathbf{j}_{F}$ is $H$-invariant, $\mathbf{j}_{F}$ is a $KK^{H}$-equivalence (see [5, Remark 3.3.16]). Thus, to show that $\iota_{F}\rtimes_{r}H:\>D_{F}\rtimes_{r}H\to\left(\bigotimes_{F}A\right)\rtimes_{r}H$ induces an isomorphism in $K_{*}$, it suffices to show that $[\iota_{F}]\cdot\mathbf{j}_{F}\in KK^{H}(D_{F},\bigotimes_{F}eAe)$ induces an isomorphism $K_{*}(D_{F}\rtimes_{r}H)\to K_{*}(\left(\bigotimes_{F}eAe\right)\rtimes_{r}H)$, for which in turn it is enough to prove that $[\iota_{F}]\cdot\mathbf{j}_{F}$ is a $KK^{H}$-equivalence.

Now both $D_{F}$ and $\bigotimes_{F}eAe$ are finite dimensional commutative C*-algebras with an $H$-action, so that we are exactly in the setting of [4, Appendix]. It is straightforward to check that $[\iota_{F}]\cdot\mathbf{j}_{F}=x^{H}_{M_{F}}$, where $x^{H}_{M_{F}}$ is the element in $KK^{H}(D_{F},\bigotimes_{F}eAe)$ corresponding to the matrix $M_{F}$, as constructed in [4, Appendix]. By [4, Lemma A.2], $x^{H}_{M_{F}}$ is a $KK^{H}$-equivalence because $M_{F}$ is an invertible matrix. The inverse of $x^{H}_{M_{F}}$ is given by $x^{H}_{M_{F}^{-1}}$. ∎

Remark 2.2**.**

Note that our assumption on $A$ that $\mathbb{C}$ appears as a direct summand is really necessary. For instance, if $A=M_{2}$, then $\bigotimes_{\Gamma}A$ would be the UHF algebra $M_{2^{\infty}}$ (as soon as $\Gamma$ is infinite). But we have $K_{0}(M_{2^{\infty}})\cong\mathbb{Z}[\frac{1}{2}]$, while our method would always yield a free abelian group for $K_{0}$. Hence our method fails.

Let us now compare with the topological full shift $\Gamma\curvearrowright\left\{0,\dotsc,n\right\}^{\Gamma}$. For a finite subset $F\subseteq\Gamma$, let $\pi_{F}$ be the canonical projection $\left\{0,\dotsc,n\right\}^{\Gamma}\twoheadrightarrow\left\{0,\dotsc,n\right\}^{F}$. Given $\varphi\in\left\{0,\dotsc,n\right\}^{F}$, we have the cylinder set $\pi_{F}^{-1}(\varphi)$ and its characteristic function $1_{\pi_{F}^{-1}(\varphi)}\in C(\left\{0,\dotsc,n\right\}^{\Gamma})$. The following is now easy to see.

Lemma 2.3**.**

The $\Gamma$-equivariant isomorphism $D\cong C(\left\{0,\dotsc,n\right\}^{\Gamma}),\,e_{\varphi}\mapsto 1_{\pi_{F}^{-1}(\varphi)}$ induces an isomorphism $D\rtimes_{r}\Gamma\cong C(\left\{0,\dotsc,n\right\}^{\Gamma})\rtimes_{r}\Gamma$.

We now compute K-theory for $C(\left\{0,\dotsc,n\right\}^{\Gamma})\rtimes_{r}\Gamma$.

Proposition 2.4**.**

If $\Gamma$ satisfies the Baum-Connes conjecture with coefficients, then

[TABLE]

Here we use the same notation as in Theorem 1.1.

Proof.

First of all, the same arguments as for [4, Examples 2.13 & 3.1] show that the family

[TABLE]

is a $\Gamma$-invariant regular basis for the compact open sets in $\left\{0,\dotsc,n\right\}^{\Gamma}$. Here $\Gamma$ acts via $\gamma.\pi_{F}^{-1}(\varphi)=\pi_{\gamma\cdot F}^{-1}(\gamma.\varphi)$, where $\gamma.\varphi\in\left\{1,\dotsc,n\right\}^{\gamma\cdot F}$ is given by $(\gamma.\varphi)(x)=\varphi(\gamma^{-1}x)$. Therefore, using the bijection

[TABLE]

and the observation that for $\gamma\in\Gamma$ and $\varphi\in\left\{1,\dotsc,n\right\}^{F}$, we have $\gamma.\pi_{F}^{-1}(\varphi)=\pi_{F}^{-1}(\varphi)$ if and only if $\gamma\cdot F=F$ and $\gamma.\varphi=\varphi$, we may apply [4, Corollary 3.14], and obtain

[TABLE]

Here ${\rm FIN}$ is the set of all finite subsets of $\Gamma$, and $\Gamma\backslash{\rm FIN}$ is the set of orbits of the left translation action $\Gamma\curvearrowright{\rm FIN}$. Moreover, ${\rm Stab}_{\Gamma}(F)$ and ${\rm Stab}_{\Gamma}(\varphi)$ denote the stabilizer groups ${\rm Stab}_{\Gamma}(F)=\left\{\gamma\in\Gamma\text{: }\gamma\cdot F=F\right\}$ and ${\rm Stab}_{\Gamma}(\varphi)=\left\{\gamma\in\Gamma\text{: }\gamma.\varphi=\varphi\right\}$, and $C^{*}_{\lambda}({\rm Stab}_{\Gamma}(\varphi))$ is the reduced group C*-algebra of ${\rm Stab}_{\Gamma}(\varphi)$.

Now we analyse $\Gamma\backslash{\rm FIN}$ and ${\rm Stab}_{\Gamma}(F)$ for $[F]\in\Gamma\backslash{\rm FIN}$. For $F=\emptyset$, we have ${\rm Stab}_{\Gamma}(\varphi)=\Gamma$. This yields $K_{*}(C^{*}_{\lambda}(\Gamma))$ as one direct summand on the right-hand side of (2). We set ${\rm FIN}^{\times}\mathrel{:=}{\rm FIN}\setminus\left\{\emptyset\right\}$. Let us describe $\Gamma\backslash{\rm FIN}^{\times}$. Let $\mathcal{C}$, $F(C)$ and $N_{C}$ be as in Theorem 1.1. Then we claim that

[TABLE]

is a bijection, and that for every $[C]\in\mathcal{C}$, $[X]\in N_{C}\backslash F(C)$, we have

[TABLE]

First note that the map (3) is well-defined. Moreover, this map is surjective because every $F\in{\rm FIN}^{\times}$ with ${\rm Stab}_{\Gamma}(F)=C$ is of the form $F=C\cdot X$ for some finite, non-empty subset $X\subseteq C\backslash\Gamma$. Now, $X$ must lie in $F(C)$. Suppose not, i.e., $X=\pi^{-1}(Y)$ for a finite subgroup $D\subseteq\Gamma$ with $C\subsetneq D$ and $Y\subseteq D\backslash\Gamma$, where $\pi:\>C\backslash\Gamma\twoheadrightarrow D\backslash\Gamma$ is the canonical projection. Then $F=C\cdot X=D\cdot Y$, so that $D\subseteq{\rm Stab}_{\Gamma}(F)$, in contradiction to ${\rm Stab}_{\Gamma}(F)=C$. Not only does this show surjectivity, but it also proves (4). To see injectivity of (3), assume that $X\in F(C)$ and $X^{\prime}\in F(C^{\prime})$ satisfy $[C\cdot X]=[C^{\prime}\cdot X^{\prime}]$, say $C^{\prime}\cdot X^{\prime}=\gamma\cdot C\cdot X$. It follows that $C^{\prime}={\rm Stab}_{\Gamma}(C^{\prime}\cdot X^{\prime})=\gamma{\rm Stab}_{\Gamma}(C\cdot X)\gamma^{-1}=\gamma C\gamma^{-1}$. Hence $[C]=[C^{\prime}]$, and since we are taking one representative out of each class, we must actually have $C=C^{\prime}$. But then $\gamma$ must lie in $N_{C}$, and we must have $C\cdot X^{\prime}=\gamma\cdot C\cdot X=C\cdot\gamma\cdot X$, so that $X^{\prime}=\gamma\cdot X$, i.e., $[X^{\prime}]=[X]$ in $N_{C}\backslash F(C)$. This shows injectivity.

We now complete the proof by plugging the bijections (3), (4) into (2) and observing that for $X\in F(C)$ and $\varphi\in\left\{1,\dotsc,n\right\}^{C\cdot X}$, we have ${\rm Stab}_{\Gamma}(\varphi)\subseteq{\rm Stab}_{\Gamma}(C\cdot X)=C$. ∎

Combining Proposition 2.1, Lemma 2.3 and Proposition 2.4, and using the concrete construction in [4, § 3] for our following assertion on $K_{1}$, we obtain

Corollary 2.5**.**

In the situation of Proposition 2.1, if $\Gamma$ satisfies the Baum-Connes conjecture with coefficients, then we have

[TABLE]

In $K_{1}$, the canonical map $C^{*}_{\lambda}(\Gamma)\hookrightarrow(\bigotimes_{\Gamma}A)\rtimes_{r}\Gamma$ induces an isomorphism

[TABLE]

If $\Gamma$ is in addition torsion-free, then we obtain

[TABLE]

Now let us apply our $K$-theory formula to generalized Lamplighter groups. Consider the case $A=C^{*}_{\lambda}(\Sigma)$ for a finite group $\Sigma$. Our assumption on $A$ that $\mathbb{C}$ appears as a direct summand is satisfied because the trivial representation gives rise to a direct summand $\mathbb{C}$ in $C^{*}_{\lambda}(\Sigma)$. The remaining direct summands of $A$ are in one-to-one correspondence with ${\rm con}^{\times}\,\Sigma$. Hence we obtain

Corollary 2.6**.**

Let $\Sigma$ be a finite group. If $\Gamma$ satisfies the Baum-Connes conjecture with coefficients, then we have

[TABLE]

In $K_{1}$, the canonical inclusion $\Gamma\hookrightarrow\Sigma\rtimes\Gamma$ induces an isomorphism

[TABLE]

If $\Gamma$ is in addition torsion-free, then we obtain

[TABLE]

This completes the proofs of Theorem 1.1, Corollary 1.2 and Corollary 1.3.

Remark 2.7**.**

As in [4, Corollary 3.14], we get $KK$-equivalences in Proposition 2.4, Corollary 2.5 and Corollary 2.6 if $\Gamma$ satisfies the strong Baum-Connes conjecture.

Remark 2.8**.**

Moreover, as in [4, Corollary 3.14], we could allow coefficients in Proposition 2.4, Corollary 2.5 and Corollary 2.6. However, in Corollary 2.6, we would only get a $K$-theory formula for crossed products $B\rtimes_{r}\left((\bigoplus_{\Gamma}\Sigma)\rtimes\Gamma\right)$ where the $(\bigoplus_{\Gamma}\Sigma)$-action on the C*-algebra $B$ is trivial.