Abstract
For a large class of \(C^*\)algebras A, we calculate the Ktheory of reduced crossed products \(A^{\otimes G}\rtimes _rG\) of Bernoulli shifts by groups satisfying the Baum–Connes conjecture. In particular, we give explicit formulas for finitedimensional \(C^*\)algebras, UHFalgebras, rotation algebras, and several other examples. As an application, we obtain a formula for the Ktheory of reduced \(C^*\)algebras of wreath products \(H\wr G\) for large classes of groups H and G. Our methods use a generalization of techniques developed by the second named author together with Joachim Cuntz and Xin Li, and a trivialization theorem for finite group actions on UHF algebras developed in a companion paper by the third and fourth named authors.
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1 Introduction
Let G be a countable discrete group and let A be a separable unital \(C^*\)algebra. Equip the infinite tensor product \(A^{\otimes G}\) with the natural Bernoulli shift action (see Sect. 2 for the definition). The objective of this paper is to compute the Ktheory group of the reduced crossed product \(A^{\otimes G}\rtimes _r G\) in as many cases as possible.
Building up on the work in [6], Xin Li [20] computed this when A is a finitedimensional \(C^*\)algebra of the form \(\mathbb {C}\oplus \bigoplus _{1\le k\le N} M_{p_k}\), assuming the Baum–Connes conjecture with coefficients (referred to as BCC below) for G [1]. His motivation was to compute the Ktheory of the reduced group \(C^*\)algebra of the wreath product \(H\wr G\) for an arbitrary finite group H. For \(A=\mathbb {C}\oplus \bigoplus _{1\le k\le N} M_{p_k}\), the Ktheory groups \(K_*\left( A^{\otimes G}\rtimes _rG\right) \) are computed in [20] as
where \(\textrm{FIN}\) is the set of all the finite subsets of G equipped with the lefttranslation Gaction and \(G_F:=\textrm{Stab}_G(F)\), resp. \(G_S:=\textrm{Stab}_G(S)\), are the stabilizer groups at F, resp. at S, for the action of G on \(\textrm{FIN}\). Note that when G is nontrivial and torsion free, (1.1) is the direct sum of \(K_*(C^*_r(G))\) and infinitely many copies of \(K_*(\mathbb {C})\). For wreath products \(H\wr G\) with respect to a finite group H, we have \(C^*_r(H\wr G)\cong C^*_r(H)^{\otimes G}\rtimes _r G\). In this case, the number N in (1.1) corresponds to the number of nontrivial conjugacy classes of H. Our results were also motivated in part by the paper [22] by Issei Ohhashi, where he gives Ktheory computations for crossed products \(A^{\otimes \mathbb {Z}}\rtimes \mathbb {Z}\) for Bernoulli shifts by the integer group \(\mathbb {Z}\).
Our first result computes \(K_*\left( A^{\otimes Z}\rtimes _rG\right) \) for an arbitrary finitedimensional \(C^*\)algebra A and for an arbitrary infinite countable Gset Z with the action of G on \(A^{\otimes Z}\) induced by the Gaction on Z. In what follows we let \(\textrm{FIN}(Z)\) denote the collection of finite subsets of Z and we put \(\textrm{FIN}^\times (Z):=\textrm{FIN}(Z){\setminus }\{\emptyset \}\). In the case \(Z=G\) we shall simply write \(\textrm{FIN}\) and \(\textrm{FIN}^\times \) as above.
Theorem A
(Theorem 3.6) Let G be a discrete group satisfying BCC and let Z be a countably infinite Gset. Let \(A=\bigoplus _{0\le j \le N}{M_{k_j}}\) where \(k_0, \ldots , k_N\) (\(N\ge 1\)) are positive integers with \(\gcd (k_0, \ldots , k_N)=n\). We have an explicit isomorphism
For \(N=0\), we have
Our strategy of the proof is as follows: First, we consider the case \(n=1\) and construct a unitpreserving KKequivalence \(\mathbb {C}^{N+1}\sim _{KK} A\) using an arithmetic argument about matrices in \(\textrm{SL}(n,\mathbb {Z})\) with nonnegative entries. By an argument borrowed from [14, 25], this allows us to replace A by \(\mathbb {C}^{N+1}\) and apply Xin Li’s formula (1.1). Second, we reduce the general case to the case \(n=1\) with the help of a trivialization theorem for finite group actions on UHF algebras developed in the companion paper [18].
One of our main techniques is the following theorem which is inspired by the regular basis technique from [6].
Theorem B
(Theorem 2.8, Corollary 2.10) Let G be a discrete group satisfying BCC. Let A be a separable, unital \(C^*\)algebra that satisfies the Universal Coefficient Theorem (UCT) and such that the unital inclusion \(\iota :\mathbb {C}\rightarrow A\) induces a splitinjection \(K_0(\mathbb {C}) \rightarrow K_0(A)\). Then \(K_*\left( A^{\otimes G} \rtimes _rG\right) \) only depends on G and the cokernel \({\tilde{K}}_*(A)\) of \(\iota _*:K_*(\mathbb {C}) \rightarrow K_*(A)\). For any countable Gset Z we have
where B is any \(C^*\)algebra satisfying the UCT with Ktheory isomorphic to \({\tilde{K}}_*(A)\)^{Footnote 1} and where \(G_F=\textrm{Stab}_G(F)\). In particular, if G is torsionfree and the action of G on Z is free, we have
Note that the second formula can be computed more explicitly using the Künneth theorem. In the important case that \(Z=G\) with the left translation action (or, more generally, if G acts properly on Z) the stabilizers \(G_F\) for \(F\in \textrm{FIN}^\times (Z)\) are all finite and the formula becomes more explicit once we can compute \(K_*\left( B^{\otimes H}\rtimes _rH\right) \) for finite groups H and for relevant building blocks for B like \(\mathbb {C}\), \(C_0(\mathbb {R})\), the Cuntzalgebras \(\mathcal {O}_n\), and \(C_0(\mathbb {R})\otimes \mathcal {O}_n\). For \(H=\mathbb {Z}/2\), these computations have been done by Izumi [14]. For general H, computing these Kgroups is a nontrivial, indeed challenging task. For \(B=C_0(\mathbb {R})\) however, they are nothing but the equivariant topological Ktheory \(K^*_H\left( \mathbb {R}^H\right) \) for the HEuclidean space \(\mathbb {R}^{H}\), i.e., \(\mathbb {R}^{\vert H\vert }\) with Haction induced by translation of coordinates. The groups \(K^*_H\left( \mathbb {R}^H\right) \) are quite wellstudied [10, 15]. Using these results, we give a more explicit formula for \(A=C(S^1)\) (Example 4.2) and for the rotation algebras (or noncommutative tori) \(A=A_\theta \) (Example 4.4). More explicitly, we have
Theorem C
(Example 4.2) Let G be a discrete group satisfying BCC. We have
As another application of Theorem B, we obtain a formula for the Ktheory of reduced \(C^*\)algebras of many wreath products \(H\wr G\):
Theorem D
(Theorem 4.14) Let G be a discrete group satisfying BCC and let H be a discrete group such that \(C^*_r(H)\) satisfies the UCT and such that the inclusion \(\mathbb {C}\rightarrow C^*_r(H)\) induces a split injection on \(K_0\). Then we have
where B is any \(C^*\)algebra satisfying the UCT with Ktheory isomorphic to \({\tilde{K}}_*(C^*_r(H))\). In particular, if G is torsionfree, we have
We note that both the UCT assumption on \(C_r^*(H)\) and the splitinjectivity of the map \(K_0(\mathbb {C})\rightarrow K_0(C_r^*(H))\) hold for every aTmenable (in particular every amenable) group H (see Remark 4.15 below).
In Sect. 4, we also obtain formulas for several \(C^*\)algebras A that are not covered in Theorem B, in particular for Cuntz algebras.
Theorem E
(Corollary 4.10, Proposition 4.11) Let G be a discrete group satisfying BCC and let \(n\ge 2\). Then we have
where \(\mathcal {O}_{n+1}\) is the Cuntz algebra on \(n+1\) generators. If G is finite and \(n=p\) prime, then \(K_*\left( \mathcal {O}_{p+1}^{\otimes G}\rtimes _r G\right) \) is a finitely generated pgroup, that is a group of the form \(\bigoplus _{1\le j\le N}\mathbb {Z}/p^{k_j}\mathbb {Z}\).
In Sect. 5, we obtain a Ktheory formula for Bernoulli shifts on unital AFalgebras in terms of colimits over the orbit category \(\textrm{Or}_{\mathcal {FIN}}(G)\). The result also applies to more general examples, in particular to many unital ASHalgebras (see Remark 5.5).
Theorem F
(Theorem 5.4) Let A be a unital AFalgebra, let G be an infinite discrete group satisfying BCC, let Z be a countable proper Gset, and let \(S\subseteq \mathbb {Z}\) be the set of all positive integers n such that \([1_A]\in K_0(A)\) is divisible by n. Then, the natural inclusions
induce the following pushout diagram
In particular, if G is torsionfree, this pushout diagram reads
where \({\tilde{K}}_*(C^*_r(G))\) denotes the cokernel of the map \(K_*(\mathbb {C})\rightarrow K_*(C_r^*(G))\) induced from the unital inclusion \(\mathbb {C}\hookrightarrow C_r^*(G)\).
The paper is structured as follows: In Sect. 2, we develop our main machinery, including Theorem B. We apply this machinery in Sect. 3 to prove Theorem A. In Sect. 4, we compute the Ktheory of many more examples, including Theorems C, D, and E. Bernoulli shifts on unital AFalgebras are investigated in Sect. 5 where we prove Theorem F. Based on similar ideas, we obtain in Sect. 6 some very general Ktheory formulas up to inverting an integer k or up to tensoring with the rationals \(\mathbb {Q}\) (see Theorems 6.3, 6.4). The rational Ktheory computations apply to all unital stably finite \(C^*\)algebras satisfying the UCT.
2 General strategy
To avoid technical complications with KKtheory, we assume throughout that all \(C^*\)algebras are separable except for the algebra \({\mathcal {B}}(H)\) of bounded operators on a Hilbert space H. For a \(C^*\)algebra A and a finite set F, we write \(A^{\otimes F}\) to denote the minimal tensor product \(\otimes _{x\in F}A\). If A is moreover unital and Z is a (not necessarily finite) countable set, we denote by \(A^{\otimes Z}\) the filtered colimit^{Footnote 2} taken over all finite subsets \(F\subseteq Z\) with respect to the connecting maps \(A^{\otimes F}\ni x\mapsto x\otimes 1\in A^{\otimes F}\otimes A^{\otimes F' F}\cong A^{\otimes F'}\), for finite sets \(F, F'\) with \(F\subseteq F'.\) Hence \(A^{\otimes Z}\) is the closed linear span of the elementary tensors \(\bigotimes _{z \in Z} a_z\), where \(a_z \in A\) for all \(z \in Z,\) and \(a_z=1\) for all but finitely many z. If G is a discrete group acting on Z, we call the Gaction on \(A^{\otimes Z}\) given by permutation of the tensor factors the Bernoulli shift. More explicitly, \(g\bigg (\bigotimes _{z \in Z} a_z\bigg )=\bigotimes _{z \in Z} a_{g^{1} z}\) for an elementary tensor \(\bigotimes _{z \in Z} a_z \in A^{\otimes Z}\).
For a countable group G and a G\(C^*\)algebra A, the Baum–Connes conjecture with coefficients (BCC) predicts a formula for the Ktheory \(K_*(A\rtimes _r G)\) of the reduced crossed product \(A\rtimes _r G\), see [1]. The precise formulation of the conjecture is not too important for us. We mostly need one of its consequences recalled in Theorem 2.3 below. Note that the Baum–Connes conjecture with coefficients has been verified for many groups, including all aTmenable groups [12] and all hyperbolic groups [19].
We refer the reader to [16] for the definition and basic properties of the equivariant KKgroups \(KK^G(A,B)\). Recall from [21] that \(KK^G\) can be organized into a triangulated category^{Footnote 3} with G\(C^*\)algebras as objects, the groups \(KK^G(A,B)\) as morphism sets, and composition given by the Kasparov product. The construction of \(KK^G\)elements from G\(*\)homomorphisms may be interpreted as a functor from the category of G\(C^*\)algebras to the category \(KK^G\). Furthermore, \(A\mapsto K_*(A\rtimes _r G)\) factorizes through this functor.
Definition 2.1
A morphism \(\phi \) in \(KK^G(A, B)\) is a weak Kequivalence in \(KK^G\) if its restrictions to \(KK^H(A,B)\) induce isomorphisms
for all finite subgroups H of G.
Remark 2.2
Weak Kequivalences are in general weaker than weak equivalences in \(KK^G\) in the sense of [21]. The latter ones are required to induce a \(KK^H\)equivalence for all finite subgroups H of G.
The following theorem has been shown in [5] in the setting of locally compact groups (see also [21]). A detailed proof in the (easier) discrete case is given in [7, Section 3.5].
Theorem 2.3
Suppose that the Baum–Connes conjecture holds for G with coefficients in A and B. Then, any weak Kequivalence \(\phi \) in \(KK^G(A, B)\) induces an isomorphism
Let A be a unital \(C^*\)algebra and let G be a countable group satisfying BCC. Our general strategy to compute \(K_*(A^{\otimes Z}\rtimes _r G)\) is to replace \(A^{\otimes Z}\) by a weakly Kequivalent G\(C^*\)algebra with computable Ktheory and then apply Theorem 2.3 above.
The following lemma and corollary are straightforward generalizations of [14, Theorem 2.1] and [25, Corollary 6.9].
Lemma 2.4
(see [14, Theorem 2.1]) Let A and B be not necessarily unital \(C^*\)algebras, let G be a countable group, and let F be a finite Gset. Then there is a map from KK(A, B) to \(KK^G(A^{\otimes F}, B^{\otimes F})\) which sends the class of a \(*\)homomorphism \(\phi :A\rightarrow B\) to the class of \(\phi ^{\otimes F}\). Furthermore, this map is compatible with compositions (i.e. Kasparov products) and in particular sends KKequivalences to \(KK^G\)equivalences. In particular, the Bernoulli shifts on \(A^{\otimes F}\) and \(B^{\otimes F}\) are \(KK^G\)equivalent if A and B are KKequivalent. The analogous statement holds if we replace the minimal tensor product \(\otimes \) by the maximal one.
Proof
We recall the description of \(KK^G\) in terms of asymptotic morphisms which admit an equivariant c.c.p. lift [26] (see also Appendix in [17]). For any G\(C^*\)algebras \(A_0\) and \(B_0\), let \([[A_0, B_0]]^G_{\textrm{cp}}\) be the set of homotopy equivalence classes of completely positive equivariant asymptotic homomorphisms from \(A_0\) to \(B_0\), see [26, Section 2]. The usual composition law for asymptotic homomorphisms restricts to \([[A_0, B_0]]^G_{\textrm{cp}}\), [26, Theorem 2]. Let S be the suspension functor \(SA_0=C_0(\mathbb {R})\otimes A_0\). Denote by \({\tilde{\mathcal {K}}}_G\) the \(C^*\)algebra of compact operators on \(\ell ^2(G\times \mathbb {N})\). It is a G\(C^*\)algebra with respect to the regular representation on \(\ell ^2(G)\). The assignment \((A_0, B_0) \mapsto {\widetilde{KK}}^G(A_0, B_0) = [[SA_0\otimes {\tilde{\mathcal {K}}}_G, SB_0\otimes {\tilde{\mathcal {K}}}_G]]^G_{\textrm{cp}}\), \((\phi :A_0\rightarrow B_0)\mapsto S\phi \otimes \textrm{id}_{ {\tilde{\mathcal {K}}}_G}\) defines a bifunctor from G\(C^*\)algebras to abelian groups. Thomsen showed that there is a natural isomorphism \({\widetilde{KK}}^G(A_0, B_0)\cong KK^G(A_0, B_0)\) of bifunctors that sends the composition product of asymptotic morphisms to the Kasparov product, see [26, Theorem 4.8]. Now, given \(\phi \in KK(A,B)\), we represent \(\phi \) as a completely positive asymptotic homomorphism from \(SA\otimes \mathcal {K}(\ell ^2(\mathbb {N}))\) to \(SB\otimes \mathcal {K}(\ell ^2(\mathbb {N}))\). Taking the (pointwise) minimal tensor product of \(\phi \) with itself over F, we obtain a completely positive equivariant asymptotic homomorphism \(\phi ^{\otimes F}\) from \((SA\otimes \mathcal {K}(\ell ^2(\mathbb {N})))^{\otimes F}\) to \((SB\otimes \mathcal {K}(\ell ^2(\mathbb {N})))^{\otimes F}\). This construction clearly respects homotopy equivalences. Therefore, \([\phi ]\mapsto [\phi ^{\otimes F}]\) defines a map \(KK(A, B)\rightarrow KK^G((SA\otimes \mathcal {K}(\ell ^2(\mathbb {N})))^{\otimes F}, (SB\otimes \mathcal {K}(\ell ^2(\mathbb {N})))^{\otimes F})\). By construction, this map is compatible with compositions and sends a \(*\)homomorphism \(\phi :A\rightarrow B\) to \((S\phi \otimes \textrm{id}_{\mathcal {K}(\ell ^2(\mathbb {N}))})^{\otimes F}\). By the stabilization theorem and by Kasparov’s Bottperiodicity theorem^{Footnote 4} (see [3, Theorem 20.3.2]), the exterior tensor product by the identity on \(C_0(\mathbb {R})^{\otimes F}\otimes \mathcal {K}(\ell ^2(\mathbb {N}))^{\otimes F}\) induces a natural isomorphism
for all G\(C^*\)algebras \(A_0\) and \(B_0\). Therefore, the map \([\phi ]\mapsto [\phi ^{\otimes F}]\) can be naturally regarded as a map from KK(A, B) to \(KK^G(A^{\otimes F}, B^{\otimes F})\). By construction, this map is compatible with compositions and sends the class of a \(*\)homomorphism \(\phi :A\rightarrow B\) to the class of \(\phi ^{\otimes F}\). The case of the maximal tensor product is proven in the exact same way. \(\square \)
We emphasize that the map \(\phi \mapsto \phi ^{\otimes F}\) is not a group homomorphism. For example, it maps n in \(KK(\mathbb {C}, \mathbb {C})\cong \mathbb {Z}\) to the class \([\pi _n]\) in \(KK^G(\mathbb {C}, \mathbb {C})\) of the finitedimensional unitary representation \(\pi _n\) of G on \(\ell ^2(\{1, \ldots , n\}^F)\) defined by permutation on the set \(\{1, \ldots , n\}^F\).
Corollary 2.5
(see [25, Corollary 6.9]) Let G be a discrete group and let \(\phi :A\rightarrow B\) be a unital \(*\)homomorphism which is a KKequivalence. Then for any countable Gset Z, the map
is a weak Kequivalence in \(KK^G\). In particular, \(\phi ^{\otimes Z}\) induces an isomorphism
whenever G satisfies BCC.
Proof
For the first statement we may assume that G is finite. Since \(K(\rtimes _r G)\) commutes with filtered colimits, we may as well assume that Z is finite and apply Lemma 2.4. The second statement follows from Theorem 2.3. \(\square \)
Definition 2.6
Let G be a discrete group, let Z be a countable Gset and let \(A_0\) and B be \(C^*\)algebras with \(A_0\) unital. We define a G\(C^*\)algebra \(\mathcal {J}^Z_{A_0,B}\) as
The Gaction is defined so that a group element \(g\in G\) sends \(A_0^{\otimes ZF} \otimes B^{\otimes F}\) to \(A_0^{\otimes ZgF} \otimes B^{\otimes gF}\) by the obvious \(*\)homomorphism.
In the above definition, \(A_0\) should be thought of as either the complex numbers \(A_0=\mathbb {C}\) or a UHFalgebra (see Sect. 6). For \(A_0=\mathbb {C}\), we just write \(\mathcal {J}^Z_B:=\mathcal {J}^Z_{\mathbb {C},B}\). Note that \(\mathcal {J}^Z_{A_0,B}\) is a G\(C_0(\textrm{FIN}(Z))\)algebra in a natural way. The following lemma is crucial for our computations:
Lemma 2.7
Let \(G,Z,A_0\) and B be as above. Then \(\mathcal {J}^Z_{A_0,B}\) can naturally be identified with the filtered colimit taken over all finite subsets \(S\subseteq Z\), with respect to the obvious connecting maps.^{Footnote 5}
Proof
Just observe that for all finite subsets \(S\subseteq Z\) we have canonical isomorphisms
\(\square \)
Theorem 2.8
(Theorem B) Let A, \(A_0\) and B be \(C^*\)algebras with A and \(A_0\) unital and let \(\iota :A_0\rightarrow A\) be a unital \(*\)homomorphism. Let \(\phi \in KK(B, A)\) be an element such that \(\iota \oplus \phi \in KK(A_0 \oplus B,A)\) is a KKequivalence. Then for each countable Gset Z, there is a weak Kequivalence in \(KK^G(\mathcal {J}^Z_{A_0,B},A^{\otimes Z})\). If G moreover satisfies BCC, there is an isomorphism
where \(\textrm{FIN}(Z)\) denotes the set of finite subsets of Z and where \(G_F\) denotes the stabilizer of F for the action of G on \(\textrm{FIN}(Z)\).
For the proof, we need the following wellknown lemma:
Lemma 2.9
Let Z be a countable Gset and let \(A_Z\) be a G\(C_0(Z)\)algebra. For any G\(C^*\)algebra D and any choice of representatives z for \([z]\in G{\setminus } Z\), there is a natural isomorphism
where \(G_z\) is the stabilizer of z and \(A_z\) is the fiber of \(A_Z\) at \(z\in Z\). The map \(\Psi \) is independent of the choice of representatives in the sense that for any \(z\in Z\) and \(g\in G\), the diagram
commutes where \(g:KK^{G_z}(A_z, D)\rightarrow KK^{G_{gz}}(A_{gz},D)\) is given by conjugation with g.
Proof
Recall that for a subgroup \(H\subseteq G\), and an H\(C^*\)algebra B, the induced G\(C^*\)algebra \({\textrm{Ind}}_H^GB\) is defined as
equipped with the lefttranslation Gaction (see [4, Section 2] for example). Since Z is discrete, we have a natural isomorphism
where the last isomorphism follows from [7, Theorem 3.4.13]. Now let
be the map with components
where \(\iota _z:A_z\hookrightarrow A_Z\) is the inclusion. The second part of the lemma follows from the construction of \(\Psi \) and the fact that conjugation by any \(g\in G\) acts trivially on \(KK^G(A_Z,D)\). We show that \(\Psi \) is an isomorphism. By (2.2) and [16, Theorem 2.9], we can identify \(\Psi \) with the product, taken over all \([z]\in G\backslash Z\), of the compression maps
These are isomorphisms by [4, Proposition 5.14] (see also [21, (20)]). \(\square \)
Proof of Theorem 2.8
For each orbit [F] in \(G \backslash \textrm{FIN}(Z)\) with stabilizer \(G_F\), we define an element
as the composition
Here the first and the third map are the obvious maps and the second map is the tensor product of the identity on \(A_0^{\otimes ZF}\) and the \(KK^{G_F}\)equivalence \((\iota \oplus \phi )^{\otimes F}\) obtained in Lemma 2.4. When F is the empty set, we define \(\Phi _{F}\) as the unital map \(\iota ^{\otimes Z}:A_0^{\otimes Z} \rightarrow A^{\otimes Z}\). By Lemma 2.9 the family \(\{\Phi _F: [F]\in G\backslash \textrm{FIN}(Z)\}\) defines an element \(\Phi \in KK^G(J_{A_0,B}^Z, A^{\otimes Z})\) that does not depend on the choice of representatives F for each [F] (since we have \(\Phi _{gF}=g(\Phi _F)\) for every \(g\in G\)).
We show that \(\Phi \) is a weak Kequivalence. By construction, for any finite subgroup H of G and for any finite Hsubset S of Z, the element \(\Phi \) may be restricted to
in \(KK^H\). Using Lemma 2.7, \(\Phi _S\) can be identified with the tensor product of the identity on \(A_0^{\otimes ZS}\) and the \(KK^H\)equivalence \((\iota \oplus \phi )^{\otimes S} \in KK^H((A_0 \oplus B)^{\otimes S}, A^{\otimes S})\) via the isomorphism
Thus each \(\Phi _S\) induces an isomorphism
By taking the filtered colimit over finite Hsubsets \(S\subseteq Z\), we see that \(\Phi \) is a weak Kequivalence.
The Ktheory computation follows from the fact that \(\mathcal {J}^Z_{A_0,B}\) is a G\(C_0(\textrm{FIN}(Z))\)algebra for the discrete Gset \(\textrm{FIN}(Z)\) together with Green’s imprimitivity Theorem (e.g., see [6, Remark 3.13] for details). \(\square \)
The following special case of Theorem 2.8 will be used to compute examples in Sect. 4. Its main advantage is that we do not have to construct the element \(\phi \) in order to apply it.
Corollary 2.10
Let G be a discrete group satisfying BCC. Let Z be a countable Gset and let A be a unital \(C^*\)algebra satisfying the UCT such that the unital inclusion \(\mathbb {C}\rightarrow A\) induces a split injection \(K_*(\mathbb {C})\rightarrow K_*(A)\). Denote by \({\tilde{K}}_*(A)\) its cokernel and let B be any \(C^*\)algebra satisfying the UCT with \(K_*(B)\cong \tilde{K}_*(A)\). Then we have
In particular, if \(Z=G\) equipped with the left translation, then \(K_*\left( A^{\otimes G}\rtimes _r G\right) \) only depends on G and \({\tilde{K}}_*(A)\).
Note that B exists and is uniquely determined up to KKequivalence (see [3, Corollary 23.10.2]).
Proof
By the UCT, the inclusion map \(K_*(B)\cong {\tilde{K}}_*(A)\rightarrow K_*(A)\) is induced by an element \(\phi \in KK(B,A)\). By construction, \(\iota \oplus \phi \in KK(\mathbb {C}\oplus B,A)\) is a KKequivalence, so that we can apply Theorem 2.8. \(\square \)
As another special case of Theorem 2.8, we recover
Corollary 2.11
([6, Example 3.17], [20, Proposition 2.4]) Let G be a discrete group satisfying BCC and let Z be a countable Gset. Then for \(n\ge 1\), we have
Moreover, if \(Z=G\) with the left translation action, we get
Here \({\mathcal {C}}\) denotes the set of all conjugacy classes of finite subgroups of G, F(C) the nonempty finite subsets of \(C\backslash G\), \(N_C=\{g\in G: gCg^{1}=C\}\) the normalizer of C in G, and \(C_S=G_S\cap C\) the stabilizer of S in C.
Proof
Let \(A=C(\{0,\dotsc ,n\})\) and \(B=C(\{1,\dotsc ,n\})\) and let \(\phi :B\rightarrow A\) be the canonical inclusion. The first isomorphism follows from Theorem 2.8. The second isomorphism is obtained by analyzing the orbit structure of \(\bigsqcup _{F\in \textrm{FIN}(G)}\{1,\dotsc ,n\}^F\) (see [20, Proposition 2.4] for details). \(\square \)
3 Finitedimensional algebras
In this section we compute the Ktheory of crossed products of the form \(A^{\otimes Z}\rtimes _r G\) where \(A=\bigoplus _{0\le j \le N}{M_{k_j}}\) is a finitedimensional \(C^*\)algebra, Z is a countable Gset, and where G is a group satisfying BCC. This generalizes the case \(k_0=1\) from [20]. We denote by \(\gcd (k_0,\dotsc ,k_N)\) the greatest common divisor of \(k_0,\dotsc ,k_N\). We believe that the following theorem is known to experts. In lack of a reference, we give a detailed proof here.
Theorem 3.1
Let \(k_0,\dotsc ,k_N\) be positive integers with \(\gcd (k_0,\dotsc ,k_N)=1\). Then there is a unital \(*\)homomorphism
that induces a KKequivalence. Moreover when \(N=1\), there are exactly two such \(*\)homomorphisms up to unitary equivalence.
The main ingredient for the proof is the following arithmetic fact:
Proposition 3.2
For any pair of positive integers \(k_1,k_2\in \mathbb {N}\), there is a unique matrix X in \(\textrm{SL}(2, \mathbb {Z})\) with nonnegative entries such that
where \(n=\gcd (k_1,k_2)\). If we allow X to be in \(\textrm{GL}(2, \mathbb {Z})\), then there are exactly two such X: the one \(X_0\) in \(\textrm{SL}(2, \mathbb {Z})\) and \(X_0 \begin{bmatrix} 0 &{} 1 \\ 1 &{} 0 \end{bmatrix}\).
Proof
Existence: Let \(f:{\mathbb {N}}_+^2\rightarrow {\mathbb {N}}_+^2\) be given by
The Euclidian algorithm precisely says that there exists a \(k \in {\mathbb {N}}\) such that \(f^k\begin{bmatrix}k_1\\ k_2\end{bmatrix}=\begin{bmatrix}n\\ n\end{bmatrix}.\) Let \(A=\begin{bmatrix}1 &{} 1 \\ 0 &{} 1\end{bmatrix}\) and \(B=\begin{bmatrix}1 &{} 0 \\ 1 &{} 1\end{bmatrix}.\) Then the map f defined above is given by
Now the above formulation of the Euclidean algorithm gives us integers \(a_1,a_2,\dotsc ,a_l, b_1,b_2,\dotsc ,b_l\ge 1\) such that
Then \(X:=A^{a_1} B^{b_1} A^{a_2} B^{b_2} \cdots A^{a_l} B^{b_l}\in {\textrm{SL}}(2,\mathbb {Z})\) is the required matrix.
Uniqueness: Suppose there are two such matrices \(X_1\) and \(X_2\). Then, \(Y=X_2^{1}X_1\in \textrm{SL}(2, \mathbb {Z})\) satisfies
From this, we have
for \(a\in \mathbb {Z}\). Now, we get
but since both \(X_1\) and \(X_2\) have nonnegative entries and since they are nonsingular, it is not hard to see that a must be 1. Thus, \(X_1=X_2\). The last assertion is immediate. \(\square \)
Remark 3.3
It follows from the proof that the subsemigroup in \(\textrm{SL}(2, \mathbb {Z})\) consisting of matrices with nonnegative entries is the free monoid of two generators A and B.
Corollary 3.4
For any positive integers \(k_0, \ldots , k_N\) with \(\gcd (k_0, \ldots , k_N)=n\), there is a matrix X in \(\textrm{SL}(N+1, \mathbb {Z})\) with nonnegative entries such that
Proof
We give the proof by induction on N. The case \(N=0\) is clear. Let \(N\ge 1\) and let \(k_0,\ldots , k_N\ge 1\) be positive integers with \(\gcd (k_0,\ldots , k_N)=n\). Set \(l:=\gcd (k_0,\ldots , k_{N1})\). By induction, we may assume that there exists a matrix \(\tilde{X}\in \textrm{SL}(N,\mathbb {Z})\) with nonnegative entries such that \({\tilde{X}\begin{bmatrix} l\\ \vdots \\ l\end{bmatrix}=\begin{bmatrix} k_0\\ \vdots \\ k_{N1}\end{bmatrix}}\). Since \(\gcd (l, k_N)=\gcd (l,n)=n\), it follows from Proposition 3.2 that there are matrices \(X_0, X_{N}\in \textrm{SL}(2,\mathbb {Z})\) with nonnegative entries such that\({X_0\begin{bmatrix} n\\ n\end{bmatrix}=\begin{bmatrix} l\\ n\end{bmatrix}}\) and \(X_{N}\begin{bmatrix} n\\ n\end{bmatrix}=\begin{bmatrix} l\\ k_N\end{bmatrix}\). Now for \(i\in \{1,\ldots , N1\}\), let \(Y_i\) denote the matrix with \(X_0\) as the \((i, i+1)\)th diagonal block, with ones in all other diagonal entries and with zeros elsewhere, and let \(Y_{N}=\begin{bmatrix} I_{N1} &{}0\\ 0&{} X_N\end{bmatrix}\). Write \(Y:=\begin{bmatrix} \tilde{X}&{}0\\ 0&{}1\end{bmatrix}\). Then \(X:=Y\cdot Y_N\cdots Y_1\in SL(N+1,\mathbb {Z})\) has nonnegative entries and
as desired. \(\square \)
Proof of Theorem 3.1
Unital \(*\)homomorphisms from \(\mathbb {C}^{N+1}\) to \(\bigoplus _{0\le j \le N}{M_{k_j}}\) are classified up to unitary equivalence by their induced maps on ordered \(K_0\)groups with units. If we identify the \(K_0\)groups of \(\mathbb {C}^{N+1}\) and \(\bigoplus _{0\le j \le N}{M_{k_j}}\) with \(\mathbb {Z}^{N+1}\), we may represent the induced maps on \(K_0\)groups by \((N+1)\)square matrices with nonnegative integer entries that send \(\begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix}\) to \(\begin{bmatrix} k_0 \\ \vdots \\ k_N \end{bmatrix}\). Such a matrix is an isomorphism on Ktheory if and only if it is in \(\textrm{GL}(N+1, \mathbb {Z})\). All the assertions now follow from Proposition 3.2 and Corollary 3.4. \(\square \)
We are now ready to prove Theorem A. For the proof, we need a result from [18]. We formulate it in full generality here since this will be needed in later sections of the paper. Recall that a supernatural number is a formal product \({\mathfrak {n}}=\prod _p p^{n_p}\) of prime powers with \(n_p\in \{0,1,\dotsc ,\infty \}\). We denote by \(M_{\mathfrak {n}}:=\otimes _p M_p^{\otimes n_p}\) the associated UHFalgebra. \(M_{\mathfrak {n}}\) and \({\mathfrak {n}}\) are called of infinite type if \(n_p\in \{0,\infty \}\) for all p and \(n_p\ne 0\) for at least one p. For an abelian group L, we denote by
the localization at \({\mathfrak {n}}\) where \((p_1,p_2,\dotsc )\) is a sequence of primes containing every p with \(n_p\ge 1\) infinitely many times. In other words, \(L[1/{\mathfrak {n}}]\) is the localization at the set of all primes dividing \({\mathfrak {n}}\). If \({\mathfrak {n}}\) is finite, this definition recovers the usual localization at a natural number.
Theorem 3.5
([18, Corollary 2.11]) Let G be a discrete group satisfying BCC, Z a countable Gset, A a G\(C^*\)algebra and \(M_{\mathfrak {n}}\) a UHFalgebra. Assume that Z is infinite or that \({\mathfrak {n}}\) is of infinite type. Then the inclusion \(A\rightarrow A\otimes M_\mathfrak n^{\otimes Z}\) induces an isomorphism
In particular, the righthand side is a \(\mathbb {Z}[1/{\mathfrak {n}}]\)module.
Theorem 3.6
(Theorem A) Let G be a discrete group satisfying BCC. Let Z be a countably infinite Gset and let \(A=\bigoplus _{0\le j \le N}{M_{k_j}}\) where \(k_0, \ldots , k_N\) (\(N\ge 1\)) are positive integers with \(\gcd (k_0, \ldots , k_N)=n\). Then
Proof
Write \(B=\bigoplus _{0\le j \le N}{M_{k_j/n}}\) so that B satisfies the assumptions of Theorem 3.1 and so that \(A\cong B\otimes M_n\). By Theorem 3.5, the inclusion \(B^{\otimes Z}\hookrightarrow A^{\otimes Z}\) induces an isomorphism
By Theorem 3.1 and Corollary 2.5, we furthermore have
This proves the isomorphism in the first line of the theorem. The isomorphism in the second line follows from Corollary 2.11. \(\square \)
4 More examples
In this section, we compute the Ktheory of Bernoulli shifts in more examples. We mostly restrict ourselves to the case \(Z=G\) with the left translation action, but some of the results have straightforward generalizations to arbitrary countable Gsets. Recall that for \(Z=G\) we write \(\textrm{FIN}=\textrm{FIN}(G)\) for the collection of finite subsets of G and we put \(\mathcal {J}_B:=\mathcal {J}_{\mathbb {C},B}^G\) as in Definition 2.6 for any \(C^*\)algebra B. We start with some easy applications of Corollary 2.5.
Example 4.1
Let \(\mathcal {A}\) be the Fibonacci algebra [8, Example III 2.6], which is the filtered colimit of \((M_{m_k}\oplus M_{n_k})_{k\in \mathbb {N}}\) where \(m_1=n_1=1\) and where the connecting maps are given by repeated use of the partial embedding matrix \(\begin{bmatrix} 1 &{} 1 \\ 1 &{} 0 \end{bmatrix}\). Since this matrix belongs to \(\textrm{GL}(2,\mathbb {Z})\), the unital embedding \(\mathbb {C}\oplus \mathbb {C}\hookrightarrow \mathcal {A}\) is a KKequivalence.
Let \({\mathcal {K}}^+\) be the unitization of the algebra of compact operators on \(\ell ^2(\mathbb {N})\) and let \(p\in {\mathcal {K}}\) be a rank1 projection. Then the unital embedding \({\mathbb {C}\oplus \mathbb {C}\hookrightarrow {\mathcal {K}}^+}\) given by \((\lambda ,\mu )\mapsto \lambda (1p)+\mu p\) is a KKequivalence.
Now let A be either \(\mathcal {A}\) or \({\mathcal {K}}^+\) and let G be a discrete group satisfying BCC. Then by Corollaries 2.5 and 2.11, we have
In particular, if G is torsion free, we have
Example 4.2
(Theorem C) Consider \(A=C(S^1)\). Note that the canonical inclusion \({\phi :C_0(\mathbb {R}) \rightarrow C(S^1)}\) together with the unital inclusion \(\iota :\mathbb {C}\rightarrow C(S^1)\) induces a KKequivalence \(\iota \oplus \phi \in KK(\mathbb {C}\oplus C_0(\mathbb {R}), A)\). By Theorem 2.8, we obtain a weak Kequivalence \( \Phi :\mathcal {J}_{C_0(\mathbb {R})} \rightarrow A^{\otimes G}. \) We can compute the Ktheory of \(\mathcal {J}_{C_0(\mathbb {R})} \rtimes _rG\) as
In this expression, each nonempty finite subset \(F\subseteq G\) can be written as \(F=G_F\cdot L_F\) where \(L_F\) is a complete set of representatives for \(G_F\backslash F\). If the cardinality of \(L_F\) is even, the \(G_F\)action on \(C_0(\mathbb {R})^{\otimes F}=C_0(\mathbb {R}^{F})\) is \(KK^{G_F}\)equivalent to the trivial action on \(\mathbb {C}\) by Kasparov’s Bottperiodicity theorem (see [3, Theorem 20.3.2]). When the cardinality of \(L_F\) is odd, then
is \(KK^{G_F}\)equivalent to \(C_0(\mathbb {R})^{\otimes G_F}=C_0\left( \mathbb {R}^{G_F}\right) \). Therefore, we have
In general, for any finite group H and for any orthogonal representation \(H\rightarrow O(V)\) on a finitedimensional Euclidean space V, the Ktheory of \(C_0(V)\rtimes _rH\) is the wellstudied equivariant topological Ktheory \(K_H^*(V)\) of V (see [10, 15]). It is known to be a finitely generated free abelian group with \(\textrm{rank}_\mathbb {Z}K_H^*(V)\) equal to the number of conjugacy classes \(\langle g\rangle \) of H which are oriented and even/odd respectively (see [15, Theorem 1.8]). Here, a conjugacy class \(\langle g\rangle \) of H is oriented if the centralizer \(C_g\) of g acts on the gfixed points \(V^{g}\) of V by oriented automorphisms. The class \(\langle g\rangle \) is even/odd if the dimension of \(V^{g}\) is even/odd respectively. For example for any cyclic group \(\mathbb {Z}/m\mathbb {Z}\), using [10, Example 4.2], we have
for odd \(m\ge 1\), and
for even \(m\ge 2\). We summarize our discussion as follows:
Theorem 4.3
Let G be a discrete group satisfying BCC. We have
Example 4.4
Let \(A_\theta \) be the rotation algebra for \(\theta \in \mathbb {R}\), the universal \(C^*\)algebra generated by two unitaries u and v satisfying \(uv=vue^{2\pi i\theta }\). It is wellknown that \(K_0(A_\theta )\cong \mathbb {Z}^2\), that \(K_1(A_\theta )\cong \mathbb {Z}^2\), and that the unital inclusion \(\iota :\mathbb {C}\rightarrow A_\theta \) induces a split injection on \(K_0\). Since \(A_\theta \) satisfies the UCT, we can apply Corollary 2.10 to \(B=\mathbb {C}\oplus C_0(\mathbb {R}) \oplus C_0(\mathbb {R})\) and obtain
Using Theorem 2.8 for \(B=C_0(\mathbb {R})\oplus C_0(\mathbb {R})\), each summand for \(F\in \textrm{FIN}^\times \) may be computed as
where the sum is taken over all ordered equivalence classes \([X_1, X_2]\) of pairs of ordered disjoint subsets \(X_1, X_2\) of F modulo the action of the stabilizer \(G_F\) of F, and where we set \(G_{X_{1,2}}:=G_F\cap G_{X_1}\cap G_{X_2}\). As in Example 4.2, the computation of \(K_*\left( \left( C_0\left( \mathbb {R}^{X_1}\right) \otimes C_0\left( \mathbb {R}^{X_2}\right) \right) \rtimes _r G_{X_{1,2}}\right) \) either reduces to \(K_*(C_r^*(G_{X_{1,2}}))\) or to that of \(K_*\left( C_0\left( \mathbb {R}^{G_{X_{1,2}}}\right) \rtimes _rG_{X_{1,2}}\right) =K^*_{G_{X_{1,2}}}\left( \mathbb {R}^{G_{X_{1,2}}}\right) \) in general. We summarize the discussion as follows.
Theorem 4.5
Let G be a discrete group satisfying BCC and let \(\theta \in \mathbb {R}\). We have
where for all \(F\in \textrm{FIN}^\times \) we set
4.1 Cuntz algebras
For \(n\in \{2,3,\dotsc ,\infty \}\) we denote by \({\mathcal {O}}_n\) the Cuntz algebra on n generators.
Example 4.6
[25, Corollary 6.9] Let G be a discrete group satisfying BCC. Then the unital inclusion \(\mathbb {C}\rightarrow {\mathcal {O}}_\infty \) induces an isomorphism
To see this combine Corollary 2.5 and the fact that the unital inclusion \(\mathbb {C}\rightarrow \mathcal {O}_\infty \) is a KKequivalence.
Example 4.7
Let G be a discrete group satisfying BCC. Then we have
For this combine Lemma 2.4, Theorem 2.3 and the fact that \(\mathcal {O}_2\) is KKequivalent to 0.
Example 4.8
Let G be a discrete group satisfying BCC. Let \(A=\mathbb {C}\oplus \mathcal {O}_n\) for \(n\ge 3\). By Theorem 2.8, we have
Each summand for \(F=G_F\cdot L_F\) becomes \(K_*\left( \left( \mathcal {O}_n^{\otimes L_F}\right) ^{\otimes G_F} \rtimes _rG_F\right) \). It follows from the UCT and an inductive application of the Künneth theorem that \(\mathcal {O}_n^{\otimes L_F}\) is KKequivalent to \(\mathcal {O}_n\otimes (C_0(\mathbb {R})\oplus \mathbb {C})^{\vert L_F\vert 1}\). Thus, by Lemma 2.4, we may express \(K_*(\mathcal {O}_n^{\otimes F} \rtimes _rG_F)\) explicitly in terms of
for finite subgroups H of \(G_F\). These groups for \(H=\mathbb {Z}/2\) are nicely computed in [14]. According to [14], we have
for odd n,
for even n, and
for odd n,
for even n. Using similar methods as in [14], we can compute the case \(H=\mathbb {Z}/3\). We omit the very technical computations. For general H, the computations become increasingly complicated as the order of H increases.
Question 4.9
Is \(K_*\left( \mathcal {O}_n^{\otimes H} \rtimes _rH\right) \) computable for all finite groups H or at least for all cyclic groups?
Although we do not know how to compute \(K_*\left( \mathcal {O}_n^{\otimes H}\rtimes _r H\right) \) in general, we can say something about its structure. The following corollary is a combination of Theorem 3.5 and Corollary 2.5.
Corollary 4.10
Let G be a discrete group satisfying BCC, Z a countable Gset, and A a \(C^*\)algebra. Let \(M_{\mathfrak {n}}\) be a UHFalgebra of infinite type such that \(A\otimes M_{\mathfrak {n}}\) is KKequivalent to zero (for instance \(A=\mathcal {O}_{n+1}\) and \(M_\mathfrak n=M_{n^{\infty }}\) for some \(n\ge 2\)). Then
For \(A=\mathcal {O}_{n+1}\) and a finite group H, we can say a bit more:
Proposition 4.11
Let H be a finite group, let Z be a finite Hset and let \(n\ge 2\). Then \(K_*\left( \mathcal {O}_{n+1}^{\otimes Z}\rtimes _r H\right) \) is a finitely generated abelian group L such that \(L[1/n]=0\). That is, any element in L is annihilated by \(n^k\) for some k. In particular, if \(n=p\) is prime, then L is isomorphic to the direct sum of finitely many pgroups \(\mathbb {Z}/p^k\mathbb {Z}\).
Proof
The method used in [14], at an abstract level, tells us that \(K_*\left( \mathcal {O}_{n+1}^{\otimes Z} \rtimes _rH\right) \) is finitely generated. To see this, let \({\mathcal {T}}_{n+1}\) be the universal \(C^*\)algebra generated by isometries \(s_1\ldots , s_{n+1}\) with mutually orthogonal range projections \(q_i=s_is_i^*\) and we let \(p=1\sum _{1\le j \le n+1}q_i\). The ideal generated by p is isomorphic to \(\mathcal {K}\) and we have the following exact sequences
where for \(1\le m\le \vert Z \vert \), an Hideal \(I_m\) of \({\mathcal {T}}_{n+1}^{\otimes Z}\) is defined as the ideal generated by \(\mathcal {K}^{\otimes F}\otimes {\mathcal {T}}_{n+1}^{\otimes ZF}\) for subsets F of H with \(\vert F \vert =m\). In particular \(I_{\vert Z \vert }= \mathcal {K}^{\otimes Z}\). By Lemma 2.4, or by the stabilization theorem, \(K_*\left( \mathcal {K}^{\otimes Z}\rtimes H\right) \cong K_*(C^*(H))\). Moreover, for each \(1\le m<\vert Z \vert \), the quotient \(I_m/I_{m+1}\) is the direct sum of \(\mathcal {K}^{\otimes F}\otimes \mathcal {O}_{n+1}^{\otimes ZF}\) over the subsets F of Z with \(\vert F \vert =m\). Thus, \(I_m/I_{m+1}\rtimes H\) is Moritaequivalent to the direct sum of \(\mathcal {K}^{\otimes F}\otimes \mathcal {O}_{n+1}^{\otimes ZF}\rtimes H_F\) over \([F]\in H\backslash \{ F \subset Z \mid \vert F \vert = m \}\) where \(H_F\) is the stabilizer of F in H. By induction on the size \(\vert Z \vert \) of Z (for all finite groups H at the same time), \(K_*\left( \mathcal {K}^{\otimes F}\otimes \mathcal {O}_{n+1}^{\otimes ZF}\rtimes H_F\right) \cong K_*\left( \mathcal {O}_{n+1}^{\otimes ZF}\rtimes H_F\right) \) is finitely generated. Using the sixterm exact sequences on Ktheory associated to (4.2), we see that \(K_*(I_m \rtimes H)\) are all finitelygenerated. By Lemma 2.4, \(K_*\left( {\mathcal {T}}_{n+1}^{\otimes Z}\rtimes H \right) \cong K_*(C^*(H))\) since \({\mathcal {T}}_{n+1}\) is KKequivalent to \(\mathbb {C}\) by [24]. Using the sixterm exact sequence on Ktheory associated to (4.1), we now see that \(K_*\left( \mathcal {O}_{n+1}^{\otimes Z}\rtimes H \right) \) is finitelygenerated.
On the other hand, we have
by Theorem 3.5. The assertion follows from Lemma 2.4 since \(\mathcal {O}_{n+1} \otimes M_{n^\infty }\) is KKequivalent to 0 by the UCT. \(\square \)
For an infinite Gset Z, Corollary 4.10 for \(A=\mathcal {O}_{n+1}\) may also be deduced from the combination of Theorem 3.5 and the following result (for \(A=\mathcal {O}_{n+1}\otimes M_n\)).
Theorem 4.12
Let G be a discrete group satisfying BCC and let Z be a countably infinite Gset. Let A be any unital \(C^*\)algebra such that \([1_{A^{\otimes r}}]=0\in K_0(A^{\otimes r})\) for some \(r\ge 1\). Then we have
Proof
By Theorem 2.3 it is enough to show \(K_{*}(A^{\otimes Z}\rtimes _rH) \cong 0\) for all finite subgroups H of G. Since Z is infinite, it contains infinitely many orbits of type \(H/H_0\) for some fixed subgroup \(H_0\subset H\). Denote \(Z_0\) the union of orbits of type \(H/H_0\). Let L be a (necessarily infinite) complete set of representatives for \(H\backslash Z_0\). We have
By assumption, the unital inclusion \(\mathbb {C}\rightarrow A^{\otimes r}\) induces the zero element in \(K_0(A^{\otimes r})=KK(\mathbb {C}, A^{\otimes r})\). In particular, the maps
induce the zero map in KKtheory since, on the level of KKtheory, they are given by the exterior Kasparov product with \([0]=[1]\in KK(\mathbb {C},A^{\otimes r})\). It follows from Lemma 2.4, that the unital inclusions
are zero in \(KK^H\). Writing \(A^{\otimes Z}=A^{\otimes Z_0}\otimes A^{\otimes ZZ_0}\), we have
The righthand side is zero by the preceding argument. \(\square \)
Remark 4.13
Theorem 4.12 can be applied whenever A is Morita equivalent to \(\mathcal {O}_{n^r+1}\) such that \([1_A]=n\in K_0(A)\cong \mathbb {Z}/{n^r}\), or whenever \(K_0(A)\cong \mathbb {Q}/\mathbb {Z}\). In the second case, this is due to the fact that \([1]^{\otimes 2}\in K_0(A^{\otimes 2})\) is in the image of \(K_0(A)\otimes _\mathbb {Z}K_0(A)\cong \mathbb {Q}/\mathbb {Z}\otimes _\mathbb {Z}\mathbb {Q}/\mathbb {Z}=0\).
4.2 Wreath products
For discrete groups G and H, the wreath product \(H\wr G\) is defined as the semidirect product \((\bigoplus _{g\in G} H)\rtimes G\), where G acts by left translation. We have a canonical isomorphism
An application of Corollary 2.10 gives the following result which generalizes [20].
Theorem 4.14
(Theorem D) Let G be a group satisfying BCC and let H be a group for which \(C^*_r(H)\) satisfies the UCT and for which the unital inclusion \(\mathbb {C}\rightarrow C^*_r(H)\) induces a split injection \(K_*(\mathbb {C})\rightarrow K_*(C^*_r(H))\). Denote its cokernel by \({\tilde{K}}_*(C^*_r(H))\). Then we have
where B is any \(C^*\)algebra satisfying the UCT with \(K_*(B)\cong {\tilde{K}}_*(C^*_r(H))\). In particular, if G is torsionfree, we have
Remark 4.15
The assumptions on H in the above theorem are not very restrictive: If H is a discrete group for which the Baum–Connes assembly map is splitinjective (for instance if H satisfies BCC, or if H is exact by [11, Theorem 1.1]), then the unital inclusion \(\mathbb {C}\rightarrow C^*_r(H)\) induces a split injection \(K_*(\mathbb {C})\rightarrow K_*(C^*_r(H))\) since the corresponding map \(K^{\textrm{top}}_*(\{e\})\rightarrow K_*^{\textrm{top}}(H)\) always splits.
On the other hand, it follows from Tu’s [27, Proposition 10.7] that \(C_r^*(H)\) satisfies the UCT for every aTmenable (in particular every amenable) group H, or, more generally, if H satisfies the strong Baum–Connes conjecture in the sense of [21].
Example 4.16
Let G be a group which satisfies BCC and consider the wreath product \({\mathbb {F}}_n\wr G\) with \({\mathbb {F}}_n\) the free group in n generators. Since \({\mathbb {F}}_n\) is known to be aTmenable it follows that Theorem 4.14 applies. Since \(K_0(C_r^*({\mathbb {F}}_n))=\mathbb {Z}\) and \(K_1(C_r^*({\mathbb {F}}_n))=\mathbb {Z}^n\) we may choose \(B=\bigoplus _{i=1}^n C_0(\mathbb {R})\) so that Theorem 4.14 implies
where each summand \(K_*(B^{\otimes F}\rtimes _r G_F)\) decomposes into a direct sum of equivariant Ktheory groups of the form \(K^*_{H}(V)\) for certain subgroups H of \(G_F\) and certain euclidean Hspaces V. A more precise analysis can be done, at least for \(n=2\), along the lines of Example 4.4.
In particular, if G is torsion free, we get
with \(B^{\otimes F} \cong \bigoplus _{i=1}^{n^{\vert F \vert }} C_0(\mathbb {R}^{\vert F \vert })\). Therefore each Gorbit of a nonempty finite set \(F\subseteq G\) provides \(n^{\vert F \vert }\) copies of \(\mathbb {Z}\) as direct summands of \(K_0\) if \(\vert F \vert \) is even and of \(K_1\) if \(\vert F \vert \) is odd.
We close this section with
Corollary 4.17
Suppose that G and H are as in Theorem 4.14, such that the Ktheory of both \(C^*_r(G)\) and \(C^*_r(H)\) is free abelian. Then the Ktheory of \(C^*_r(H\wr G)\) is free abelian as well.
Proof
Note that \({\tilde{K}}_*(C^*_r(H))\) is free abelian as it is the direct summand of the free abelian group \(K_*(C^*_r(H))\). Therefore, in Theorem 4.14, B can be taken as the direct sum of, possibly infinitely many, \(\mathbb {C}\) and \(C_0(\mathbb {R})\). The assertion follows from the fact that the equivariant Ktheory \(K_{G_0}^*(\mathbb {R}^{G_0})\) (more generally \(K^*_{G_0}(V)\) for any \(G_0\)Euclidean space V) is a (finitelygenerated) free abelian group for any finite group \(G_0\) by [15] (or [10]). \(\square \)
5 AFalgebras
Let be a unital AFalgebra (with unital connecting maps) and let G be a discrete group satisfying BCC. If Z is a countable Gset, then in principle, we can try to compute
using the decomposition from Theorem 3.6. In general, such a computation can be challenging, even in relatively simple cases like \(A=M_2\oplus M_{3^\infty }\). Instead of trying to compute the connecting maps, we now provide an abstract approach to calculating \(K_*(A^{\otimes Z}\rtimes _r G)\) for an arbitrary unital AFalgebra A, a discrete group G satisfying BCC and a countable proper Gset Z (for example \(Z=G\) with the left translation action).
We first need some preparation. Let \({\mathcal {F}}\) be a family of subgroups, i.e. a nonempty set of subgroups of G closed under taking conjugates and subgroups. The orbit category \(\textrm{Or}_{\mathcal {F}}(G)\) has as objects homogeneous Gspaces G/H for each \(H\in {\mathcal {F}}\) and as morphisms Gmaps (see [9, Definition 1.1]). We will mainly use the family \(\mathcal {FIN}\) of finite subgroups. For any G\(C^*\)algebra A, we have a functor from \(\textrm{Or}_{\mathcal {F}}(G)\) to the category of graded abelian groups that sends G/H to \(K_*(A\rtimes _rH)\) and a morphism \(G/H_0\rightarrow G/{H_1}\) given by \(H_0 \mapsto gH_1\) (so that \(H_0\subseteq gH_1g^{1}\)) to the map \(K_*(A\rtimes _rH_0) \rightarrow K_*(A\rtimes _rH_1)\) induced by the composition
where the second map is given by conjugation with \(g^{1}\) inside \(A\rtimes _rG\).^{Footnote 6} We denote by
the colimit of the functor \( G/H \mapsto K_*(A\rtimes _rH)\) from \(\textrm{Or}_{\mathcal {F}}(G)\) to the category of graded abelian groups. The inclusions \(A\rtimes _rH\rightarrow A\rtimes _rG\) induce a natural homomorphism
Remark 5.1
The map (5.1) should not be confused with the assembly map
which corresponds to taking the homotopy colimit at the level of Ktheory spectra instead of taking the ordinary colimit at the level of Ktheory groups, see [9, Section 5.1]. By the involved universal properties, there is a natural commuting diagram
but the vertical map is neither injective nor surjective in general. One can think of the map (5.1) as the best approximation of \(K_*(A\rtimes _rG)\) using 0dimensional G\({\mathcal {F}}\)CWcomplexes. Likewise, the best approximation by rdimensional G\({\mathcal {F}}\)CWcomplexes can be defined by taking the colimit of \(H^G_*(X, {\mathbb {K}}^{\textrm{top}}_A)\) over the category of rdimensional G\({\mathcal {F}}\)CWcomplexes.
Let us give two examples of how to compute \({\underset{G/H\in \textrm{Or}_{\mathcal {FIN}}(G)}{{\text {colim}}}K_*(A\rtimes _r H)}\) when the structure of \(\textrm{Or}_{\mathcal {FIN}}(G)\) is understood. Recall that the coinvariants of a Gmodule L are given by
Here the group G is considered as a category with one object with morphisms given by the elements of G, and L is considered as a functor from G to the category of (graded) abelian groups.
Example 5.2
If G is torsionfree, then \(\underset{G/H \in \textrm{Or}_{\mathcal {FIN}}(G)}{{\text {colim}}}K_*(A\rtimes _rH)\) can be identified with the coinvariants \(K_*(A)_G\).
Example 5.3
If G has only one nontrivial conjugacy class [H] of finite subgroups and if \(N_G(H)\) denotes the normalizer of H in G, then the colimit \({\underset{G/H \in \textrm{Or}_{\mathcal {FIN}}(G)}{{\text {colim}}} K_*(A\rtimes _rH)}\) is given by the pushout
For a Gmodule M and a set S of positive integers, we denote by
its localization at S, where \((s_1,s_2,\dotsc )\) is a sequence containing every element of S infinitely many times. Note that localization at S commutes with taking coinvariants since both constructions are colimits.
Theorem 5.4
(Theorem F) Let A be a unital \(C^*\)algebra of the form (with unital connecting maps) where \((k_n)_{n\in \mathbb {N}}\) is a sequence of positive integers and where each \(A_n\) is a unital \(C^*\)algebra satisfying the assumptions of Theorem 2.8 for the unital inclusion \(\mathbb {C}\hookrightarrow A_n\). Let G be an infinite discrete group satisfying BCC, let Z be a countable proper Gset, and let \(S\subseteq \mathbb {Z}\) be the set of all positive integers n such that \([1_A]\in K_0(A)\) is divisible by n. Then, the natural inclusions
induce the following pushout diagram
In particular, if G is torsionfree, this pushout diagram reads
where \({\tilde{K}}_*(C^*_r(G))\) denotes the cokernel of \(K_*(\mathbb {C})\rightarrow K_*(C_r^*(G))\) induced from the unital inclusion \(\mathbb {C}\hookrightarrow C_r^*(G)\).
Remark 5.5
By Theorem 3.1, Theorem 5.4 applies to all unital AFalgebras. More generally, Theorem 5.4 applies whenever A is of the form (with unital connecting maps) where each \(A_n\) is one of the following examples:

1.
The unitization \(B^+\) of a \(C^*\)algebra B, e.g. \(\mathbb {C}\oplus B\) for unital B;

2.
A \(C^*\)algebra of the form \(\bigoplus _{1\le j\le N}C(X_j)\otimes M_{l_j}\) for nonempty compact metric spaces \(X_j\) and \(\gcd (l_1,\dotsc ,l_N)=1\) (use Theorem 3.1);

3.
The reduced group \(C^*\)algebra \(C^*_r(\Gamma )\) of a countable group \(\Gamma \) that satisfies the UCT and such that the map \(K_*(\mathbb {C})\rightarrow K_*(C^*_r(\Gamma ))\) is a splitinjection (see Remark 4.15 and Theorem 4.14).
It would be interesting to know if Theorem 5.4 also holds for unital ASHalgebras, i.e. when A is a filtered colimit of algebras of the form \(p(C(X)\otimes M_n) p\) for a projection \(p\in C(X)\otimes M_n\).
For the proof of Theorem 5.4, we need the following Lemma.
Lemma 5.6
Let G be any discrete group, let \(\mathcal {F}\) be a family of subgroups of G, let \(H_i\in \mathcal {F}\), and let \(A_i\) be \(H_i\)\(C^*\)algebras for \(i\in I\). Denote by \(A=\oplus _{i \in I}{\textrm{Ind}}_{H_i}^G(A_i)\) the direct sum of the induced G\(C^*\)algebras. Then, the natural map
is an isomorphism.
Proof
By additivity we may assume \(A={\textrm{Ind}}_{H_0}^GA_0\) for \(H_0\in \mathcal {F}\) and an \(H_0\)\(C^*\)algebra \(A_0\). Write
where \(A_z\) is the fiber of A at \([z] \in G/H_0\). Note that \(A_z\) is a \(zH_0z^{1}\)\(C^*\)algebra, in particular, \(A_e\) is the \(H_0\)\(C^*\)algebra \(A_0\). By [7, Proposition 2.6.8], the inclusion \(A_e\rightarrow A\) induces an isomorphism
This isomorphism factors through \({\text {colim}}_{G/H \in \textrm{Or}_{\mathcal {F}}(G)}K_*(A\rtimes _rH)\) as
Our claim follows once we show that the map
is surjective. Fix a subgroup \(H\in {\mathcal {F}}\). Decomposing \(G/H_0\) into Horbits and using the decomposition of (2.2), we obtain a decomposition
where \(H_z:=H\cap zH_0z^{1}\in \mathcal {F}\). In (5.4), the inclusions \(K_*(A_z\rtimes _r H_z)\subseteq K_*(A\rtimes _r H)\) are induced by the natural inclusions \(A_z\rtimes _r H_z\subseteq A\rtimes _r H\). In the colimit \({\text {colim}}_{G/H \in \textrm{Or}_{\mathcal {F}}(G)}K_*(A\rtimes _rH)\), the summand
gets via conjugation with \(z^{1}\) identified with the summand
corresponding to \([e]\in G/H_0\) and \(H_z'=z^{1}Hz\cap H_0\). But the elements in \({\text {colim}}_{G/H \in \textrm{Or}_{\mathcal {F}}(G)}K_*(A\rtimes _rH)\) coming from \(K_*(A_e\rtimes _r H'_z)\) are certainly in the image of the map in (5.3) since
factors through the Ktheory map of the inclusion \(A_e\rtimes _rH_z' \subseteq A_e\times _rH_0\). \(\square \)
As a direct consequence of Lemma 5.6, we get
Corollary 5.7
Let G be any discrete group, let Z be a countable Gset, and let \(A_0\) and B be \(C^*\)algebras with \(A_0\) unital. Let \({\dot{\mathcal {J}}}_{A_0,B}^{Z}\) be the G\(C^*\)algebra
so that \(\mathcal {J}_{A_0,B}^Z=A_0^{\otimes Z}\oplus {\dot{\mathcal {J}}}^Z_{A_0,B}\) as in Definition 2.6. Then the natural map
is an isomorphism for any family \(\mathcal {F}\) of subgroups of G containing all the stabilizers of the Gaction on \(\textrm{FIN}^\times (Z)\). This applies in particular when Z is a proper Gset and \(\mathcal {F}= \mathcal {FIN}\) is the family of finite subgroups. \(\square \)
Proof of Theorem 5.4
We prove Theorem 5.4 by considering three successively more general cases:
Case 1
A satisfies the assumptions of Theorem 2.8 for the unital inclusion \(\iota :\mathbb {C}\rightarrow A\).
By assumption, there is a \(C^*\)algebra B and an element \(\phi \in KK(B,A)\) such that \(\iota \oplus \phi \in KK(\mathbb {C}\oplus B,A)\) is a KKequivalence. Using the weak Kequivalence of \(\mathcal {J}^Z_B\) and \(A^{\otimes Z}\) constructed in Theorem 2.8, we may identify the maps in (5.2) with the natural maps
where the first map is induced from the (nonunital) inclusion \(\mathbb {C}\hookrightarrow \mathcal {J}^Z_B\) corresponding to \(F=\emptyset \). We may therefore replace \(A^{\otimes Z}\) by \(\mathcal {J}^Z_B\) throughout the proof. The corresponding statement for \(\mathcal {J}^Z_B\) then follows from Corollary 5.7 since by the decomposition \(\mathcal {J}^Z_B=\mathbb {C}\oplus {\dot{\mathcal {J}}}^Z_B\), we have
and consequently a pushout diagram
In the torsion free case this becomes
Case 2
\(A=M_n\otimes D\) where D is as in Case 1.
By applying Case 1 and localizing at n, we see that the maps in (5.2) (for D instead of A) induce a pushout diagram
and in the torsionfree case an isomorphism
Here we have used that localization at n commutes with taking colimits. By Theorem 3.5, the unital inclusion \(D\rightarrow M_n\otimes D=A\) induces an isomorphism \(K_*\left( D^{\otimes Z}\rtimes _r H\right) [1/n]\cong K_*\left( A^{\otimes Z}\rtimes _r H\right) \) for every subgroup H of G. This finishes the proof of Case 2.
Case 3
where each \(A_k\) is as in Case 2.
For each k, denote by \(S_k\subseteq S\) the set of positive integers n such that the unit \([1]\in K_0(A_k)\) is divisible by n. Note that we have \(S=\bigcup _k S_k\). By Case 2, the conclusion of the theorem holds if we replace A by \(A_k\) and S by \(S_k\). Now the general case follows by taking the filtered colimit along k. \(\square \)
6 Rational and kadic computations
In this section we give systematic tools to compute the Ktheory of noncommutative Bernoulli shifts up to localizing at a supernatural number \(\mathfrak {n}\) (i.e. at the set of prime factors of \(\mathfrak {n}\)). The results apply to unital \(C^*\)algebras A for which the inclusion \(\iota :\mathbb {C}\rightarrow A\) does not induce a split injection \(K_0(\mathbb {C})\rightarrow K_0(A)\) integrally, but a split injection
after localizing at a supernatural number \({\mathfrak {n}}\). Important special cases are \({\mathfrak {n}}=k^\infty \) for \(k\in \mathbb {N}\) (Example 6.1), or when \({\mathfrak {n}}=\prod _p p^{\infty }\) (Example 6.2). The latter case amounts to rational Ktheory computations since in this case we have \(L[1/\mathfrak n]\cong L\otimes _\mathbb {Z}\mathbb {Q}\) for any abelian group L. We give two examples of when one of these situations naturally occurs:
Example 6.1
(\(M_{\mathfrak {n}}=M_{k^{\infty }}\)) Let A be a unital \(C^*\)algebra that admits a finitedimensional representation \(A \rightarrow M_k(\mathbb {C})\) for some k. Then the unital inclusion \(\iota :\mathbb {C}\rightarrow A\) induces a splitinjection \(K_*(\mathbb {C})[1/k] \rightarrow K_*(A)[1/k]\). Concrete examples are unital continuous trace \(C^*\)algebras or subhomogeneous \(C^*\)algebras.
Example 6.2
(\(M_{\mathfrak {n}}={\mathcal {Q}}\)) Let A be a unital \(C^*\)algebra for which the unit \([1]\in K_0(A)\) is not torsion, for instance let A be unital and stably finite. Then the inclusion \(\iota :\mathbb {C}\rightarrow A\) induces a split injection \(K_*(\mathbb {C})\otimes _\mathbb {Z}\mathbb {Q}\rightarrow K_*(A)\otimes _\mathbb {Z}\mathbb {Q}\).
Theorem 6.3
(c.f. Theorem 2.8) Let G be a discrete group satisfying BCC, let Z be a countable Gset and let \({\mathfrak {n}}\) be a supernatural number. Let A be a unital \(C^*\)algebra satisfying the UCT such that the unital inclusion \(\mathbb {C}\rightarrow A\) induces a split injection \(K_*(\mathbb {C})[1/\mathfrak n]\rightarrow K_*(A)[1/{\mathfrak {n}}]\). Denote by \({\tilde{K}}_*(A)\) the cokernel of the injection \(K_*(\mathbb {C}) \rightarrow K_*(A)\) and let B be any \(C^*\)algebra satisfying the UCT with \(K_*(B)\cong {\tilde{K}}_*(A)\). Then we have
Proof
Replacing \({\mathfrak {n}}\) by \({\mathfrak {n}}^\infty \), we may assume that \({\mathfrak {n}}\) is of infinite type. By the UCT, there is a UCT \(C^*\)algebra B whose Ktheory is isomorphic to the cokernel \({\tilde{K}}_*(A)\) of \(K_*(\iota )\), and an element \(\phi \in KK(B\otimes M_{\mathfrak {n}},A\otimes M_{\mathfrak {n}})\) which together with the inclusion \(\iota _{\mathfrak {n}}:M_{\mathfrak {n}}\rightarrow A\otimes M_{\mathfrak {n}}\) induces a KKequivalence
We can thus apply Theorem 2.8 for \(A\otimes M_{{\mathfrak {n}}}\) in place of A, for \(A_0=M_{{\mathfrak {n}}}\) and for \(B\otimes M_{{\mathfrak {n}}}\) in place of B. We get
where the first and last isomorphisms are obtained from Theorem 3.5. \(\square \)
Theorem 5.4 has a counterpart as well:
Theorem 6.4
Let G be a discrete group satisfying BCC, let Z be a countable, proper Gset, and let \({\mathfrak {n}}\) a supernatural number. Let A be a unital \(C^*\)algebra which is a unital filtered colimit of \(C^*\)algebras \(A_k\) satisfying the assumptions of Theorem 6.3. Then, the natural inclusions
induce a pushout diagram
In particular, if G is torsionfree, this pushout diagram reads
Proof
Without loss of generality we may assume that \({\mathfrak {n}}\) is of infinite type and A satisfies the assumption of Theorem 6.3. Let \(B,\phi \) and \(\iota _{\mathfrak {n}}\) be as in the proof of Theorem 6.3. In particular, Theorem 2.8 applies so that \((A\otimes M_{\mathfrak n})^{\otimes Z}\) is weakly Kequivalent to \(\mathcal J_{M_{\mathfrak {n}},B\otimes M_{\mathfrak {n}}}^Z\). As in the proof of Theorem 5.4, the theorem now follows from the combination of Corollary 5.7 and Theorem 3.5. \(\square \)
Example 6.5
Let \(p\in C(X)\otimes M_n\) be a rankk projection over C(X) corresponding to some vector bundle on X. Let \(A=p(C(X)\otimes M_n) p\) be the associated homogeneous \(C^*\)algebra. Of course, A is Moritaequivalent to C(X) but the unitinclusion \(\iota :\mathbb {C}\rightarrow A\) is not splitinjective in general; it corresponds to the inclusion \(\mathbb {Z}[p] \rightarrow K^0(X)\). On the other hand, A has a kdimensional irreducible representation and therefore satisfies the assumptions of Theorem 6.3. Therefore, we get
where B is any UCT \(C^*\)algebra whose Ktheory is isomorphic to the cokernel of \(K_*(\iota )\).
As an application to Theorem 6.3, we obtain a proof of the fact that Bernoulli shifts by finite groups rarely have the Rokhlin property (see [13, Definition 3.1]). This result is possibly known to experts and we would like to thank N. C. Phillips for mentioning it to us.
Corollary 6.6
Let A be a unital \(C^*\)algebra satisfying the UCT such that \([1]\in K_0(A)\) is not torsion (for instance, suppose that A is stably finite). Let \(G\ne \{e\}\) be a finite group and let Z a Gset. Then the Bernoulli shift of G on \(A^{\otimes Z}\) does not have the Rokhlin property.
Proof
It follows from Theorem 6.3 that the inclusion \(C^*_r(G)\hookrightarrow A^{\otimes Z}\rtimes _r G\) induces a split injection
Now assume that the action of G on \(A^{\otimes Z}\) has the Rokhlin property. Using [23, Theorem 2.6], we can find a unital equivariant \(*\)homomorphism \(C(G)\rightarrow A^{\otimes Z}\). But then we can factor the map in (6.2) as the composition
which is never injective unless \(G=\{e\}\). \(\square \)
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Notes
Note that this determines B uniquely up to KKequivalence.
We use the term ‘filtered colimit’ (which is the standard categorical notion) instead of the terms ‘direct limit’ or ‘inductive limit’ which seem to be more commonly used in \(C^*\)algebra theory. We do this to be consistent with the the use of more general colimits of functors in Sect. 5.
Recently, \(KK^G\) was even refined to a stable \(\infty \)category [2].
Since the Gaction on F factors through the finite group \(\textrm{Sym}(F)\), the theorem is applicable even if G itself is not finite.
If \(S\subseteq S'\) the connecting map is the tensor product of the identity on \((A_0\oplus B)^{\otimes S}\) with the canonical inclusion \(A_0^{\otimes ZS}=A_0^{\otimes ZS'}\otimes A_0^{\otimes S'S}\hookrightarrow A_0^{\otimes ZS'}\otimes (A_0\oplus B)^{\otimes S'S}\).
This is a welldefined functor since different choices for g give the same map on Ktheory. On the other hand, \(G/H\mapsto A\rtimes _rH\) does not define a functor. This is why it requires more care to upgrade this to a functor taking values in the category of spectra, see [9, Section 2].
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Acknowledgements
The authors would like to thank Arthur Bartels and Eusebio Gardella for helpful discussions and N. Christopher Phillips for pointing out Corollary 6.6 to them.
Funding
Open Access funding enabled and organized by Projekt DEAL. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) ProjectID 427320536 SFB 1442 and under Germany’s Excellence Strategy EXC 2044 390685587, Mathematics Münster: Dynamics, Geometry, Structure. The first author was also supported by DST, Government of India under the DSTINSPIRE Faculty Scheme with Faculty Reg. No. IFA19MA139.
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Chakraborty, S., Echterhoff, S., Kranz, J. et al. Ktheory of noncommutative Bernoulli shifts. Math. Ann. 388, 2671–2703 (2024). https://doi.org/10.1007/s0020802302587w
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DOI: https://doi.org/10.1007/s0020802302587w