Crossed products and twisted k-graph algebras

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New York Journal of Mathematics

New York J. Math.22(2016) 229–250.

Crossed products and twisted k-graph algebras

Nathan Brownlowe, Valentin Deaconu, Alex Kumjian and David Pask

Abstract. An automorphismβof ak-graph Λ induces a crossed prod- uctC^∗(Λ)o^βZwhich is isomorphic to a (k+1)-graph algebraC^∗(Λ×βZ).

In this paper we show how this process interacts with k-graph C^∗- algebras which have been twisted by an element of their second cohomology group. This analysis is done using a long exact sequence in cohomology associated to this data. We conclude with some examples.

Contents

1. Background 231

1.1. Higher-rank graphs 231

1.2. Cubical cohomology ofk-graphs 232

1.3. Crossed product graphs 233

1.4. The long exact sequence of cohomology 234

1.5. Twistedk-graph C^∗-algebras 236

2. Main results 237

3. Examples 242

References 248

A higher-rank graph (ork-graph) is a countable category Λ together with a functor d : Λ → N^k satisfying a factorisation property. For k = 1, Λ is the path category of a directed graph EΛ. In general we view ak-graph as a higher dimensional analog of a directed graph. In [14] it was shown how to associate aC^∗-algebra to ak-graph in such a way that fork= 1 we have C^∗(Λ) =C^∗(E_Λ).

The universal property of a k-graph C^∗-algebra C^∗(Λ) implies that an automorphismβ of Λ induces an automorphismβ ofC^∗(Λ) and hence gives rise to a crossed productC^∗(Λ)oβZ. The results of [8] show that there is

Received January 14, 2015.

2010Mathematics Subject Classification. Primary 46L05; Secondary 18G60, 55N10.

Key words and phrases. Higher-rank graph;C^∗-algebra; cohomology; crossed product.

This research was supported by the Australian Research Council and the University of Wollongong Research Committee.

ISSN 1076-9803/2016

229

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N. BROWNLOWE, V. DEACONU, A. KUMJIAN AND D. PASK

a (k+ 1)-graph Λ×_βZ such thatC^∗(Λ)oβZis isomorphic toC^∗(Λ×_βZ).

The purpose of this paper is to examine how this situation generalizes in the setting of twistedk-graph C^∗-algebras.

Recent attention has been drawn to the homological properties of a k- graph Λ which are nontrivial when k ≥ 2. Specifically in [16, 17] two cohomology theories for a k-graph Λ are described: cubical and categorical. Twisted versions of k-graph C^∗-algebras are introduced in both cases using 2-cocycles. Here we work with the cubical cohomology which is more tractable. Ifϕis a (cubical)T-valued 2-cocycle on Λ, the twistedC^∗-algebra is denoted byC_ϕ^∗(Λ).

In [17] it is shown that the cubical and categorical cohomologies for a k-graph agree forH⁰, H¹ andH². An isomorphism between the two twisted versions ofk-graphC^∗-algebras compatible with the isomorphism inH² was also proven in [17].

If β is an automorphism of a k-graph Λ, then [16] gives a long exact sequence for the homology of the (k+ 1)-graph Λ×_β Z in the categorical context. In this paper we describe the analogous cohomology sequence in the cubical context (see Proposition 1.8) and use it to generalise the result in [8] in three different ways.

Our main result, Theorem 2.1 shows that if we twist the C^∗-algebra of Λ×_β Z by a 2-cocycle ϕ then the resulting C^∗-algebra C_ϕ^∗(Λ×_βZ) is isomorphic to the crossed product of a certain twisted C^∗-algebra of Λ (with twisting cocycle obtained by restrictingϕ) by an automorphism associated to β and ϕ. Applying this result in different contexts, associated to the exact sequence outlined in Proposition 1.8 yields Corollary 2.3which deals with the case where the class of the restriction of ϕis trivial; Corollary2.4 asserts that ifψis a 2-cocycle on Λ whose cohomology class is left invariant by β, then there is an automorphism of C_ψ^∗(Λ) which is compatible with β for which the crossed product is isomorphic to a twisted C^∗-algebra of Λ×_βZ. The case when β is trivial which motivated this work is discussed in Corollary2.5.

We conclude with a section of examples of twisted k-graph C^∗-algebras arising as crossed products. In each case the twistedk-graphC^∗-algebra lies in a classifiable class of C^∗-algebras. In Example 3.1 we consider quasifree automorphisms on Cuntz algebras and show how they arise in the setting of Corollary 2.3. In Example3.2we use Theorem2.1 to compute the cohomology of a 2-graph with infinitely many vertices which arises as a crossed product. In Example 3.3we use other techniques to compute the cohomology of a 3-graph with one vertex which arises as a crossed product. We also consider a family of 2-cocycles on a 3-graph for which the associated C^∗-algebra is isomorphic toO₂.

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1. Background

We begin by giving some background onk-graphs, their cubical cohomology, and crossed-product graphs induced by automorphisms ofk-graphs. We then prove the existence of a long exact sequence involving the cohomology groups of a k-graph and a crossed product graph. We finish with recalling the twistedk-graph C^∗-algebras introduced in [16].

1.1. Higher-rank graphs. We adopt the conventions of [14, 15, 21] for k-graphs. Given a nonnegative integer k, a k-graph is a nonempty countable small category Λ equipped with a functor d : Λ → N^k satisfying the factorisation property: for all λ∈Λ and m, n∈N^k such thatd(λ) =m+n there exist unique µ, ν ∈ Λ such that d(µ) = m, d(ν) = n, and λ = µν.

Whend(λ) =nwe sayλhasdegree n. We will typically usedto denote the degree functor in any k-graph in this paper.

For k ≥ 1, the standard generators of N^k are denoted e1, . . . , e_k, and for n ∈ N^k and 1 ≤ i ≤ k we write n_i for the i^th coordinate of n. For n= (n₁, . . . , n_k) ∈N^k let |n|:=Pk

i=1n_i; for λ∈ Λ we define|λ|:= |d(λ)|.

Form, n∈N^k, we writem∨nfor the coordinatewise maximum of the two, and write m≤nifm_i≤n_i fori= 1, . . . , k.

Forn∈N^k, we write Λⁿford⁻¹(n). Thevertices of Λ are the elements of Λ⁰. The factorisation property implies that o7→id_o is a bijection from the objects of Λ to Λ⁰. We will frequently and without further comment use this bijection to identify Obj(Λ) with Λ⁰. The domain and codomain maps in the category Λ then become maps s, r: Λ→Λ⁰. More precisely, forα ∈Λ, thesource s(α) is the identity morphism associated with the object dom(α) and similarly, r(α) = id_cod(α). An edge is a morphism f with d(f) =e_i for somei= 1, . . . , k.

Let λ be an element of a k-graph Λ and suppose m, n ∈ N^k satisfy 0 ≤ m ≤ n ≤ d(λ). By the factorisation property there exist unique elements α, β, γ∈Λ such that

λ=αβγ, d(α) =m, d(β) =n−m, and d(γ) =d(λ)−n.

We defineλ(m, n) :=β. In particularα =λ(0, m) andγ =λ(n, d(λ)).

Forα, β ∈Λ and E⊂Λ, we write αE for{αλ:λ∈E, r(λ) =s(α)} and Eβfor{λβ :λ∈E, s(λ) =r(β)}. So foru, v∈Λ⁰, we haveuE =E∩r⁻¹(u), Ev=E∩s⁻¹(v) and uEv=uE∩Ev.

Suppose that Λ is row-finite with no sources, that is, for all v ∈ Λ⁰ and n∈N^k we have 0<|vΛⁿ|<∞. By [18, Remark A.3], Λ is cofinal if for all v, w∈Λ⁰ there is N ∈N^k such that for all α∈vΛ^N we have wΛs(α)6=∅.

And by [27, Lemma 3.2 (iv)], Λ is aperiodic (or satisfies the aperiodicity condition) if for every v ∈ Λ⁰ and each pair m 6= n∈ N^k, there is λ∈ vΛ such thatd(λ)≥m∨nand

(1) λ(m, m+d(λ)−(m∨n))6=λ(n, n+d(λ)−(m∨n)).

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Ak-graph Λ can be visualized by its 1-skeleton: This is a directed graph EΛ with vertices Λ⁰ and edges ∪^k_i=1Λ^eⁱ which have range and source inEΛ

determined by their range and source in Λ. Each edge in E_Λ with degree e_i is assigned the same colour, soE_Λ is a coloured graph. It is common to call edges with degree e1 in Λ blue edges in EΛ and draw them with solid lines; edges with degreee₂ in Λ are then called red edges and are drawn as dashed lines. In practice, along with the 1-skeleton we give a collection of commuting squares or factorisation rules which relate the edges of E_Λ that occur in the factorisation of morphisms of degree ei+ej (i6=j) in Λ. For more information about 1-skeletons we refer the reader to [25].

A functor β : Λ→ Γ between k-graphs is a k-graph morphism if it pre- serves degree, that isdΓ◦β=dΛ. If Γ = Λ andβ is invertible thenβ is an automorphism. The collection Aut Λ of automorphisms of Λ forms a group under composition.

LetT₁ be the categoryNregarded as a 1-graph with degree functor given by the identity map.

1.2. Cubical cohomology of k-graphs. Fork≥0 define 1_k:=

k

X

i=1

ei ∈N^k.

By conventionN⁰ ={0}and 10 = 0.

Definition 1.1. Let Λ be ak-graph. Forr ≥0 let Qr(Λ) ={λ∈Λ :d(λ)≤1_k,|λ|=r}.

We haveQ₀(Λ) = Λ⁰,Q₁(Λ) =S_k

i=1Λ^eⁱ the set of edges in Λ andQ_r(Λ) =∅ ifr > k. For 0< r≤k the setQ_r(Λ) consists of the morphisms in Λ which may be expressed as the composition of a sequence of r edges with distinct degrees. We regard elements ofQ_r(Λ) as unitr-cubes in the sense that each one gives rise to a commuting diagram of edges in Λ shaped like anr-cube.

In particular, when r ≥ 1, each element of Q_r(Λ) has 2r faces in Qr−1(Λ) defined as follows.

Definition 1.2. Fix λ ∈ Qr(Λ) and write d(λ) = ei1 +· · ·+eir where i₁ < · · · < i_r. For 1 ≤ j ≤ r, define F_j⁰(λ) and F_j¹(λ) to be the unique elements of Qr−1(Λ) such that there existµ, ν ∈Λ^e^ij satisfying

F_j⁰(λ)ν=λ=µF_j¹(λ).

In [16] the cubical homology of Λ is identified with the homology of the complex (ZQ∗, ∂∗) where the boundary map ∂r :ZQr → ZQr−1 is determined by

∂_rλ=

r

X

j=1 1

X

`=0

(−1)^j+`F_j^`(λ).

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Remark 1.3. If β ∈ Aut(Λ), then it is straightforward to check that the induced action of β on ZQr commutes with∂r. We first observe that

F_j⁰(βλ) =βF_j⁰(λ) andF_j¹(βλ) =βF_j¹(λ);

and hence

∂r(βλ) =

r

X

j=1 1

X

`=0

(−1)^j+`F_j^`(βλ) =

r

X

j=1 1

X

`=0

(−1)^j+`βF_j^`(λ) =β∂r(λ).

Notation 1.4. Let Λ be a k-graph and let A be an abelian group. For r ≥0, we writeC^r(Λ, A) for the collection of all functions f :Q_r(Λ) →A.

Identify C^r(Λ, A) with Hom(ZQr(Λ), A) in the usual way. Define maps δ^r:C^r(Λ, A)→C^r+1(Λ, A) by

δ^r(f)(λ) :=f(∂r+1(λ)) =

r+1

X

j=1 1

X

`=0

(−1)^j+`f(F_j^`(λ)).

Then (C^∗(Λ, A), δ^∗) is a cochain complex.

Definition 1.5. We define thecubical cohomology H^∗(Λ, A) of the k-graph Λ with coefficients inAto be the cohomology of the complex (C^∗(Λ, A), δ^∗);

that is H^r(Λ, A) := ker(δ^r)/Im(δ^r−1). Forr ≥0, we write Z^r(Λ, A) := ker(δ^r)

for the group ofr-cocycles, and for r >0, we write B^r(Λ, A) = Im(δ^r−1) for the group ofr-coboundaries.

Remark 1.6. For each 0≤ r ≤k we define β^∗ :C^r(Λ, A)→ C^r(Λ, A) by β(f) =f ◦β. For eachf ∈C^r(Λ, A) andλ∈Q_r(Λ) we have

δ^rβ^∗(f)(λ) =

r+1

X

j=1 1

X

`=0

f(F_j^`(β(λ))) =δ^r(f)(βλ) =β^∗δ^r(f)(λ), and so β^∗◦δ^r =δ^r◦β^∗. Henceβ^∗ induces a homomorphism

β^∗ :H^∗(Λ, A)→H^∗(Λ, A).

1.3. Crossed product graphs. Recall from [8] that if Λ is a row-finite k-graph with no sources andβ ∈Aut Λ, then there is a (k+ 1)-graph Λ×_βZ with morphisms Λ×N, range and source maps given by r(λ, n) = (r(λ),0), s(λ, n) = (β⁻ⁿ(s(λ)),0), degree map given by d(λ, n) = (d(λ), n) and composition given by

(λ, m)(µ, n) := (λβ^m(µ), m+n).

Evidently, Λ×_βZis also row-finite with no sources and (Λ×_βZ)⁰= Λ⁰×{0}.

Remark 1.7. If β= id, note that Λ×_βZ= Λ×T1.

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Recall from [16,§4] that we may describe ther-cubes of Λ×_βZin terms of the cubes of Λ. The 0-cubes are given byQ0(Λ×_βZ) =Q0(Λ)× {0}. For each 0≤r≤k the (r+ 1)-cubes are given by

Qr+1(Λ×_βZ) ={(λ,1) :λ∈Qr(Λ)} ∪ {(λ,0) :λ∈Qr+1(Λ)}.

Observe that forλ∈Qr(Λ),F_j^`(λ,0) = (F_j^`(λ),0) and (2) F_j^`(λ,1) =







(F_j^`(λ),1) ifj≤r,

(λ,0) ifj=r+ 1, `= 0, (β⁻¹(λ),0) ifj=r+ 1, `= 1.

So for f ∈C^r(Λ×_βZ, A), we have δ^r(f)(λ,0) =

r+1

X

j=1 1

X

`=0

(−1)^j+`f(F_j^`(λ),0) forλ∈Q_r+1(Λ) and

δ^r(f)(λ,1) = (−1)^r+1(f(λ,0)−f(β⁻¹(λ),0)) (3)

+

r

X

j=1 1

X

`=0

(−1)^j+`f(F_j^`(λ),1), for each λ∈Q_r(Λ).

1.4. The long exact sequence of cohomology. Supposeβ is an automorphism of a k-graph Λ. In [16, Theorem 4.13] the authors presented a long exact sequence relating the homology groups of Λ and Λ×_βZ. In the next result we present the corresponding long exact sequence of cohomology.

Proposition 1.8. Suppose β is an automorphism of ak-graphΛ, andA is an abelian group. There is a long exact sequence

0−→H⁰(Λ×_βZ, A) ⁱ

∗

−→H⁰(Λ, A) ^1−β

∗

−−−→H⁰(Λ, A)

j^∗

−→H¹(Λ×_β Z, A) ⁱ

∗

−→ · · ·^1−β

∗

−→ H^r(Λ, A)

j^∗

−→H^r+1(Λ×_βZ, A) ⁱ

∗

−→H^r+1(Λ, A)^1−β

∗

−→ H^r+1(Λ, A)

j^∗

−→ · · · ^j

∗

−→H^k(Λ×_βZ, A) ⁱ

∗

−→H^k(Λ, A)

1−β^∗

−−−→H^k(Λ, A) ^j

∗

−→H^k+1(Λ×_βZ, A)−→0, where

i^∗(f)(λ) =f(λ,0) for each f ∈Z^r(Λ×_βZ, A), and

j^∗(f)(λ,0) = 0 and j^∗(f)(λ,1) =f(λ) for each f ∈Z^r−1(Λ, A).

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Proof. For each 0 ≤ r ≤ k the maps i : ZQ_r(Λ) → ZQ_r(Λ×_β Z) and j:ZQr+1(Λ×_βZ)→ZQr(Λ) determined by

i(λ) = (λ,0) forλ∈Qr(Λ) j(λ, `) =

(0 if`= 0

λ if`= 1 for (λ, `)∈Q_r+1(Λ×_βZ) induce maps

i^∗ :C^r(Λ×_βZ, A)→C^r(Λ, A) and j^∗ :C^r(Λ, A)→C^r+1(Λ×_βZ, A) given by

i^∗(f)(λ) =f(λ,0) forf ∈C^r(Λ×_βZ, A) j^∗(f)(λ, `) =

(0 if`= 0

f(λ) if`= 1 forf ∈C^r(Λ, A).

Using the description of the cubes in Λ×_βZwe obtain a short exact sequence of complexesE where

(4) E_r: 0→C^r−1(Λ, A) ^j

∗

→C^r(Λ×_βZ, A) ⁱ

∗

→C^r(Λ, A)→0, sincei^∗ and j^∗ commute with the coboundary maps. Indeed,

(δ^ri^∗)f(λ) =

r+1,1

X

j=1,`=0

(i^∗f)(F_j^`(λ)) =

r+1,1

X

j=1,`=0

f(F_j^`(λ),0) =δ^rf(λ,0)

= (i^∗δ^r)f(λ) forf ∈C^r(Λ×_βZ, A), and

(δ^rj^∗)f(λ,1) = (−1)^r+1 (j^∗f)(λ,0)−(j^∗f)(β⁻¹(λ),0) +

r,1

X

j=1,`=0

(−1)^j+`(j^∗f)(F_j^`(λ),1)

=

r,1

X

j=1,`=0

(−1)^j+`f(F_j^`(λ)) =δ^r−1f(λ) = (j^∗δ^r−1)f(λ,1) and

(δ^rj^∗)f(λ,0) =

r+1,1

X

j=1,`=0

(−1)^j+`(j^∗f)(F_j^`(λ),0) = 0 = (j^∗δ^r−1)f(λ,0) forf ∈C^r−1(Λ, A).

Using the long exact sequence associated to a short exact sequence of homology complexes (see [19, Theorem II.4.1]) applied to a short exact sequence of cohomology complexes with the appropriate reindexing, we obtain

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the long exact sequence

· · ·^δ

r

→E H^r(Λ, A) ^j

∗

→H^r+1(Λ×_βZ, A)

i^∗

→H^r+1(Λ, A)^δ

r+1

→E H^r+1(Λ, A)→ · · · . Indeed, the boundary mapδE (see [19, II.4]) is defined as follows. Start with m∈Z^r(Λ, A) and take the liftn∈C^r(Λ×_β Z, A) given by n(λ,0) =m(λ) and n(λ,1) = 0. Thenδ^rn(λ,0) = 0 for all λ, so there is c∈Z^r(Λ, A) such that δ^rn=j^∗c, that is δ^rn(λ,1) =c(λ). Then δE takes the class ofm into the class ofc. Using (2) and (3) we get

c(λ) =j^∗(c)(λ,1) =δ^rn(λ,1) = (−1)^r+1(n(λ,0)−n(β⁻¹(λ),0))

= (−1)^r+1(m−m◦β⁻¹)(λ).

Hence we see that δ^r_E = (−1)^r+1(1−(β⁻¹)^∗). By using a similar argument as in [16, Theorem 4.13], the sequence remains exact after replacingδE with

1−β^∗.

Ifβ= id, then as noted in Remark1.7Λ×_βZmay be identified with Λ×T₁ and 1−β^∗ = 0. Hence, for allr, we have the short exact sequence

0→H^r(Λ, A) ^j

∗

−→H^r+1(Λ×T1, A) ⁱ

∗

−→H^r+1(Λ, A)→0.

Moreover there is a mapσ:C^r(Λ, A)→C^r(Λ×T₁, A) such thatσ(f)(λ,0) = f(λ) for λ ∈ Q_r(Λ) and, if r ≥ 1, σ(f)(λ,1) = 0 for λ ∈ Qr−1(Λ). It is straightforward to check that σ intertwines boundary maps and that i^∗σ(f) =f for all f ∈C^r(Λ, A). Hence, the map

(5) Ξ : (f, g)∈Z^r(Λ, A)⊕Z^r+1(Λ, A)7→j^∗(f) +σ(g)∈Z^r+1(Λ×T1, A) is an isomorphism which intertwines the boundary maps. We thereby obtain the following result:

Corollary 1.9. If β = id, then H⁰(Λ×T₁, A) ∼= H⁰(Λ, A) and for r ≥ 0 the map on cohomology induced byΞ is an isomorphism

Ξ :H^r(Λ, A)⊕H^r+1(Λ, A)∼=H^r+1(Λ×_βZ, A).

1.5. Twisted k-graph C^∗-algebras.

Definition 1.10. Let Λ be a row-finite k-graph with no sources and fix ϕ ∈ Z²(Λ,T). A Cuntz–Krieger ϕ-representation of Λ in a C^∗-algebra A is a set {p_v : v ∈ Λ⁰} ⊆ A of mutually orthogonal projections and a set {s_λ :λ∈Q₁(Λ)} ⊆A satisfying:

(TG1) for all 1≤i≤kand λ∈Λ^eⁱ,s^∗_λsλ=p_s(λ);

(TG2) for all 1≤i < j ≤kandµ, µ⁰ ∈Λ^eⁱ,ν, ν⁰ ∈Λ^e^j such thatµν =ν⁰µ⁰, s_ν⁰s_µ⁰ =ϕ(µν)sµsν; and

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(TG3) for all v∈Λ⁰ and alli= 1, . . . , k such thatvΛ^eⁱ 6=∅, p_v = X

λ∈vΛ^ei

s_λs^∗_λ.

Definition 1.11. Let Λ be a row-finite k-graph with no sources and let ϕ∈Z²(Λ,T). We defineC_ϕ^∗(Λ) to be the universalC^∗-algebra generated by a Cuntz–Kriegerϕ-representation of Λ.

Remark 1.12. Ifϕ∈Z²(Λ,T) is the trivial cocycle, thenϕ∈B²(Λ,T) and so by [17, Proposition 5.3] and [17, Proposition 5.6] we haveC_ϕ^∗(Λ)∼=C^∗(Λ).

Remark 1.13. In the context of twisted k-graph C^∗-algebras, we shall be particularly interested in the following part of the exact sequence forA=T

· · · →H¹(Λ,T) ^1−β

∗

−−−→H¹(Λ,T) ^j

∗

−→H²(Λ×_βZ,T)

i^∗

−→H²(Λ,T) ^1−β

∗

−−−→H²(Λ,T)→ · · · 2. Main results

In this section we present ourC^∗-algebraic results. In our main result we generalise the isomorphismC^∗(Λ×_βZ)∼=C^∗(Λ)oβ˜Zfrom [8, Theorem 3.4]

(in the case l = 1) to the twisted setting. Note that for A = T we use multiplicative notation; inverses are given by conjugation, and the identity element is 1∈T.

Theorem 2.1. LetΛbe a row-finitek-graph with no sources, letβ ∈Aut(Λ) and let ϕ∈Z²(Λ×_βZ,T). Then

(i) There is an automorphism β_ϕ of C_i^∗∗(ϕ)(Λ) such that (6) β_ϕ(p_v) =p_βv and β_ϕ(s_e) =ϕ(βe,1)s_βe,

for all v∈Q0(Λ)and e∈Q1(Λ).

(ii) Let (F_n)n∈Nbe an increasing family of finite subsets ofΛ⁰ such that

∪_n∈_NFn= Λ⁰. The sequence(P

v∈Fns_(v,1))n∈Nconverges strictly to a unitary U ∈ MC_ϕ^∗(Λ×_βZ) satisfying

(7) U p_(v,0)U^∗ =p_(βv,0) and U s_(e,0)U^∗ =ϕ(βe,1)s_(βe,0), for all v∈Q0(Λ)and e∈Q1(Λ).

(iii) There is a homomorphismπ :C_i^∗∗(ϕ)(Λ)→C_ϕ^∗(Λ×_βZ)which forms a covariant pair (π, U) whose integrated form

π×U :C_i^∗∗(ϕ)(Λ)oβϕ Z→C_ϕ^∗(Λ×_βZ) is an isomorphism.

Remark 2.2. In the proof of this theorem we need to calculate the faces of a cube (βλ,1)∈Q₃(Λ×_β Z), where λ∈Q₂(Λ). Supposeλ=ef =f⁰e⁰, where e, e⁰ ∈Λ^eⁱ and f, f⁰ ∈Λ^e^j such that 1≤i < j≤k. We can factorise (βλ,1) according to the following diagram, and then calculate its faces.

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. .

(βe,0) (βe⁰,0)

(e,0) (e⁰,0)

(βf⁰,0) (βf,0)

(f⁰,0) (f,0)

(βr(f⁰),1) (βr(f),1) (βs(f⁰),1) (βs(f),1)

F₁⁰(βλ,1) = (βf⁰,1) F₁¹(βλ,1) = (βf,1) F₂⁰(βλ,1) = (βe,1) F₂¹(βλ,1) = (βe⁰,1) F₃⁰(βλ,1) = (βλ,0) F₃¹(βλ,1) = (λ,0) Proof of Theorem 2.1. Let λ ∈ Q2(Λ). Write λ = ef = f⁰e⁰, where e, e⁰ ∈ Λ^eⁱ and f, f⁰ ∈ Λ^e^j such that 1 ≤ i < j ≤ k. Using (3) and the identities in Remark2.2 we get

(8) 1 = δ²(ϕ)(βλ,1) =ϕ(βλ,0)ϕ(λ,0)ϕ(βf⁰,1)ϕ(βe,1)ϕ(βe⁰,1)ϕ(βf,1).

For eachv∈Q0(Λ) letPv :=pβv and for each e∈Q1(Λ) let S_e:=ϕ(βe,1)s_βe.

We claim that {P, S} defines a i^∗(ϕ)-representation of Λ in C_i^∗∗(ϕ)(Λ). We check condition (TG2) using (8):

S_f⁰S_e⁰ =ϕ(βf⁰,1)ϕ(βe⁰,1)s_βf⁰s_βe⁰

=ϕ(βf⁰,1)ϕ(βe⁰,1)i^∗(ϕ)(βeβf)s_βes_βf

=ϕ(βf⁰,1)ϕ(βe⁰,1)ϕ(β(ef),0)sβesβf

=ϕ(βf⁰,1)ϕ(βe⁰,1)ϕ(β(ef),0)ϕ(βe,1)ϕ(βf,1)S_eS_f

=ϕ(ef,0)S_eS_f =i^∗(ϕ)(ef)S_eS_f.

Conditions (TG1) and (TG3) follow easily. By the universal property of C_i^∗∗(ϕ)(Λ) we obtain a homomorphism β_ϕ satisfying

βϕ(pv) =p_βv and βϕ(se) =ϕ(βe,1)s_βe.

Similar calculations show that the collection{p_β−1v, ϕ(e,1)s_β⁻¹_e} is also an i^∗(ϕ)-representation of Λ in C_i^∗∗(ϕ)(Λ), and the corresponding homomorphism coming from the universal property of C_i^∗∗(ϕ)(Λ) is the inverse ofβ_ϕ. So βϕ is an automorphism, and (i) holds.

For any finite subsetF ⊆Λ⁰ we denote by P(F) :=X

v∈F

p_(v,0)∈C^∗(Λ×_βZ).

To see that (ii) holds, first let (Fn)n∈N be an increasing sequence of finite subsets of Λ⁰ such that ∪_n∈_NF_n = Λ⁰. Then a standard argument shows thatP(Fn)→1 strictly inMC_ϕ^∗(Λ×_βZ). Forn≥1 letUn:=P

v∈Fns_(v,1).

(11)

Since the elements in the sum defining U_n have the same degree, by (TG1) we have

U_n^∗U_n= X

v,w∈Fn

s^∗_(v,1)s_(w,1) = X

v∈Fn

s^∗_(v,1)s_(v,1) = X

v∈Fn

p_(β⁻¹_v,0) (9)

=P(β⁻¹(F_n)), and by (TG3) we have

(10) UnU_n^∗= X

v,w∈Fn

s_(v,1)s^∗_(w,1) = X

v∈Fn

s_(v,1)s^∗_(v,1) = X

v∈Fn

p_(v,0) =P(Fn).

HenceU_n is a partial isometry, with initial projectionP(β⁻¹(F_n)) and final projection P(Fn).

For (w,0)∈(Λ×_βZ)⁰ and (e,0)∈(Λ×_β Z)^e^j, 1≤j ≤kwe have Unp_(w,0)=s_(βw,1) ifβw∈Fn, and zero otherwise, and (11)

U_ns_(e,0)=s_(βr(e),1)s_(e,0) ifβr(e)∈F_n, and zero otherwise;

and

p_(w,0)Un=s_(w,1) ifw∈Fn and zero otherwise, and (12)

s_(e,0)Un=s_(e,0)s_(s(e),1) ifs(e)∈Fn, and zero otherwise.

Hence Un multiplied on the left or right of any product of generators of C_ϕ^∗(Λ×_βZ) is eventually constant as n→ ∞. A standard argument shows that U_n converges strictly to an element U ∈ MC^∗(Λ×_β Z). Moreover, U is independent of the choice of Fn, and from (10) and (9) we see that U U^∗ = 1 =U U^∗. Finally from (11) and (12) it follows that for w, βw∈F_n we have U_np_(w,0)=p_(βw,0)U_n, and for βr(e), βs(e)∈F_n we have

Uns_(e,0) =s_(e,0)s_(βr(e),1)=ϕ(βe,1)s_(βe,0)s_(βs(e),1)=ϕ(βe,1)s_(βe,0)Un

We see that the identities (7) hold by taking n → ∞. This completes the proof of (ii).

For (iii) we first claim that {p_(v,0), s_(e,0)} is an i^∗(ϕ)-representation of Λ in C_ϕ^∗(Λ×_β Z). We check (TG2): for e, e⁰ ∈Λ^eⁱ and f, f⁰ ∈Λ^e^j such that ef =f⁰e⁰, where 1≤i < j ≤k, we have

s_(f⁰_,0)s_(e⁰_,0) =ϕ(ef,0)s_(e,0)s_(f,0).

Checking conditions (TG1) and (TG3) is straightforward. The universal property of C_i^∗∗(ϕ)(Λ) now gives a homomorphism

π:C_i^∗∗(ϕ)(Λ)→C_ϕ^∗(Λ×_βZ) satisfying π(pv) =p_(v,0) and π(se) =s_(e,0).

The homomorphismπ and the unitaryU from (ii) satisfy U π(s_e)U^∗ =U s_(e,0)U^∗ =ϕ(βe,1)S_βe⁰ =π ϕ(βe,1)s_βe

=π β_ϕ(s_e) ,

(12)

for each e ∈ Q₁(Λ). It follows that (π, U) is a covariant representation of (C_i^∗∗(ϕ)(Λ), βϕ), and hence by the universal property of the full crossed productC_i^∗∗(ϕ)(Λ)oβϕ Zwe get a homomorphism

π×U :C_i^∗∗(ϕ)(Λ)oβϕZ→C_ϕ^∗(Λ×_βZ).

If we denote the universal covariant pair by (iΛ, i_Z(1)), then we know that (π ×U)◦iΛ = π and π×U(i_Z(1)) = U, where π×U is the extension of π×U to the multiplier algebra MC_ϕ^∗(Λ×_βZ).

We claim thatπ×U is an isomorphism. To find the inverse we construct a ϕ-representation of Λ×_βZ inC_i^∗∗(ϕ)(Λ)oβϕZ. For each

(v,0)∈Q₀(Λ×_βZ)

let P_(v,0) :=iΛ(pv), for each (e,0)∈Q1(Λ×_βZ) let S_(e,0) :=iΛ(se), and for each (v,1)∈ Q1(Λ×_βZ) let S_(v,1) := iΛ(pv)i_Z(1). We claim that{P, S} is a Cuntz–Krieger ϕ-representation of Λ×_βZ in C_i^∗∗(ϕ)(Λ)oβϕ Z. To check (TG2) we have two cases to consider. For

(βe,1) = (βr(e),1)(e,0) = (βe,0)(βs(e),1)∈(Λ×_β Z)^eⁱ^+e^k+1 we have

S_(βr(e),1)S_(e,0) =i_Λ(p_βr(e))i_Z(1)i_Λ(s_e)

=iΛ(p_βr(e))iΛ(βϕ(se))i_Z(1)

=ϕ(βe,1)iΛ(p_βr(e))iΛ(sβe))i_Z(1)

=ϕ(βe,1)i_Λ(s_βe)i_Λ(p_βs(e))i_Z(1)

=ϕ(βe,1)S_(βe,0)S_(βs(e),1). The other case is whenef =f⁰e⁰, where

e, e⁰ ∈(Λ×_β Z)^eⁱ and f, f⁰ ∈(Λ×_β Z)^e^j and 1≤i < j ≤k. Then

(ef,0) = (e,0)(f,0) = (f⁰,0)(e⁰,0) = (f⁰e⁰,0)∈Λ×_βZ, and

S_(f⁰_,0)S_(e⁰_,0) =i_Λ(s_f⁰s_e⁰) =i^∗(ϕ)(ef)i_Λ(s_es_f) =ϕ(ef,0)S_(e,0)S_(f,0). Properties (TG1) and (TG3) follow more easily. The universal property of C_ϕ^∗(Λ×_βZ) now gives a homomorphismρP,S:C_ϕ^∗(Λ×_βZ)→C_i^∗∗(ϕ)(Λ)oβϕZ such that ρ_P,S(p_(v,0)) =P_(v,0),ρ_P,S(s_(e,0)) =S_(e,0), and ρ_P,S(s_(v,1)) =S_(v,1). One checks on generators that ρ_P,S is the inverse ofπ×U. Corollary 2.3. Let Λ be a row-finite k-graph with no sources, let β ∈ Aut(Λ) and letc∈Z¹(Λ,T). There is an automorphism β^c of C^∗(Λ) satisfying

(13) β^c(pv) =p_βv and β^c(se) =c(βe)s_βe,

(13)

for all v∈Q₀(Λ) and e∈Q₁(Λ), and an isomorphism C^∗(Λ)oβ^cZ∼=C_j^∗∗(c)(Λ×_βZ).

Proof. We apply Theorem 2.1 with ϕ := j^∗(c) ∈ Z²(Λ×_β Z,T). Then β^c is just β_j^∗_(c), and (13) follows because ϕ(e,1) = j^∗(c)(e,1) = c(e) for all e ∈Q₁(Λ). The isomorphism C^∗(Λ)oβ^c Z∼= C_j^∗∗(c)(Λ×_β Z) follows by Theorem2.1 and realising thati^∗(ϕ) =i^∗(j^∗(c)) = 1.

Corollary 2.4. Let Λ be a row-finite k-graph with no sorces and let β ∈ Aut(Λ). Suppose that ψ∈Z²(Λ,T) such that [ψ]∈ker(1−β^∗). Then there is ϕ ∈Z²(Λ×_β Z,T) such that ψ =i^∗(ϕ) and so Theorem 2.1 applies. In particular there is an automorphism β_ϕ of C_ψ^∗(Λ) such that

C_i^∗∗(ϕ)(Λ)oβϕZ∼=C_ϕ^∗(Λ×_β Z).

Proof. Since [ψ] ∈ ker(1−β^∗), [ψ] = [β^∗ψ] and so there is a map b : Q₁(Λ)→Tsuch that (β^∗ψ)ψ=δ¹(b). So forλ=ef =f⁰e⁰wheree, e⁰ ∈Λ^eⁱ and f, f⁰∈Λ^e^j such that 1≤i < j≤k, we get

ψ(βλ)ψ(λ) =b(e)b(f)b(f⁰)b(e⁰).

We defineϕ∈Z²(Λ×_βZ,T) by ϕ(λ, `) =

(

ψ(λ) ifλ∈Q2(Λ), `= 0, b(β⁻¹λ) ifλ∈Q1(Λ), `= 1.

Then by a computation as in Equation (8) we haveδ²(ϕ)(βλ,1) = 1 for all λ∈Q2(Λ); moreover, forλ∈Q3(Λ), we haveδ²(ϕ)(λ,0) = (δ²(ψ)(λ),0) = 1 sinceψ∈Z²(Λ,T) . Hence,ϕ∈Z²(Λ×_βZ,T). By constructionψ=i^∗(ϕ)

and so Theorem2.1gives the result.

Recall that ifβ is the identity automorphism of Λ then Λ×_βZ∼= Λ×T1 by Remark 1.7 and Ξ :Z¹(Λ,T)⊕Z²(Λ,T) ∼=Z²(Λ×T1,T) by Equation (5).

In this case Theorem2.1 reduces to the following result:

Corollary 2.5. Let Λ be a row-finite k-graph with no sources and let ϕ∈ Z²(Λ×T₁,T). Then ϕ= Ξ(ϕ₁, ϕ₂) where (ϕ₁, ϕ₂) ∈ Z¹(Λ,T)⊕Z²(Λ,T).

Moreover,

(i) We have ϕ(e,1) =ϕ1(e) for all e∈Q1(Λ) and ϕ(λ,0) =ϕ2(λ) for all λ∈Q₂(Λ).

(ii) There is an automorphism α=αϕ of C_ϕ^∗₂(Λ) such that α(p_v) =p_v and α(s_e) =ϕ₁(e)s_e for allv∈Q₀(Λ) and e∈Q₁(Λ).

(iii) There is an isomorphism C_ϕ^∗₂(Λ)oαZ→C_ϕ^∗(Λ×T1).

(14)

3. Examples

We consider some examples of automorphisms and the associated crossed products. In Example 3.1 we consider quasifree automorphisms on Cuntz algebras. In Examples 3.2 and 3.3 we compute cohomology of the crossed product; in Example 3.3 we adduce conditions under which the twisted crossed product C^∗-algebra is simple and purely infinite and use classification results to show that it is isomorphic toO₂.

Example 3.1. Forn >1 letB_n denote the 1-graph which is the path category of the directed graph with a single vertex v and edges f1, . . . , fn. It is well-known that C^∗(B_n) ∼= O_n. It is straightforward to see that Z¹(Bn,T) ∼= Tⁿ, since we may label each edge fi with an independent element ofT. It is also straightforward to see that AutB_n is isomorphic to Sn, the symmetric group of order n, which acts by permuting the edgesfi. Following [7], an automorphism α of O_n is said to be quasifree if it is determined by a unitary matrix u∈U(n) in the following sense

α(s_f_i) =

n

X

j=1

ui,js_f_j fori= 1, . . . , n.

We write α =α_u. Given u, u⁰ ∈U(n), we have α_uu⁰ =α_u◦α_u⁰. Moreover, ifu, u⁰ ∈U(n) are conjugate, the corresponding automorphisms αu,α_u⁰ are conjugate.

Evans notes on [7, Page 917] (citing an argument of Archbold from [1]) that α_u is outer if and only if u6= 1. Hence by [13, Lemma 10] the crossed product O_noαuZ is simple and purely infinite if and only if u^m 6= 1 for all m 6= 0. By the Pimsner–Voiculescu six-term exact sequence we have K_i(O_noαuZ) = Z/(n−1)Z for i = 0,1. Hence if u^m 6= 1 for all m 6= 0, the Kirchberg–Phillips Theorem [11,24] yields that the isomorphism class of O_noαuZ is independent of u.

We consider the situation covered in Corollary 2.3 in the case Λ = Bn. If the action β on B_n is induced by the identity permutation and c = (c1, . . . , cn) ∈Z¹(Bn,T), then the automorphism β^c of Corollary 2.3is the quasifree automorphismαu of O_n arising from then×ndiagonal matrix u with entries determined byc (see [10,7,12]). By Remark 1.7we have that Bn×_βZ∼=Bn×T1 and so by Corollary2.3we have

O_noβ^cZ∼=C_j^∗∗(c)(B_n×T₁).

Moreover, u^m 6= 1 for all m 6= 0 if and only if c_i is not a root of unity for some 1 ≤ i ≤ n, and hence in this case by the above paragraph we have C_j^∗∗(c)(B_n×T₁) simple and purely infinite with

Ki(C_j^∗∗(c)(Bn×T1)) =Z/(n−1)Z fori= 0,1.

(15)

Now let α be a quasifree automorphism of O_n; then α = α_u for some unitary matrixu∈U(n). Then since every unitary is conjugate to a diagonal unitary,α is conjugate toβ^cfor somec∈Z¹(B_n,T) (whereβ is the identity permutation). Hence,

O_noαZ∼=O_noβ^c Z∼=C_j^∗∗(c)(B_n×T₁).

Let γ be an arbitrary permutation of the edges of Bn. Then the induced automorphism γ of O_n is the quasifree automorphism associated to the unitary corresponding to the permutation. So if c ∈Z¹(Bn,T), then γ^c = β^c◦γ is also a quasifree automorphism.

Example 3.2. Consider the infinite 1-graph Λ with vertices vn forn ≥ 1 and edges e_nj,n≥1, j = 1, . . . , n, wheres(e_nj) =v_n+1 and r(e_nj) =v_n for j= 1, . . . , n. ThenC^∗(Λ) is strongly Morita equivalent to the UHF-algebra with K₀-group isomorphic to Q. Let β be the automorphism of Λ which fixes the vertices and cyclically permutes the edges between two successive vertices, i.e.

β(enj) =en,j+1, j = 1, . . . , n−1, β(enn) =en1.

The crossed product graph Λ×_βZ has degree (1,0) edgese⁰_nj = (e_nj,0) for n≥1, j= 1, . . . , nand it has a degree (0,1) loopfn= (vn,1) at each vertex v_n (we identify (v_n,0) withv_n), with commuting squares

(14) e⁰_njf_n+1 =f_ne⁰_n,j+1, forj= 1, . . . , n−1, e⁰_nnf_n+1 =f_ne⁰_n1. It is a rank-2 Bratteli diagram (see [22, Definition 4.1]) with 1-skeleton shown below

. . . .

v1 v2 v3 v4

f1 f2 f3 f4

e⁰₂₁

e⁰₂₂

e⁰₃₁

e⁰₃₃

e⁰₁₁ e⁰₃₂

By [22, Corollary 3.12, Theorem 4.3 and Lemma 4.8] theC^∗-algebra C^∗(Λ×_βZ)

is strongly Morita equivalent to an AT-algebra, the inductive limit of C(T)→C(T)⊗M_2! →C(T)⊗M_3!→ · · ·

with K-theory groups isomorphic to Q. Fix m ≤ n. Then for all α ∈ vn(Λ×_β Z)^(m,0) we have vm(Λ×_β Z)s(α) 6= ∅ and so Λ×_β Z is cofinal.

Fixα∈v_n(Λ×_βZ)^(N,0), then by repeated use of (14) it follows thato(α) = n(n+ 1)· · ·(n+N) whereo(α) is the order ofα(see [22,§5]). It is then easy to see that Λ×_β Zhas large-permutation factorisations (see [22, Definition 5.6]) and so C^∗(Λ×_β Z) is simple and has real-rank zero by [22, Theorem 5.7].

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We now turn our attention to computingH¹(Λ×_βZ, A) forAan abelian group; first we computeH¹(Λ, A). Observe that

δ⁰(f)(λ) =f(s(λ))−f(r(λ))

where f ∈C⁰(Λ, A), λ∈Λ and δ¹ :C¹(Λ, A)→ C²(Λ, A) is the zero map.

Hence H¹(Λ, A) = C¹(Λ, A)/imδ⁰ and, identifying elements of C¹(Λ, A) with “double sequences” a = (aⁿ_j)n≥1,1≤j≤n with aⁿ_j ∈ A, then H¹(Λ, A) may be identified with the set of equivalence classes of such elements where a∼b if there are c_n ∈A with bⁿ_j =aⁿ_j +c_n, n ≥1, j = 1, . . . , n. It follows thatH¹(Λ, A)∼= (Q

n≥1Aⁿ)/∼.

To compute the cohomologyH¹(Λ×_βZ, A), observe that

δ¹(ϕ)(e⁰_njfn+1 =fne⁰_n,j+1) =ϕ(fn+1) +ϕ(e⁰_nj)−ϕ(e⁰_n,j+1)−ϕ(fn), so forϕ∈Z¹(Λ×_β Z, A) we have

ϕ(e⁰_n,j+1)−ϕ(e⁰_nj) =ϕ(f_n+1)−ϕ(f_n) =ϕ(e⁰_n1)−ϕ(e⁰_nn),

forn≥1,j= 1, . . . , n−1. Summing overjwe obtainn(ϕ(f_n+1)−ϕ(f_n)) = 0 for n≥1. If A is torsion free, then ϕ(fn) is constant and therefore ϕ(e⁰_nj) is constant. In this case,H¹(Λ×_β Z, A)∼=A. IfAhas torsion, then

H¹(Λ×_βZ, A)∼=A×T₂(A)×T₃(A)× · · · , whereT_n(A) denotes then-torsion subgroup of A forn≥2.

The last part of the long exact sequence

· · · →H¹(Λ, A)^1−β

∗

−→ H¹(Λ, A) ^j

∗

−→H²(λ×_βZ, A)→0, implies that

H²(Λ×_βZ, A)∼= coker(1−β^∗).

Since β cyclically permutes the edges at each stage, the map β^∗ :C¹(Λ, A)→C¹(Λ, A)

isQ β_n:Q

n≥1Aⁿ→Q

n≥1Aⁿ, where

β_n:Aⁿ→Aⁿ, β_n(a₁, . . . , a_n) = (a_n, a₁, . . . , an−1), an automorphism of order n. Therefore, the map

1−β^∗ :C¹(Λ, A)→C¹(Λ, A) is determined by

(1−βn)(a1, . . . , an) = (a1−an, a2−a1, . . . , an−an−1).

Since H¹(Λ, A) =C¹(Λ, A)/imδ⁰, we conclude that coker(1−β^∗)∼= Y

n≥1

Aⁿ

!

/(im(1−β^∗) + imδ⁰).

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First observe that



 Y

n≥1

Aⁿ



/im(1−β^∗)∼= Y

n≥1

A

by the map (aⁿ_j)7→(Pn

j=1aⁿ_j). Since (aⁿ_j)∈ imδ⁰ iff there are c_n∈A with aⁿ_j =cn for all j= 1, . . . , n, we conclude that imδ⁰ inQ

n≥1A isQ

n≥1nA.

It follows that

H²(Λ×_βZ, A)∼= Y

n≥1

A

!

/imδ¹ ∼= Y

n≥1

A/nA.

IfA is divisible, in particular if A=T, thenH²(Λ×_βZ,T)∼= 0.

Given ϕ ∈ Z²(Λ×_β Z,T), both [ϕ] and [i^∗(ϕ)] are trivial since both H²(Λ×_βZ,T) and H²(Λ,T) are trivial. It follows by Theorem2.1 that

C^∗(Λ)oβϕ Z∼=C_ϕ^∗(Λ×_βZ)∼=C^∗(Λ×_β Z).

Example 3.3. For n ≥ 1 let n = {0, . . . , n−1}. Define a bijection θ : 2×3→2×3 byθ(i, j) = (i+1 (mod 2), j+1 (mod 3)). Consider the 2-graph F⁺_θ defined in [2]: F⁺_θ is the unital semigroup generated by f₀, f₁, g₀, g₁, g₂ subject to the relations figj =gj⁰fi⁰ whereθ(i, j) = (i⁰, j⁰); that is,

f₀g₀ =g₁f₁, f₁g₀=g₁f₀, f₀g₁ =g₂f₁, (15)

f1g1 =g2f0, f0g2=g0f1, f1g2 =g0f0, where the degree off0, f1 is e1 and the degree ofg0, g1, g2 ise2.

If β1 = (01) ∈ S2 and β2 = (012) ∈ S3, then it is easy to check that θ◦(β₁×β₂) = (β₁×β₂)◦θ and soβ =β₁×β₂ induces an automorphism of F⁺_θ.

SinceF⁺_θ has only one vertexv, it follows thatvF⁺_θs(α)6=∅for allα∈F⁺_θ and soF⁺_θ is cofinal. Furthermore F⁺_θ is aperiodic by [3, Corollary 3.2] since log 3 and log 2 are rationally independent. Hence C^∗(F⁺_θ) is simple by [27, Theorem 3.1]. Moreover, the loop f₀ has an entrancef₁g₀ and since there is only one vertex, it follows by [28, Proposition 8.8] thatC^∗(F⁺_θ) is purely infinite. By [6, Proposition 3.16] it follows that

K₀(C^∗(F⁺_θ)) =K₁(C^∗(F⁺_θ)) = 0

sinceM1 = [2] andM2 = [3]. SinceC^∗(F⁺_θ) is a Kirchberg algebra,C^∗(F⁺_θ)∼= O₂ by the Kirchberg–Phillips Theorem [11,24].

To compute the cohomology of Λ =F⁺_θ and of Λ×_βZ, we first compute the homology H_n(Λ). Next we determine the maps β∗ : H_n(Λ) → H_n(Λ) induced by the automorphism β and then use the exact sequence (see [16, Theorem 4.13]) to computeHn(Λ×_βZ). Thereafter, we apply the Universal