Simplicity of twisted C*-algebras of higher-rank graphs and crossed products by quasifree actions
aa r X i v : . [ m a t h . OA ] J un SIMPLICITY OF TWISTED C ∗ -ALGEBRAS OF HIGHER-RANKGRAPHS AND CROSSED PRODUCTS BY QUASIFREEACTIONS ALEX KUMJIAN, DAVID PASK, AND AIDAN SIMS
Dedicated to George A. Elliott on the occasion of his 70th birthday.
Abstract.
We characterise simplicity of twisted C ∗ -algebras of row-finite k -graphs with no sources. We show that each 2-cocycle on a cofinal k -graphdetermines a canonical second-cohomology class for the periodicity group ofthe graph. The groupoid of the k -graph then acts on the cartesian product ofthe infinite-path space of the graph with the dual group of the centre of anybicharacter representing this second-cohomology class. The twisted k -graphalgebra is simple if and only if this action is minimal. We apply this result tocharacterise simplicity for many twisted crossed products of k -graph algebrasby quasifree actions of free abelian groups. Introduction
Higher-rank graphs, or k -graphs, are combinatorial objects introduced by thefirst two authors in [22] as graph-based models for the higher-rank Cuntz–Kriegeralgebras studied by Robertson and Steger [42]. These C ∗ -algebras have beenwidely studied [4, 7, 11, 12], and their fundamental structure theory is by nowfairly well understood. They provide interesting examples in noncommutativegeometry [33, 48] and have been used to establish weak semiprojectivity [49] andcalculate nuclear dimension [44] for UCT Kirchberg algebras.Recently [24], we introduced a cohomology theory for k -graphs, and studiedthe associated twisted k -graph algebras C ∗ (Λ , c ) [26, 25]. These include manyexamples that do not arise naturally from untwisted k -graph C ∗ -algebras (see [26,Example 7.10], [34], and Theorem 5.1 below). So twisted k -graph C ∗ -algebrascould serve as useful models of various classes of classifiable C ∗ -algebras, particu-larly as they are always nuclear and belong to the UCT class [26, Corollary 8.7].It is therefore important to understand when twisted k -graph C ∗ -algebras aresimple. Simplicity of untwisted k -graph algebras was characterised in [41], and [26,Corollary 8.2] shows that if C ∗ (Λ) is simple, so is every C ∗ (Λ , c ). But the conversefails: regarding N as a 2-graph T , the associated 2-graph algebra C ∗ ( T ) ∼ = C ( T ) Date : August 31, 2018.2010
Mathematics Subject Classification.
Primary 46L05.
Key words and phrases. C ∗ -algebra; graph algebra; k -graph; simplicity; twisted C ∗ -algebra;groupoid; cocycle; cohomology; crossed product; quasifree action.This research was supported by the Australian Research Council. We thank Becky Armstrongfor bringing a number of typographical errors to our attention. is not simple, but for each irrational number θ there is a cocycle c θ such that C ∗ ( T , c θ ) is the irrational-rotation algebra A θ . This elementary example indi-cates that characterising simplicity of C ∗ (Λ , c ) is a subtle problem—an indicationconfirmed by the partial results in [45]. In this paper, we present a complete so-lution to this problem: Theorem 3.4 gives a necessary and sufficient condition for C ∗ (Λ , c ) to be simple.The broad strokes of our solution are as follows. We show that C ∗ (Λ , c ) isisomorphic to the C ∗ -algebra of a Fell bundle B Λ over a topologically principalamenable ´etale quotient H Λ of the k -graph groupoid introduced in [22]. Resultsof Ionescu and Williams [16] show that if B is Fell bundle over a groupoid H , then H acts on the primitive ideal space of the C ∗ -subalgebra C ∗ ( H (0) ; B ) ⊆ C ∗ ( H ; B )sitting over the unit space of H . We prove that if H is topologically principal,amenable and ´etale, then C ∗ ( H ; B ) is simple if and only if this action is minimal.In particular, C ∗ ( H Λ ; B Λ ) is simple if and only if the Ionescu-Williams action of H Λ is minimal. Our task is then to identify the action; it turns out that this isquite intricate.The k -graph has a periodicity group Per(Λ) ⊆ Z k [6]. We use groupoid tech-nology to show that c determines a class [ ω ] ∈ H (Per(Λ) , T ). We then identify amap from H Λ to the primitive ideal space of the noncommutative torus A ω thatbecomes a homomorphism when Prim( A ω ) is regarded as a quotient of T k . Themap H Λ → Prim( A ω ) determines an action θ of H Λ on H (0)Λ × Prim( A ω ). Weprove that there is a homeomorphism between this cartesian-product space andPrim( C ∗ ( H (0)Λ ; B Λ )) which intertwines the action θ with the action described inthe preceding paragraph, and deduce our main result.Even the action θ is not easy to compute in a generic example. So we developsome key examples where it can be computed. Firstly, if A ω is simple, then θ isminimal precisely when Λ is cofinal. So we show how to decide efficiently whether A ω is simple. Secondly, we show how to recover many twisted crossed productsof k -graph algebras by quasifree actions of Z l as twisted ( k + l )-graph algebras,and show that our main theorem yields a very satisfactory characterisation ofsimplicity of many such twisted crossed products.The paper is organised as follows. In Section 2 we establish the background thatwe need regarding k -graphs and their groupoids. This includes a fairly generaltechnical result giving a sufficient condition for the interior of the isotropy of aHausdorff ´etale groupoid to be closed. The main point is that the interior of theisotropy in the groupoid G Λ of a cofinal k -graph Λ is always closed, and relativelyeasy to describe: Corollary 2.2 says that if Per(Λ) is the periodicity group of the k -graph discussed in [6], then the interior I Λ of the isotropy in G Λ can be identifiedcanonically with G (0)Λ × Per(Λ). Moreover, we can form the quotient groupoid H Λ := G Λ / I Λ , and this quotient is itself a topologically principal amenable ´etalegroupoid.In Section 3, we study carefully the relationship between cocycles on a k -graphΛ, and dynamics associated to the corresponding k -graph groupoid. In [22], the k -graph algebra C ∗ (Λ) is identified with the C ∗ -algebra of a groupoid G Λ withunit space Λ ∞ , the space of infinite paths in Λ. We showed in [26] that each IMPLICITY OF TWISTED k -GRAPH ALGEBRAS 3 C ∗ (Λ , c ) can be realised as a twisted C ∗ -algebra C ∗ ( G Λ , σ c ) of the same groupoid.The restriction of the groupoid cocycle σ c to each fibre of I Λ determines a 2-cocycle of Per(Λ). We show that these 2-cocycles are all cohomologous, and deducein Proposition 3.1 that σ c is cohomologous to a cocycle σ whose restriction to I Λ is of the form 1 Λ ∞ × ω for some bicharacter ω of Per(Λ). We then have C ∗ (Λ , c ) ∼ = C ∗ ( G Λ , σ ), and we seek to characterise simplicity of the latter. Asin [29], the primitive-ideal space of each fibre of C ∗ ( I Λ , σ ) can be identified withthe character space of the kernel Z ω ⊆ Per(Λ) of the antisymmetric bicharacterassociated to ω . So Prim C ∗ ( I Λ , σ ) can be identified with Λ ∞ × b Z ω . We constructfrom σ a Per(Λ) b -valued 1-cocycle r σ on G Λ . We then have the wherewithal tostate our main theorem, Theorem 3.4, although the proof must wait until the endof the subsequent section. We show in Lemma 3.5 how to compute the bicharacter ω , and hence the group Z ω appearing in the main theorem, without passing to thegroupoid G Λ . We then show that r σ determines an action θ of the quotient H Λ of G Λ by the interior of its isotropy on Λ ∞ × b Z ω . Theorem 3.4 can be recast assaying that C ∗ (Λ , c ) is simple if and only if θ is minimal (see Corollary 4.8).In Section 4 we prove Theorem 3.4 using the technology of Fell bundles. Weuse ideas from [8] (see also [37]) to recover C ∗ (Λ , c ) as the C ∗ -algebra of a Fellbundle B Λ over the quotient H Λ of G Λ discussed above. We show that the re-striction C ∗ ( H (0)Λ ; B Λ ) of C ∗ ( H Λ ; B Λ ) to the unit space of H Λ is isomorphic to C (Λ ∞ ) ⊗ C ∗ (Per(Λ) , ω ). Since Per(Λ) is a free abelian group, the C ∗ -algebra C ∗ (Per(Λ) , ω ) is a noncommutative torus, and has primitive ideal space Λ ∞ × b Z ω (see, for example, [40]). Results of Ionescu and Williams [16] show that conjuga-tion in B Λ determines an action of H Λ on the primitive-ideal space of C ∗ ( H (0)Λ ; B Λ )and hence on Λ ∞ × b Z ω . By identifying specific elements in the fibres of B Λ thatimplement the Ionescu–Williams action, we prove that it matches up with theaction θ of Section 3. In Lemma 4.6 and Corollary 4.7, we adapt the standardargument in [38] to prove that if B is a Fell bundle over a topologically principalamenable ´etale groupoid H , then C ∗ ( H ; B ) is simple if and only if the Ionescu-Williams action of H is minimal. We then prove our main theorem by applyingthis result to the bundle B Λ over H Λ .In Section 5, we investigate a broad class of examples of twisted higher-rankgraph C ∗ -algebras where the hypotheses of our main result are readily checkable.We show how a T l -valued 1-cocycle on a k -graph combined with a bicharacter ω of Z l can be combined to obtain a 2-cocycle on the Cartesian-product ( k + l )-graphΛ × N l for which the associated twisted ( k + l )-graph C ∗ -algebra is isomorphic toa twisted crossed product of C ∗ (Λ) by Z l . We demonstrate that the hypothesesof our main theorem can be effectively checked for these examples, and obtain ausable characterisation of simplicity of crossed products arising in this way whenΛ is aperiodic.We finish in Section 6 by presenting a number of concrete examples of our mainresult, showing how all of its working parts interact and demonstrating that eachof the ingredients of the statement is genuinely necessary to obtain a satisfactorycharacterisation of simplicity. We also present a somewhat unrelated examplewhich we believe is nevertheless interesting in its own right: a 3-graph all of whose ALEX KUMJIAN, DAVID PASK, AND AIDAN SIMS twisted C ∗ -algebras (including the untwisted one) are simple, but for which thetwisted C ∗ -algebras are not all mutually isomorphic.2. Background
Throughout the paper, we regard N k as a monoid under addition, with identity0 and generators e , . . . , e k . For m, n ∈ N k , we write m i for the i th coordinate of m , and define m ∨ n ∈ N k by ( m ∨ n ) i = max { m i , n i } .Given a small category C , we write C ∗ = { ( λ, µ ) ∈ C × C : s ( λ ) = r ( µ ) } forthe collection of composable pairs in C . A T -valued 2-cocycle c on C is a map c : C ∗ → T such that c ( λ, s ( λ )) = c ( r ( λ ) , λ ) = 1 for all λ and c ( µ, ν ) c ( λ, µν ) = c ( λ, µ ) c ( λµ, ν ) for composable λ , µ , ν . If b : C → T is a function with b ( α ) = 1 forevery identity morphism α of C , then δ b ( µ, ν ) := b ( µ ) b ( ν ) b ( µν ) defines a 2-cocyclecalled the 2-coboundary associated to b . Two 2-cocycles c, c ′ are cohomologous if( µ, ν ) c ( µ, ν ) c ′ ( µ, ν ) is a 2-coboundary.If C is a discrete group, then any function c : C × C → T for which the func-tions c ( · , α ) and c ( β, · ) are homomorphisms is a cocycle; such cocycles are called bicharacters. If, in addition, C is abelian, every 2-cocycle is cohomologous toa bicharacter (see [20, Theorem 7.1]). For more background on 2-cocycles andbicharacters on abelian groups see [20, 2, 29] (note that in the first two references2-cocycles are called multipliers).2.1. k -graphs and twisted C ∗ -algebras. Let Λ be a countable small categoryand d : Λ → N k be a functor. Write Λ n := d − ( n ) for n ∈ N k . Then Λ is a k -graph (see [22]) if d satisfies the factorisation property : ( µ, ν ) µν is a bijectionof { ( µ, ν ) ∈ Λ m × Λ n : s ( µ ) = r ( ν ) } onto Λ m + n for each m, n ∈ N k . We then haveΛ = { id o : o ∈ Obj(Λ) } , and so we regard the domain and codomain maps asmaps s, r : Λ → Λ . Recall from [31] that for v, w ∈ Λ and X ⊆ Λ, we write vX := { λ ∈ X : r ( λ ) = v } , Xw := { λ ∈ X : s ( λ ) = w } , and vXw = vX ∩ Xw. A k -graph Λ is row-finite with no sources if 0 < | v Λ n | < ∞ for all v ∈ Λ and n ∈ N k . See [22] for further details regarding the basic structure of k -graphs.Let Λ be a row-finite k -graph with no sources. We recall the definition of the infinite path space Λ ∞ given in [22, Definition 2.1]. We write Ω k for the k -graph { ( m, n ) ∈ N k : m ≤ n } with r ( m, n ) = ( m, m ), s ( m, n ) = ( n, n ), ( m, n )( n, p ) =( m, p ) and d ( m, n ) = n − m . We identify Ω k with N k by ( m, m ) m . We defineΛ ∞ to be the collection of all k -graph morphisms x : Ω k → Λ. For p ∈ N k , wedefine T p : Λ ∞ → Λ ∞ by ( T p x )( m, n ) := x ( m + p, n + p ) for all ( m, n ) ∈ Ω k .(Traditionally, as in [22], these shift maps T p have been denoted σ p , but we willuse σ in this paper to denote a 2-cocycle on G Λ .) For x ∈ Λ ∞ we denote x (0) by r ( x ). For λ ∈ Λ and x ∈ Λ ∞ with r ( x ) = s ( λ ), there is a unique element λx ∈ Λ ∞ such that ( λx )(0 , d ( λ )) = λ and T d ( λ ) ( λx ) = x .As in [22, Definition 4.7], we say that Λ is cofinal if, for every x ∈ Λ ∞ and every v ∈ Λ , there exists n ∈ N k such that v Λ x ( n ) = ∅ . We say that Λ is aperiodic if it In [26] these were called categorical cocycles , in contradistinction to cubical cocycles.
IMPLICITY OF TWISTED k -GRAPH ALGEBRAS 5 satisfies the “aperiodicity condition” of [22, Definition 4.3]: for every v ∈ Λ thereexists x ∈ Λ ∞ with r ( x ) = v and T m ( x ) = T n ( x ) whenever m = n .Given a k -graph Λ, the group of all 2-cocycles on Λ (as described above) isdenoted Z (Λ , T ). Let Λ be a row-finite k -graph with no sources, and fix c ∈ Z (Λ , T ). A Cuntz–Krieger (Λ , c )-family in a C ∗ -algebra B is a function t : λ t λ from Λ to B such that(CK1) { t v : v ∈ Λ } is a collection of mutually orthogonal projections;(CK2) t µ t ν = c ( µ, ν ) t µν whenever s ( µ ) = r ( ν );(CK3) t ∗ λ t λ = t s ( λ ) for all λ ∈ Λ; and(CK4) t v = P λ ∈ v Λ n t λ t ∗ λ for all v ∈ Λ and n ∈ N k . C ∗ (Λ , c ) is then defined to be the universal C ∗ -algebra generated by a Cuntz–Krieger (Λ , c )-family (see [26, Notation 5.4]).2.2. Groupoids.
A groupoid is a small category G with inverses. We use standardgroupoid notation as in, for example, [38]. So G (0) is the set of identity morphismsof G , called the unit space, and G (2) denotes the set G ∗ of composable pairs in G .The groupoid G is an ´etale Hausdorff groupoid if it has a locally compact Hausdorfftopology under which all operations in G are continuous (when G (2) ⊆ G × G isgiven the relative topology) and the range and source maps r, s : G → G (0) arelocal homeomorphisms. It then makes sense to talk about continuous cocycleson G . We write Z ( G , T ) for the group of continuous T -valued 2-cocycles on G and say that two continuous 2-cocycles are cohomologous if they differ by acontinuous 2-coboundary—that is, the coboundary δ b associated to a continuousmap b : G → T such that b | G (0) ≡
1. A 1-cocycle on G with values in a group G is a map ρ : G → G that carries composition in G to the group operation in G .Given u ∈ G (0) we write G u for { γ ∈ G : r ( γ ) = u } , G u for { γ ∈ G : s ( γ ) = u } and G uu = G u ∩G u . The isotropy of G is the set S u ∈G (0) G uu of elements of G whose rangeand source coincide. A groupoid is minimal if r ( G u ) is dense in G (0) for every unit u ∈ G (0) . It is topologically principal if { u ∈ G (0) : G uu = { u }} is dense in G (0) .It will be important later to know that the interior of the isotropy in thegroupoid associated to a cofinal k -graph is closed. This will follow from the fol-lowing fairly general result. Proposition 2.1.
Let G be an ´etale groupoid, let G be a countable discrete abeliangroup, and let c ∈ Z ( G , G ) . Suppose that G is minimal and that for all x , therestriction of c to G xx is injective. Let I denote the interior of the isotropy of G .For x, y ∈ G (0) , we have c ( I ∩G xx ) = c ( I ∩G yy ) . The set H defined by H := c ( I ∩G xx ) for any x ∈ G (0) is a subgroup of G , and s × c induces an isomorphism from I to G (0) × H . In particular the interior of the isotropy of G is closed.Proof. For x ∈ G (0) set I x := I ∩ G xx and note that H x := c ( I x ) is a subgroup of G . Fix x, y ∈ G (0) ; we prove that H x = H y . By symmetry it suffices to show that H x ⊂ H y , so we fix h ∈ H x and prove that h ∈ H y . Fix α ∈ I x such that c ( α ) = h .Since G is discrete and c is continuous, there is an open neighbourhood U of α such that U ⊆ I ∩ c − ( h ). Since c is injective on each G xx , this U is a bisection.Since G is minimal, the set s ( G y ) is dense in G (0) , and so there exists γ ∈ G y such ALEX KUMJIAN, DAVID PASK, AND AIDAN SIMS that s ( γ ) ∈ s ( U ). The unique element β of I s ( γ ) ∩ U satisfies c ( β ) = h , and then γβγ − ∈ I y satisfies c ( γβγ − ) = c ( γ ) c ( β ) c ( γ ) = h . So h ∈ H y as required.Thus the map s × c yields an isomorphism from the interior of the isotropy of G to G (0) × H . Since I is the intersection of the closed set c − ( H ) with the isotropyof G , which is also closed, we deduce that I is closed. (cid:3) Given an ´etale groupoid G and a 2-cocycle σ ∈ Z ( G , T ), it is straightforwardto check that C c ( G ) is a *-algebra under the operations( f g )( γ ) = X ηζ = γ σ ( η, ζ ) f ( η ) g ( ζ ) and f ∗ ( γ ) = σ ( γ, γ − ) f ( γ − )for f, g ∈ C c ( G ). The twisted groupoid C ∗ -algebra C ∗ ( G , σ ) is then defined to bethe closure of C c ( G ) under the maximal C ∗ -norm (see [38] for more details).2.3. k -graph groupoids. Following [22, Definition 2.7] we associate a groupoid G Λ to each row-finite k -graph Λ with no sources by putting G Λ := { ( x, l − m, y ) ∈ Λ ∞ × Z k × Λ ∞ : l, m ∈ N k , T l x = T m y } . For µ, ν ∈ Λ with s ( µ ) = s ( ν ) define Z ( µ, ν ) ⊂ G Λ by Z ( µ, ν ) := { ( µx, d ( µ ) − d ( ν ) , νx ) : x ∈ Λ ∞ , r ( x ) = s ( µ ) } . For λ ∈ Λ, we define Z ( λ ) := Z ( λ, λ ).The sets Z ( µ, ν ) form a basis of compact open sets for a locally compact Haus-dorff topology on G Λ under which it is an ´etale groupoid with structure maps r ( x, l − m, y ) = ( x, , x ), s ( x, l − m, y ) = ( y, , y ), and ( x, l − m, y )( y, p − q, z ) =( x, l − m + p − q, z ). (see [22, Proposition 2.8]). The Z ( λ ) are then a basis for therelative topology on G (0)Λ ⊆ G Λ . We identify G (0)Λ = { ( x, , x ) : x ∈ Λ ∞ } with Λ ∞ .Following [22, Proposition 2.8, Corollary 3.5] G Λ is an amenable ´etale groupoid.Moreover G Λ is minimal if and only if Λ is cofinal [22, Proof of Proposition 4.8].Suppose now that Λ is cofinal. As in [6], we define a relation on Λ by µ ∼ ν ifand only if s ( µ ) = s ( ν ) and µx = νx for all x ∈ s ( µ )Λ ∞ . This is an equivalencerelation on Λ which respects range, source and composition. By [6, Theorem4.2(1)], the set Per(Λ) := { d ( µ ) − d ( ν ) : µ, ν ∈ Λ and µ ∼ ν } ⊆ Z k is a subgroupof Z k . Since Λ is cofinal, [6, Lemma 4.6] givesPer(Λ) ⊆ { m ∈ Z k : ( x, m, x ) ∈ G Λ for all x ∈ Λ ∞ } . Since Per(Λ) is a subgroup of Z k , it is also a finitely generated free abelian groupand so Per(Λ) ∼ = Z l for some integer l ≤ k . Corollary 2.2.
Let Λ be a cofinal row-finite k -graph with no sources. Let I Λ denote the interior of the isotropy in G Λ . Then I Λ is closed and I Λ = { ( x, m, x ) : x ∈ Λ ∞ , m ∈ Per(Λ) } ∼ = Λ ∞ × Per(Λ) . Moreover, H Λ := G Λ / I Λ is an amenable, topologically principal, locally compact,Hausdorff, ´etale groupoid. IMPLICITY OF TWISTED k -GRAPH ALGEBRAS 7 Proof.
Note that G Λ is a minimal ´etale groupoid and the restriction of the canonicalcocycle c ∈ Z ( G Λ , Z k ), given by c ( x, n, y ) = n to ( G Λ ) xx is injective for each x ∈ Λ ∞ . Hence by Proposition 2.1 I Λ is closed.By definition of the topology on G Λ , the set { ( x, m, x ) : x ∈ Λ ∞ , m ∈ Per(Λ) } is contained in I Λ , the interior of the isotropy of G Λ . Conversely, if α ∈ I Λ , then α = ( x, m, x ) for some x ∈ Λ ∞ and m ∈ Z k , and Proposition 2.1 applied to thecocycle c of the preceding paragraph shows that ( y, m, y ) ∈ I Λ for every y . Inparticular m ∈ Per(Λ). So I Λ = { ( x, m, x ) : x ∈ Λ ∞ , m ∈ Per(Λ) } as claimed.It is routine to check that G Λ / I Λ is a locally compact Hausdorff ´etale groupoid(see, for example, [47, Proposition 2.5]). Since c ( I Λ ) = Per(Λ), there exists ˜ c ∈ Z ( H Λ , Z k / Per(Λ)) such that ˜ c ([( x, n, y )]) = n + Per(Λ). The groupoid ˜ c − (0) isisomorphic to c − (0) which is amenable by, for example, [52, Lemma 6.7]. Since Z k / Per(Λ) is abelian and hence amenable, it follows from [50, Proposition 9.3] that H Λ is amenable. By construction of I Λ , the interior of the isotropy of H Λ is trivial,and therefore H Λ is topologically principal by, for example, [39, Proposition 3.6]. (cid:3) We frequently identify I Λ with Λ ∞ × Per(Λ) as in Corollary 2.2.Let Λ s ∗ s Λ := { ( µ, ν ) ∈ Λ × Λ : s ( µ ) = s ( ν ) } . Lemma 6.6 of [26] shows thatthere exists P ⊆ Λ s ∗ s Λ such that(2.1) ( λ, s ( λ )) ∈ P for all λ and G Λ = G ( µ,ν ) ∈P Z ( µ, ν ) . Given such a set P , for g ∈ G Λ we write ( µ g , ν g ) for the element of P with g ∈ Z ( µ g , ν g ). Fix c ∈ Z (Λ , T ). Let ˜ d : G Λ → Z k be the canonical 1-cocycle˜ d ( x, n, y ) = n . Lemma 6.3 of [26] shows that for composable g, h ∈ G Λ , there exist α, β, γ ∈ Λ and y ∈ Λ ∞ such that ν g α = µ h β, µ g α = µ gh γ and ν h β = ν gh γ ; and g = ( µ g αy, ˜ d ( g ) ,ν g αy ) , h = ( µ h βy, ˜ d ( h ) , ν h βy ) and gh = ( µ gh γy, ˜ d ( gh ) , ν gh γy ) . (2.2)The formula(2.3) σ c ( g, h ) = c ( µ g , α ) c ( ν g , α ) c ( µ h , β ) c ( ν h , β ) c ( µ gh , γ ) c ( ν gh , γ )does not depend on the choice of α, β, γ , and determines a continuous groupoid2-cocycle on G Λ . If σ ′ c is obtained from c in the same way with respect to an-other collection P ′ , then σ c and σ ′ c are cohomologous. Corollary 7.7 of [26] showsthat there is an isomorphism C ∗ (Λ , c ) ∼ = C ∗ ( G Λ , σ c ) that carries each s λ to thecharacteristic function 1 Z ( λ,s ( λ )) .3. An action of the k -graph groupoid associated to a k -graph2-cocycle We consider a row-finite k -graph Λ with no sources. Lemma 7.2 of [45] says thatif Λ is not cofinal, then C ∗ (Λ , c ) is nonsimple for every c ∈ Z (Λ , T ). Since we areinterested here in when C ∗ (Λ , c ) is simple, we shall therefore assume henceforththat Λ is cofinal. ALEX KUMJIAN, DAVID PASK, AND AIDAN SIMS
Recall that if σ is a continuous T -valued 2-cocycle on a groupoid G , then thereis a groupoid extension G (0) × T → G × σ T → G , where G × σ T is equal to G × T as a set, with unit space ( G × σ T ) (0) = G (0) × { } , range map r ( g, z ) = ( r ( g ) , s ( g, z ) = ( s ( g ) ,
1) and operations( α, w )( β, z ) = ( αβ, σ ( α, β ) zw ) and ( α, w ) − = ( α − , σ ( α, α − ) w ) . Given σ ∈ Z ( G Λ , T ) we write I Λ × σ T for I Λ × T regarded as a subgroupoid of G Λ × σ T . We often implicitly identify ( G Λ × σ T ) (0) with G (0)Λ . Proposition 3.1.
Let Λ be a cofinal row-finite k -graph with no sources, and take c ∈ Z (Λ , T ) . Fix P ⊆ Λ s ∗ s Λ as in (2.1) , and let σ c ∈ Z ( G Λ , T ) be as in (2.3) .For x ∈ Λ ∞ , define σ xc ∈ Z (Per(Λ) , T ) by σ xc ( p, q ) = σ c (( x, p, x ) , ( x, q, x )) . Thenthere is a bicharacter ω of Per(Λ) such that σ xc is cohomologous to ω for every x ∈ Λ ∞ . For any such bicharacter ω , there exists a cocycle σ ∈ Z ( G Λ , T ) suchthat σ is cohomologous to σ c and σ | I Λ = ω × Λ ∞ . To prove the proposition, we first prove some lemmas.
Lemma 3.2.
Let Λ be a row-finite k -graph with no sources. Take σ ∈ Z ( G Λ , T ) .For α = ( x, m, y ) ∈ G Λ and p ∈ Per(Λ) , define r σα : Per(Λ) → T by (3.1) r σα ( p ) = σ (cid:0) α, ( y, p, y ) (cid:1) σ (cid:0) ( x, m + p, y ) , α − (cid:1) σ (cid:0) α, α − (cid:1) . For x ∈ Λ ∞ define σ x ∈ Z (Per(Λ) , T ) by σ x ( p, q ) = σ (( x, p, x ) , ( x, q, x )) . Foreach p ∈ Per(Λ) , the map r σ · ( p ) : G Λ → T is continuous, and we have (3.2) r σα ( p + q ) = σ r ( α ) ( p, q ) σ s ( α ) ( p, q ) r σα ( p ) r σα ( q ) . If σ x = σ y for all x, y , then α r σα is a continuous Per(Λ) b -valued -cocycle on G Λ .Proof. The map α r σα ( p ) is continuous because σ is continuous.A straightforward calculation in the central extension G Λ × σ T shows that for w, z ∈ T , we have( α, w )(( y, p, y ) , z )( α, w ) − = (( x, p, x ) , r σα ( p ) z ) . Computing further in the central extension, we have(( x, p + q, x ) ,r σα ( p + q ))= ( α, y, p + q, y ) , α, − = ( α, y, p, y ) , σ y ( p, q ))( α, − ( α, y, q, y ) , α, − = (( x, p, x ) , r σα ( p ) σ y ( p, q ))(( x, q, x ) , r σα ( q ))= (cid:0) ( x, p + q, x ) , σ x ( p, q ) σ y ( p, q ) r σα ( p ) r σα ( q ) (cid:1) , giving (3.2).Now suppose that σ x = σ y for all x, y . Then (3.2) implies immediately that r σα ∈ Per(Λ) b for all α . Take p ∈ Per(Λ) and composable α, β ∈ G Λ . Let y = s ( β ). IMPLICITY OF TWISTED k -GRAPH ALGEBRAS 9 Computing again in G Λ × σ T , we have (cid:0) ( r ( α ) , p, r ( α )) , r σα ( p ) r σβ ( p ) (cid:1) = ( α, β, (cid:0) ( y, p, y ) , (cid:1) ( β, − ( α, − = ( αβ, σ ( α, β )) (cid:0) ( y, p, y ) , (cid:1) ( αβ, σ ( α, β )) − = (cid:0) ( r ( α ) , p, r ( α )) , r σαβ ( p ) (cid:1) . So α r σα is a 1-cocycle on G Λ . (cid:3) Given a cocycle ω ∈ Z (Per(Λ) , T ), we write ω ∗ for the cocycle ( p, q ) ω ( q, p ).By [29, Proposition 3.2] (see also [20, Lemma 7.1]), the map ωω ∗ is a bicharacterof Per(Λ) which is antisymmetric in the sense that ( ωω ∗ )( p, q ) = ( ωω ∗ )( q, p ).Proposition 3.2 of [29] implies that ω ωω ∗ descends to an isomorphism of H (Per(Λ) , T ) onto the group X (Per(Λ) , T ) of all antisymmetric bicharacters ofPer(Λ). Lemma 3.3.
Let Λ be a cofinal row-finite k -graph with no sources, and suppose c ∈ Z (Λ , T ) . Let σ c be a continuous cocycle on G Λ of the form (2.3) , so that C ∗ ( G Λ , σ c ) ∼ = C ∗ (Λ , c ) . For each x ∈ Λ ∞ , let σ xc ∈ Z (Per(Λ) , T ) be the cocyclegiven by σ xc ( p, q ) = σ c (( x, p, x ) , ( x, q, x )) . Then the cohomology class of σ xc isindependent of x .Proof. The formula (2.3) shows that σ c is locally constant as a function from G Λ × G Λ → T . Restricting σ c to I Λ , we obtain cocycles on the groups ( I Λ ) x = { ( x, p, x ) : p ∈ Per(Λ) } ∼ = Per(Λ), and hence cocycles σ xc on Per(Λ) as claimed.For x ∈ Λ ∞ , let ω x be the bicharacter of Per(Λ) given by ω x ( p, q ) := σ c (cid:0) ( x, p, x ) , ( x, q, x ) (cid:1) σ c (cid:0) ( x, q, x ) , ( x, p, x ) (cid:1) . Fix free abelian generators g , . . . , g l for Per(Λ). Since each ω x is a bicharac-ter, it is determined by the values ω x ( g i , g j ). Fix x ∈ Λ ∞ . Since σ c is locallyconstant, for each i, j there is a neighbourhood U i,j of x such that σ xc ( g i , g j ) = σ c (( x, g i , x ) , ( x, g j , x )) is constant on U i,j . So for y ∈ U := T i,j U i,j we have ω y ( g i , g j ) = ω x ( g i , g j ) for all i, j , and hence ω y = ω x . Now [29, Proposition 3.2]implies that the cohomology class (in H (Per(Λ) , T )) of σ xc is locally constantwith respect to x . Since Λ is cofinal, G Λ is minimal, and so every orbit in G Λ isdense; so to see that the cohomology class of σ xc is globally constant, it sufficesto show that it is constant on orbits. For x ∈ Λ ∞ , let A x denote the subgroup { (( x, p, x ) , z ) : p ∈ Per(Λ) , z ∈ T } ⊆ ( G Λ × σ c T ) xx , which is isomorphic to the groupextension Per(Λ) × σ xc T of Per(Λ) by T . Conjugation by any α in G Λ × σ c T is anisomorphism Ad α : A s ( α ) ∼ = A r ( α ) . For γ ∈ G Λ , let r σ c γ : Per(Λ) → T be the mapof Lemma 3.2. Fix α = ( γ, w ) ∈ G Λ × σ c T , p ∈ Per(Λ) and z ∈ T , and let x = r ( γ )and y = s ( γ ). A quick calculation gives Ad α (( y, p, y ) , z ) = (( x, p, x ) , r σ c γ ( p ) z ).If p = 0, then (3.1) collapses to give Ad α (( y, , y ) , z ) = (( x, , x ) , z ) because σ c (( x, m, y ) , ( y, , y )) = 1. If q : I Λ × σ c T → Λ ∞ × Per(Λ) is the quotient map(( x, p, x ) , z ) ( x, p ), then q (Ad α (( y, p, y ) , z )) = q (cid:0) ( x, p, x ) , r σ c γ ( p ) z (cid:1) = ( x, p ) , In [29], the word “symplectic” is used instead of antisymmetric. so that Ad α descends through q to the map ( y, p ) ( x, p ). Thus conjugation by α determines an isomorphism0 T A y Per(Λ) 00 T A x Per(Λ) 0 id Ad α id of extensions. Hence σ xc is cohomologous to σ yc ; so the cohomology class of σ xc isconstant on orbits. (cid:3) Proof of Proposition 3.1.
Let σ c be a continuous cocycle on G Λ constructed inSection 2.3 so that C ∗ ( G Λ , σ c ) ∼ = C ∗ (Λ , c ). By Lemma 3.3, the cohomology classof σ xc is independent of x . So there exists a cocycle ω ∈ Z (Per(Λ) , T ) whosecohomology class agrees with that of σ xc for each x . We may assume that ω is abicharacter.The map ˜ c x : Per(Λ) × Per(Λ) → T defined by( p, q ) ˜ c x ( p, q ) := σ c (( x, p, x ) , ( x, q, x )) ω ( p, q )is a coboundary on Per(Λ) for each x . Fix free abelian generators g , . . . , g l forPer(Λ). Fix x ∈ Λ ∞ , and define b x (0) = b x ( g j ) = 1 ∈ T for all j , and then define b x on Per(Λ) inductively by(3.3) b x ( m ) b x ( m + g i ) = ˜ c x ( g i , m ) whenever m ∈ h g j : j ≤ i i .Since x ˜ c x ( p, q ) is continuous for each p, q , an induction argument shows that x b x ( p ) is continuous for each p .We claim that each δ b x = ˜ c x . To see this, choose for each x a cochain ˜ b x :Per(Λ) → T such that δ ˜ b x = ˜ c x . The map a x ( m ) := Q li =1 ˜ b x ( g i ) m i is a 1-cocycle on Per(Λ), and so δ a x = 1. Hence δ ( a x ˜ b x ) = δ ˜ b x = ˜ c x . We have( a x ˜ b x )(0) = ( a x ˜ b x )( g i ) = 1 for all i , and for i ≤ l and m ∈ h g j : j ≤ i i , we have( a x ˜ b x )( m )( a x ˜ b x )( m + g i ) = ( a x ˜ b x )( g i )( a x ˜ b x )( m )( a x ˜ b x )( m + g i )= δ ( a x ˜ b x )( g i , m ) = ˜ c x ( g i , m ) . So b x and a x ˜ b x both take 0 and each g i to 1 and satisfy (3.3). Hence b x = a x ˜ b x ,and δ b x = ˜ c x for all x .Since x b x ( p ) is continuous for each p , the map b : ( x, p, x ) b x ( p ) is acontinuous cochain on I Λ . Since I Λ is clopen in G Λ , we can extend b to a cochain˜ b on all of G Λ by setting ˜ b | G Λ \I Λ ≡ δ ˜ b is a continuous coboundary on G Λ , so σ := σ c · δ ˜ b represents thesame cohomology class as σ c . Hence C ∗ ( G Λ , σ ) ∼ = C ∗ ( G Λ , σ c ) ∼ = C ∗ (Λ , c ) [38,Proposition II.1.2]. We have σ (( x, p, x ) , ( x, q, x )) = ω ( p, q ) for p, q ∈ Per(Λ) byconstruction of ˜ b . Each σ xc is cohomologous to ω by choice of ω . (cid:3) IMPLICITY OF TWISTED k -GRAPH ALGEBRAS 11 Given an abelian group A and a cocycle ω ∈ Z ( A, T ), we write(3.4) Z ω := { p ∈ A : ( ωω ∗ )( p, q ) = 1 for all q ∈ A } for the kernel of the homomorphism p ( ωω ∗ )( p, · ) from A to b A induced by ωω ∗ .Thus Z ω is a subgroup of A .We now have all the ingredients needed to state our main theorem. Theorem 3.4.
Let Λ be a row-finite cofinal k -graph with no sources, and supposethat c ∈ Z (Λ , T ) . Suppose that σ ∈ Z ( G Λ , T ) and ω ∈ Z (Per(Λ) , T ) satisfy C ∗ ( G Λ , σ ) ∼ = C ∗ (Λ , c ) and σ | I Λ = ω × Λ ∞ (such a pair σ, ω exist by Proposi-tion 3.1). Let r σ : G Λ → Per(Λ) b be the cocycle of Lemma 3.2. Then C ∗ (Λ , c ) is simple if and only if { ( r ( γ ) , r σγ | Z ω ) : γ ∈ ( G Λ ) x } is dense in Λ ∞ × b Z ω for all x ∈ Λ ∞ . In particular, if ω is nondegenerate, then C ∗ (Λ , c ) is simple. The proof of the main theorem will occupy the remainder of this section andmost of the next. Before we get started, we provide a practical method for com-puting Z ω without reference to G Λ . To see why this is useful, observe that toapply our main theorem, it is typically necessary to compute a cocycle σ on G Λ with the required properties, and this is not so easy to do. (We discuss a classof examples where this is possible in Section 5.) But the last statement of thetheorem says that if we know that the centre of the bicharacter ω is trivial, thenno computations in G Λ are necessary.In the following result, for m ∈ Z k , we write m + and m − for m ∨ − m ) ∨ m = m + − m − and m + ∧ m − = 0. Lemma 3.5.
Let Λ be a row-finite cofinal k -graph with no sources, and supposethat c ∈ Z (Λ , T ) . Let g , . . . , g l be free generators for Per(Λ) . There exists v ∈ Λ such that T g + i ( x ) = T g − i ( x ) for all x ∈ Z ( v ) and ≤ i ≤ l . Let N = P li =1 g + i + g − i ,and fix λ ∈ v Λ N . For ≤ i ≤ l , factorise λ = µ i τ i = ν i ρ i where d ( µ i ) = g + i and d ( ν i ) = g − i . For ≤ i, j ≤ l , factorise λ = µ ij τ ij = ν ij ρ ij where d ( µ ij ) = ( g i + g j ) + and d ( ν ij ) = ( g i + g j ) − . Let ω be the bicharacter of Per(Λ) such that ω ( g i , g j ) := c ( µ i , τ i ) c ( ν i , ρ i ) c ( µ j , τ j ) c ( ν j , ρ j ) c ( µ ij , τ ij ) c ( ν ij , ρ ij ) for ≤ i, j ≤ l. Then there exist σ ∈ Z ( G Λ , T ) and a bicharacter ω ′ of Per(Λ) such that C ∗ (Λ , c ) ∼ = C ∗ ( G Λ , σ ) , σ | I Λ = ω ′ × Λ ∞ , and Z ω ′ = Z ω . In particular, if Z ω = { } then C ∗ (Λ , c ) is simple.Proof. We claim that there is a set
P ⊆ Λ s ∗ s Λ such that { ( λ, s ( λ )) : λ ∈ Λ } ∪ { ( µ i , ν i ) : i ≤ l } ∪ { ( µ ij , ν ij ) : i = j } ⊆ P , and such that G Λ = F ( µ,ν ) ∈P Z ( µ, ν ).To see this, we argue as in [26, Lemma 6.6] to see that the Z ( λ, s ( λ )) aremutually disjoint and F λ Z ( λ, s ( λ )) is clopen in G Λ . Let P := { ( λ, s ( λ )) : λ ∈ Λ } ∪ { ( µ i , ν i ) : i ≤ l } ∪ { ( µ ij , ν ij ) : 1 ≤ i ≤ j ≤ l } . For each i ≤ l either ( µ i , ν i ) = ( µ i , s ( µ i )), or d ( µ i ) − d ( ν i ) N k and so Z ( µ i , ν i ) ∩ Z ( λ, s ( λ )) = ∅ for all λ ; and likewise for each ( µ ij , ν ij ). Each Z ( µ i , ν i ) ⊆ Λ ∞ ×{ g i } × Λ ∞ , and each Z ( µ ij , ν ij ) ⊆ Λ ∞ × { g i + g j } × Λ ∞ . So the Z ( µ i , ν i ) and the Z ( µ ij , ν ij ) collectively are mutually disjoint. Hence the Z ( µ, ν ) where ( µ, ν ) ∈ P
02 ALEX KUMJIAN, DAVID PASK, AND AIDAN SIMS are mutually disjoint, and F ( µ,ν ) ∈P Z ( µ, ν ) is clopen in G Λ . Now the argument ofthe final two paragraphs of [26, Lemma 6.6] shows that we can extend P to therequired collection P .The formula (2.3) yields a cocycle σ c ∈ Z ( G Λ , T ), and the construction of P shows that σ c (( x, g i , x ) , ( x, g j , x )) = ω ( g i , g j ) for all x ∈ Z ( λ ) and i, j ≤ l .Corollary 7.9 of [26] implies that C ∗ (Λ , c ) ∼ = C ∗ ( G Λ , σ c ). Proposition 3.1 applied tothis P gives a bicharacter ω ′ of Per(Λ) and a cocycle σ on G Λ such that C ∗ ( G Λ , σ ) ∼ = C ∗ ( G Λ , σ c ) ∼ = C ∗ (Λ , c ) and ω ′ is cohomologous to σ xc and hence to ω . Thus Z ω = Z ω ′ by [29, Proposition 3.2]. The final statement follows from Theorem 3.4. (cid:3) We finish the section by showing that r σ induces an action θ of the quotient H Λ of G Λ by the interior of its isotropy on the space Λ ∞ × b Z ω . In particular, we provethat θ is minimal if and only if { ( r ( γ ) , r σγ | Z ω ) : γ ∈ ( G Λ ) x } is dense in Λ ∞ × b Z ω forall x ∈ Λ ∞ as in Theorem 3.4. In the next section we will realise C ∗ (Λ , c ) as the C ∗ -algebra of a Fell bundle over a quotient groupoid H Λ of G Λ , and show that theaction, given by [16], of H Λ on the primitive ideal space of the C ∗ -algebra overthe unit space in this bundle is isomorphic as a groupoid action to θ . We thenprove our main theorem by showing that minimality of the action described in [16]characterises simplicity of the C ∗ -algebra of the Fell bundle.Given Λ , c and ω as in Proposition 3.1, and with Z ω as in (3.4), we can formthe quotient H := Per(Λ) /Z ω . We then have b H ∼ = Z ⊥ ω ≤ Per(Λ) b , so we regard b H as a subgroup of Per(Λ) b . Lemma 3.6.
Let Λ be a row-finite cofinal k -graph with no sources, and supposethat c ∈ Z (Λ , T ) . Suppose that σ ∈ Z ( G Λ , T ) is cohomologous to σ c and that ω ∈ Z (Per(Λ) , T ) satisfies σ | I Λ = 1 Λ ∞ × ω as in Proposition 3.1. Let r σ bethe Per(Λ) b -valued -cocycle on G Λ given by Lemma 3.2. For α ∈ I Λ , we have r σα ∈ Z ⊥ ω . Let π : G Λ → H Λ := G Λ / I Λ be the quotient map. There is a continuous b Z ω -valued -cocycle ˜ r σ on H Λ such that ˜ r σπ ( α ) ( p ) = r σα ( p ) for all α ∈ G Λ and p ∈ Z ω . There is an action θ of H Λ on Λ ∞ × b Z ω such that θ α ( s ( α ) , χ ) = ( r ( α ) , ˜ r σα · χ ) for all α ∈ H Λ and χ ∈ b Z ω .Proof. Suppose that p ∈ Per(Λ) and q ∈ Z ω . Calculating in G Λ × σ T , we have(( x, p, x ) , x, q, x ) , x, p, x ) , − = (cid:0) ( x, q, x ) , σ (cid:0) ( x, p, x ) , ( x, q, x ) (cid:1) σ (cid:0) ( x, q, x ) , ( x, p, x ) (cid:1)(cid:1) = (cid:0) ( x, q, x ) , ωω ∗ ( p, q ) (cid:1) . Hence r σ ( x,p,x ) ( q ) = ( ωω ∗ )( p, q ) = 1 since q ∈ Z ω .Suppose that π ( α ) = π ( β ). Then α = βγ for some γ ∈ I Λ . So for p ∈ Z ω wehave r σα ( p ) = r σβ ( p ) r σγ ( p ) = r σβ ( p ), showing that ˜ r σ is well defined. It is continuousby definition of the quotient topology, and is a 1-cocycle because r σ is. For p, p ′ ∈ Z ω , we have˜ r σπ ( α ) ( p + p ′ ) = r σα ( p + p ′ ) = r σα ( p ) r σα ( p ′ ) = ˜ r σπ ( α ) ( p )˜ r σπ ( α ) ( p ′ ) . IMPLICITY OF TWISTED k -GRAPH ALGEBRAS 13 The final statement follows immediately. (cid:3) A Fell bundle associated to a k -graph 2-cocycle To identify the twisted k -graph algebra C ∗ (Λ , c ) with the C ∗ -algebra of a Fellbundle, we must first construct the bundle. We start by describing what willbecome the C ∗ -algebra sitting over the unit-space in this bundle: the twisted C ∗ -algebra of I Λ sitting inside that of G Λ . Lemma 4.1.
Let Λ be a row-finite cofinal k -graph with no sources, and take c ∈ Z (Λ , T ) . Suppose that σ ∈ Z ( G Λ , T ) is cohomologous to σ c and that ω ∈ Z (Per(Λ) , T ) satisfies σ | I Λ = 1 Λ ∞ × ω as in Proposition 3.1. There is a ∗ -homomorphism Φ : C c (Λ ∞ ) ⊗ C ∗ (Per(Λ) , ω ) → C c ( I Λ , σ ) such that ( f ⊗ u p )( x, q, x ) = δ p,q f ( x ) for all f ∈ C c (Λ ∞ ) and p, q ∈ Per(Λ) .This ∗ -homomorphism extends to an isomorphism Φ : C (Λ ∞ ) ⊗ C ∗ (Per(Λ , ω )) → C ∗ ( I Λ , σ ) .Proof. Recall that we identify I Λ with Λ ∞ × Per(Λ) as in Corollary 2.2. For each p ∈ Per(Λ), the characteristic function U p := 1 Λ ∞ ×{ p } is a unitary multiplierof C ∗ ( I Λ , σ ). By construction of σ , we have U p U q = ω ( p, q ) U p + q for all p, q , sothe universal property of C ∗ (Per(Λ) , ω ) gives a homomorphism C ∗ (Per(Λ) , ω ) →M C ∗ ( I Λ , σ ) satisfying u p U p . The U p clearly commute with C (Λ ∞ ), sothe universal property of the tensor product gives a homomorphism C (Λ ∞ ) ⊗ C ∗ (Per(Λ) , ω ) → C ∗ ( I Λ , σ ) satisfying f ⊗ u p U p f . For x ∈ Λ ∞ , we have( I Λ ) x ∼ = Per(Λ), and σ | ( I Λ ) x = ω by Proposition 3.1. Hence the regular rep-resentation of C ∗ ( I Λ , σ ) on ℓ (( I Λ ) x ) is unitarily equivalent to the canonicalfaithful representation of C ∗ (Per(Λ) , ω ) on ℓ (Per(Λ)). Thus the homomorphism ρ : f ⊗ u p U p f is a fibrewise-injective homomorphism of C (Λ ∞ )-algebras, andis injective on the central copy of C (Λ ∞ ). Since the norm on a C ( X )-algebrais the same as the supremum norm on the algebra of sections of the associatedupper-semicontinuous bundle [51], it follows that ρ is injective. Since its rangecontains C (Λ ∞ × { p } ) for each p , it is also surjective. (cid:3) Given a compact abelian group G , and an action β of a closed subgroup H of G on a C ∗ -algebra C , the induced algebra Ind GH C is defined byInd GH C = { f : G → C | f ( g − h ) = β h ( f ( g )) for g ∈ G and h ∈ H } with pointwise operations. Note that we use additive notation to emphasize that G must be abelian. This algebra carries an induced action lt of G given by translation:lt g ( f )( g ′ ) = f ( g ′ − g ). For h ∈ H , we have lt h ( f )( g ) = β h ( f ( g )).If C is simple, then Prim(Ind GH C ) is homeomorphic to G/H : for g + H ∈ G/H ,the corresponding primitive ideal I g + H is the ideal { f ∈ Ind GH C : f | g + H = 0 } (see,for example, [36, Proposition 6.6]).Now let A be a discrete abelian group and let ω be a T -valued 2-cocycle on A .Let Z ω ⊆ A be the centre of ω as in (3.4). The antisymmetric bicharacter ωω ∗ descends to an antisymmetric bicharacter ( ωω ∗ )˜of B := A/Z ω . There is a cocycle˜ ω ∈ Z ( B, T ) such that˜ ω ˜ ω ∗ ( a + Z ω , a ′ + Z ω ) = ( ωω ∗ )( a, a ′ ) for all a, a ′ ∈ A. By construction, the antisymmetric bicharacter ( ωω ∗ )˜ is nondegenerate in thesense that g + Z ω ( ωω ∗ )˜( g + Z ω , · ) is injective (as a map from B to its dual b B ).There is an action β A : b A → Aut( C ∗ ( A, ω )) such that β At ( U a ) = χ ( a ) U a for χ ∈ b A . Recall from [29, Lemma 5.11 and Theorem 6.3] that there is an b A -equivariant isomorphism C ∗ ( A, ω ) ∼ = Ind b A b B C ∗ ( B, ˜ ω ) , and that C ∗ ( B, ˜ ω ) is simple. Hence Prim( C ∗ ( A, ω )) ∼ = b A/ b B ∼ = c Z ω [36, Proposi-tion 6.6]. In particular, in the situation of Lemma 4.1, we havePrim( C ∗ ( I Λ , σ )) ∼ = Prim (cid:0) C (Λ ∞ ) ⊗ C ∗ (Per(Λ) , ω ) (cid:1) ∼ = Λ ∞ × c Z ω . Now resume the hypotheses and notation of Lemma 4.1. We construct a Fellbundle over H Λ . We describe the fibres of the bundle and the multiplicationand involution operations first, and then prove in Proposition 4.2 that there isa topology compatible with these operations under which we obtain the desiredFell bundle. We write C ∗ ( H ; B ) for the full C ∗ -algebra of a Fell bundle B over agroupoid H , and C ∗ r ( H ; B ) for the corresponding reduced C ∗ -algebra. We makethe convention that C ∗ ( H (0) ; B ) denotes the full C ∗ -algebra of the restriction of B to the unit space; this C ∗ ( H (0) ; B ) is a C ( H (0) )-algebra. For background on Fellbundles over ´etale groupoids, see [21, 28].We identify both H and G (0)Λ with Λ ∞ . We continue to write π : G Λ → H Λ forthe quotient map, and we often write [ γ ] for π ( γ ), and regard it as an equivalenceclass in G Λ ; that is [ γ ] = { α ∈ G Λ : π ( α ) = π ( γ ) } = γ · I Λ .We define A x = C ∗ (Per(Λ) , ω ) for x ∈ Λ ∞ , and we write A Λ for the trivialbundle Λ ∞ × C ∗ (Per(Λ) , ω ). So C ∗ ( I Λ , σ ) ∼ = C ∗ (Λ ∞ ; A Λ ), and A x is the fibre of A Λ over x . For [ γ ] ∈ H Λ , we let B ◦ [ γ ] := C c ([ γ ]) as a vector space over C , andwe define multiplication B ◦ [ α ] × B ◦ [ β ] → B ◦ [ αβ ] where s ( α ) = r ( β ), and involution B ◦ [ α ] → B ◦ [ α − ] by( f ∗ g )( γ ) = X ηζ = γ,η ∈ [ α ] ,ζ ∈ [ β ] σ ( η, ζ ) f ( η ) g ( ζ ) and f ∗ ( γ ) = σ ( γ, γ − ) f ( γ − ) . We regard each B ◦ [ x ] (where x ∈ Λ ∞ = H (0)Λ ) as a dense subspace of A x , andendow it with the norm inherited from A x . We write B x := B ◦ [ x ] ∼ = A x , and identify B x and A x . For [ γ ] ∈ H Λ , we define an A s ( γ ) -valued inner product on B ◦ [ γ ] by h f, g i s ( γ ) := f ∗ ∗ g . Then we obtain a norm on B ◦ [ γ ] by k f k = kh f, f i s ( γ ) k / , andwe write B [ γ ] for the completion of B ◦ [ γ ] in this norm. Note that B [ γ ] is a full right A s ( γ ) -Hilbert module. It is straightforward to show that k f ∗ g k ≤ k f kk g k for all There is a typographical error in the statement of [29, Theorem 6.3]:
G/G Z should read G Z . IMPLICITY OF TWISTED k -GRAPH ALGEBRAS 15 f ∈ B ◦ [ α ] and g ∈ B ◦ [ β ] when s ( α ) = r ( β ), and involution extends to an isometricconjugate linear map B [ α ] → B [ α − ] . Proposition 4.2.
With notation as above there is a unique topology on B Λ = F [ γ ] ∈H Λ B [ γ ] under which it is a Banach bundle and such that for each f ∈ C c ( G Λ , σ ) , the section [ γ ] f | [ γ ] is norm continuous. Under this topology, thespace B Λ is a saturated continuous Fell bundle over H Λ .Proof. The first assertion follows from [13, Proposition 10.4] once we show thatthe sections [ γ ] f | [ γ ] associated to elements f ∈ C c ( G Λ , σ ) are norm continuousand the ranges of these sections are pointwise dense. Let f ∈ C c ( G Λ , σ ). Thenthe restriction [ x ] (cid:13)(cid:13) f | [ x ] (cid:13)(cid:13) is continuous on H (0)Λ = Λ ∞ because C ∗ ( I Λ , σ ) ∼ = C (Λ ∞ ) ⊗ C ∗ (Per(Λ) , ω ) by Lemma 4.1. To show [ γ ] (cid:13)(cid:13) f | [ γ ] (cid:13)(cid:13) is continuous,note that k f | [ γ ] k = k ( f ∗ ∗ f ) | [ s ( γ )] k / . Since [ x ] (cid:13)(cid:13) ( f ∗ ∗ f ) | [ x ] (cid:13)(cid:13) is continuous on H (0)Λ , and since the source map [ γ ] [ s ( γ )] in H Λ is continuous, it follows that[ γ ] (cid:13)(cid:13) f | [ γ ] (cid:13)(cid:13) is continuous.It is straightforward to check that for each b ∈ B ◦ [ γ ] , there exists f ∈ C c ( G Λ , σ )such that b = f | [ γ ] . Hence, for each [ γ ], { f | [ γ ] : f ∈ C c ( G Λ , σ ) } is dense in B [ γ ] . (cid:3) We now establish that C ∗ (Λ , c ) can be identified with the C ∗ -algebra of theFell bundle we have just constructed. Recall that if B is a Fell bundle over agroupoid G , then C ∗ ( G ; B ), the C ∗ -algebra of the bundle, is a universal completionof the algebra of compactly supported sections of the bundle, and C ∗ r ( G ; B ) is thecorresponding reduced C ∗ -algebra. Theorem 4.3.
Suppose that Λ is a row-finite cofinal k -graph with no sources,and take c ∈ Z (Λ , T ) . Let H Λ denote the quotient of G Λ by the interior I Λ ofits isotropy, and let B Λ be the Fell bundle over H Λ described in Proposition 4.2.Then C ∗ ( H Λ ; B Λ ) = C ∗ r ( H Λ ; B Λ ) , and there is an isomorphism π : C ∗ (Λ , c ) ∼ = C ∗ ( H Λ ; B Λ ) such that π ( s λ )([ γ ]) = 1 Z ( λ,s ( λ )) | [ γ ] for all λ ∈ Λ and γ ∈ G Λ .Proof. Corollary 2.2 says that H Λ is amenable. Hence C ∗ ( H Λ ; B Λ ) = C ∗ r ( H Λ ; B Λ )by [46, Theorem 1]. Define elements t λ of C c ( H Λ ; B Λ ) by t λ ([ γ ]) := 1 Z ( λ,s ( λ )) | [ γ ] ∈ B ◦ [ γ ] ⊆ B [ γ ] . Theorem 6.7 of [26] shows that the 1 Z ( λ,s ( λ )) form a Cuntz–Krieger(Λ , c )-family in C ∗ ( G Λ , σ ). It follows that the t λ constitute a Cuntz–Krieger (Λ , c )-family in C ∗ ( H Λ ; B Λ ). So the universal property of C ∗ (Λ , c ) yields a homomor-phism π : C ∗ (Λ , c ) → C ∗ ( H Λ ; B Λ ).To see that π is injective, we aim to apply the gauge-invariant uniqueness the-orem [26, Corollary 7.7]. The projections { t v : v ∈ Λ } are nonzero becausethe Z ( v ) are nonempty, so we just need to show that there is an action β of T k on C ∗ ( H Λ ; B Λ ) such that β z ( t λ ) = z d ( λ ) t λ for all λ ∈ Λ. Let ˜ d : G Λ → Z k be the canonical cocycle ( x, m, y ) m . For z ∈ T k , [ γ ] ∈ H Λ and f ∈ B ◦ [ γ ] define β [ γ ] z ( f ) ∈ B ◦ [ γ ] by β [ γ ] z ( f )( α ) = z ˜ d ( α ) f ( α ). Simple calculations show that β [ γ ] z ( f ) ∗ β [ γ ′ ] z ( g ) = β [ γγ ′ ] z ( f ∗ g ) and that β [ γ − ] z ( f ∗ ) = β [ γ ] z ( f ) ∗ . For x ∈ G (0)Λ ,the map β [ x ] z : C c ( I Λ , σ ) → C c ( I Λ , σ ) extends to the canonical action of T k on A x = C ∗ (Per(Λ) , ω ), and so is isometric. It follows that the β [ γ ] z are isometric for the norms on the fibres B [ γ ] of B Λ . For f ∈ C c ( G Λ , σ ) supported on a basicopen set Z ( µ, ν ), the map [ γ ] β [ γ ] z ( f ) is clearly continuous. Since such f span C c ( G Λ , σ ), and by definition of the topology on B (see Proposition 4.2), it followsthat if p : B Λ → H Λ is the bundle map, then ξ β p ( ξ ) z ( ξ ) is continuous. So forfixed z ∈ T k , the collection β [ γ ] z determines an automorphism of the bundle B Λ ,and hence induces an automorphism β z of C ∗ ( H Λ ; B Λ ). It is routine to check that z β z is an action of T k on C ∗ ( H Λ ; B Λ ) such that β z ( f )([ γ ]) = β [ γ ] z ( f ([ γ ])) for f ∈ C c ( H Λ ; B Λ ) and z ∈ T k . In particular, each β z ( t λ ) = z d ( λ ) t λ , and so thegauge-invariant uniqueness theorem [26, Corollary 7.7] shows that π is injective asrequired.It remains to show that π is surjective. For this, it suffices to fix γ, γ ′ ∈ G Λ such that [ γ ] = [ γ ′ ], and show that the set { π ( a )([ γ ]) : a ∈ C ∗ (Λ , c ) , π ( a )[ γ ′ ] = 0 } is dense in B [ γ ] . Since π is linear, it will suffice to show that for each α ∈ [ γ ] thereexists a ∈ C ∗ (Λ , c ) such that π ( a )([ γ ′ ]) = 0 and π ( a )([ γ ]) = δ α . Fix α ∈ [ γ ], say˜ d ( α ) = m . Choose a compact open bisection U ⊆ ˜ d − ( m ). If m Per(Λ) + ˜ d ( γ ′ ),then U ∩ [ γ ′ ] = ∅ ; otherwise, since [ γ ′ ] = [ α ], either r ( γ ′ ) = r ( α ) or s ( γ ′ ) = s ( α ),and so we may shrink U to ensure that U ∩ [ γ ′ ] = ∅ . By definition of the topologyon G Λ , there exist µ, ν ∈ Λ such that α ∈ Z ( µ, ν ) ⊆ U . The function t µ t ∗ ν takesvalues in T on Z ( µ, ν ), and so there is a complex scalar z ∈ T such that a := zs µ s ∗ ν satisfies π ( a )([ γ ]) = δ α and π ( a )([ γ ′ ]) = 0 as claimed. (cid:3) To prove Theorem 3.4, we identify the action of H Λ on Prim( C ∗ ( H (0)Λ ; B Λ ))obtained from [16] with the action θ of Lemma 3.6. To do this, we first give anexplicit description of the action from [16]. Lemma 4.4.
Let B be a Fell bundle over a groupoid G with unital fibres over G (0) .Let γ ∈ G and suppose that u ∈ B γ is unitary in the sense that u ∗ u = 1 A s ( γ ) and uu ∗ = 1 A r ( γ ) . For any ideal I of A s ( γ ) , we have uIu ∗ = span { xay ∗ : x, y ∈ B γ , a ∈ I } . In particular, I uIu ∗ is the map from Prim( A s ( γ ) ) to Prim( A r ( γ ) ) describedin [16, Lemma 2.1] .Proof. We clearly have uIu ∗ ⊆ span { xay ∗ : x, y ∈ B γ , a ∈ I } . For the reverseinclusion, fix x i , y i ∈ B γ and a i ∈ I , and observe that X i x i a i y ∗ i = X i uu ∗ x i ay i uu ∗ = u (cid:16) X i u ∗ x i ay i u (cid:17) u ∗ ∈ uIu ∗ . (cid:3) We can now identify the action described in [16] with that of Lemma 3.6.
Theorem 4.5.
Let Λ be a row-finite cofinal k -graph with no sources, and supposethat c ∈ Z (Λ , T ) . Take ω as in Proposition 3.1, and let H Λ denote the quotient of G Λ by the interior I Λ of its isotropy. Let B Λ be the Fell bundle over H Λ described inProposition 4.2, and let A Λ denote the bundle of C ∗ -algebras obtained by restricting B Λ to H (0)Λ . There is a homeomorphism i : Prim( C ∗ ( H (0)Λ ; B Λ )) → Λ ∞ × c Z ω thatintertwines the action θ of H Λ on Λ ∞ × b Z ω described in Lemma 3.6 and the actionof H Λ on Prim( C ∗ ( H (0)Λ ; B Λ )) described in [16] . IMPLICITY OF TWISTED k -GRAPH ALGEBRAS 17 Proof.
Put H := Per(Λ) /Z ω . Take x ∈ Λ ∞ . Then A x = C ∗ (Per(Λ) , ω ) ∼ = span { δ ( x,p,x ) : p ∈ Per(Λ) } , and there is an isomorphism i x : A x → Ind
Per(Λ) bb H C ∗ ( H, ω ) such that i x ( δ ( x,p,x ) )( χ ) = χ ( p ) U p + Z ω for all χ ∈ Per(Λ) b .In particular i x is equivariant for the action of Per(Λ) b on A x given by χ · δ ( x,p,x ) = χ ( p ) δ ( x,p,x ) and the action lt of Per(Λ) b on Ind Per(Λ) bb H C ∗ ( H, ω ) by translation.Take γ ∈ G Λ and p ∈ Per(Λ). A straightforward calculation in B Λ shows that δ γ ∗ δ ( s ( γ ) ,p,s ( γ )) ∗ δ ∗ γ = r σγ ( p ) δ ( r ( γ ) ,p,r ( γ )) . The element δ γ is a unitary in the fibre B [ γ ] , and so Lemma 4.4 shows that conjuga-tion by δ γ in B Λ implements the homeomorphism α [ γ ] of [16] from Prim( A s ( γ ) ) ⊆ Prim( C ∗ ( H (0)Λ ; B Λ )) to Prim( A r ( γ ) ). We have i r ( γ ) ◦ Ad δ γ ◦ i − r ( γ ) = lt r σγ , and it fol-lows that for χ ∈ Per(Λ) b the primitive ideal I χ b H of Ind Per(Λ) bb H C ∗ ( H, ω ) consistingof functions that vanish on χ b H satisfies (cid:0) i r ( γ ) ◦ Ad δ γ ◦ i − r ( γ ) (cid:1) ( I χ b H ) = I r σγ · χ b H . That is, the induced map ( i r ( γ ) ) ∗ α [ γ ] ( i r ( γ ) ) − ∗ on Prim (cid:0) Ind
Per(Λ) bb H C ∗ ( H, ω ) (cid:1) istranslation by ˜ r σ [ γ ] . Lemma 4.1 yields a homeomorphism i : Prim( C ∗ ( H (0) ; B Λ )) → Λ ∞ × b Z ω such that i | Prim( A x ) = i x , and this homeomorphism does the job. (cid:3) Recall from [21, Proposition 3.6] that if B is a Fell bundle over an ´etale groupoid H , then restriction of compactly supported sections to H (0) extends to a faithfulconditional expectation P : C ∗ r ( H ; B ) → C ∗ ( H (0) ; B ). Lemma 4.6.
Suppose that H is an ´etale topologically principal groupoid and that B is a Fell bundle over H . If I is a nonzero ideal of C ∗ r ( H ; B ) , then I ∩ C ∗ ( H (0) ; B ) = { } .Proof. Our argument follows that of [1, Proposition 2.4] (see also [3, Lemma 4.2]or [23, Lemma 3.5]).Let I be a nonzero ideal of C ∗ r ( H ; B ). Let P : C ∗ r ( H ; B ) → C ∗ ( H (0) ; B ) bethe faithful conditional expectation discussed above. Choose a ∈ I + such that k P ( a ) k = 1. Choose b ∈ Γ c ( H ; B ) ∩ C ∗ r ( H ; B ) + such that k a − b k < /
4, so that k P ( b ) k > /
4. Then b − P ( b ) ∈ Γ c ( H ; B ); thus, K := supp( b − P ( b )) is compactand contained in H \ H (0) . Let U = { u ∈ H (0) : k P ( b )( u ) k > / } . By [3,Lemma 3.3], there exists an open set V such that V ⊆ U and V KV = ∅ .Fix a continuous function h : R → [0 ,
1] such that h is identically 0 on ( −∞ , / / , ∞ ). Then h ( P ( b )) P ( b ) h ( P ( b )) ≥ h ( P ( b )) . The statement in [21] says “ r -discrete,” but ´etale is meant. Choose g ∈ C c ( H (0) ) ⊆ M C ∗ r ( H ; B ) such that k g k ∞ = 1 and supp( g ) ⊆ V . Let f := gh ( P ( b )). Since k h ( P ( b ))( u ) k = 1 for all u ∈ U , and since V ⊆ U , we have f = 0.Since supp( f ) ⊆ supp( g ) ⊆ V , we have f ( b − P ( b )) f = 0. Hence f bf = f (cid:0) P ( b ) + ( b − P ( b )) (cid:1) f = f P ( b ) f = gh ( P ( b )) P ( b ) h ( P ( b )) g ≥ g h ( P ( b )) = 12 f . Thus f af ≥ f bf − f = f P ( b ) f − f ≥ f . Since f af ∈ I and I is hereditary, we deduce that f ∈ I ∩ C ∗ ( H (0) ; B ) \ { } . (cid:3) Lemma 4.6 allows us to characterise the simplicity of C ∗ ( H ; B ) when H isamenable and topologically principal. Corollary 4.7.
Suppose that H is a topologically principal amenable groupoid andthat B is a Fell bundle over H . Then C ∗ ( H ; B ) is simple if and only if the actionof H on Prim( C ∗ ( H (0) ; B )) described in [16, Section 2] is minimal.Proof. We denote the action of H on Prim( C ∗ ( H (0) ; B )) by ϑ . First suppose that ϑ is not minimal. Then there exists ρ in Prim( C ∗ ( H (0) ; B )) such that H · ρ isnot dense in Prim( C ∗ ( H (0) ; B )). Hence X := H · ρ is a nontrivial closed invariantsubspace of Prim( C ∗ ( H (0) ; B )). The set I X of sections of B | H (0) that vanish on X is a nontrivial H Λ -invariant ideal of the C ∗ -algebra C ∗ ( H (0) ; B ) sitting over theunit space of H . Thus the ideal of C ∗ ( H ; B ) generated by I X is a proper nontrivialideal [16, Theorem 3.7] and so C ∗ ( H ; B ) is not simple.Now suppose that ϑ is minimal. Let I be a nonzero ideal of C ∗ ( H ; B ). The-orem 1 of [46] shows that C ∗ ( H ; B ) = C ∗ r ( H ; B ), so Lemma 4.6 implies that I := I ∩ C ∗ ( H (0) ; B ) is nonzero. Let X denote the set of primitive ideals of C ∗ ( H (0) ; B ) that contain I . Since I is nonzero, X is nonempty, and it is closed bydefinition of the topology on Prim( C ∗ ( H (0) ; B )). Lemma 2.1 of [16] implies that X is also ϑ -invariant. Since ϑ is minimal, it follows that X = Prim( C ∗ ( H (0) ; B )),and so I = C ∗ ( H (0) ; B ). Since C ∗ ( H (0) ; B ) contains an approximate identity for C ∗ r ( H ; B ), it follows that I = C ∗ ( H ; B ) as required. (cid:3) Since we established in Corollary 2.2 that H Λ is always topologically principaland amenable, we obtain an immediate corollary. Corollary 4.8.
Let Λ be a row-finite cofinal k -graph with no sources, and supposethat c ∈ Z (Λ , T ) . Let θ be the action of H Λ on Λ ∞ × Per(Λ) b obtained from anychoice of σ in Lemma 3.6. Then C ∗ (Λ , c ) is simple if and only if θ is minimal.Proof. Theorem 4.3 shows that C ∗ (Λ , c ) is simple if and only if C ∗ ( H Λ ; B Λ ) issimple. Corollary 2.2 shows that H Λ is amenable and topologically principal, andso Corollary 4.7 implies that C ∗ ( H Λ ; B Λ ) is simple if and only if the action of H Λ on Prim( C ∗ ( H (0)Λ ; B Λ )) is minimal. Theorem 4.5 shows that this action is minimalif and only if θ is minimal, giving the result. (cid:3) We can now prove our main theorem.
IMPLICITY OF TWISTED k -GRAPH ALGEBRAS 19 Proof of Theorem 3.4.
By Corollary 4.8, it is enough to show that the set(4.1) { ( r ( γ ) , r σγ | Z ω ) : γ ∈ ( G Λ ) x } is dense in Λ ∞ × b Z ω for every x ∈ Λ ∞ if and only if the action θ of H Λ on Λ ∞ × Per(Λ) b is minimal. Clearly (4.1) is dense for every x if and only if { ( r ( γ ) , r σγ | Z ω · χ ) : γ ∈ ( G Λ ) x } is dense in Λ ∞ × Z ω for every x and every χ ∈ b Z ω , which is preciselyminimality of θ . (cid:3) Twists induced by torus-valued 1-cocycles, and crossed-productsby quasifree actions
In this section we describe a class of examples of twisted ( k + l )-graph C ∗ -algebras arising from T l -valued 1-cocycles on aperiodic k -graphs. We describe thesimplicity criterion obtained from our main theorem for these examples. We thenshow that these twisted C ∗ -algebras can also be interpreted as twisted crossed-products of k -graph algebras by quasifree actions, giving a characterisation ofsimplicity for the latter. See also [5, Theorem 2.1] for the case where l = 1.Recall that we write T l for N l when regarded as an l -graph with degree mapthe identity functor.Given a cocycle ω ∈ Z ( Z l , T ) and an action α of Z l on a C ∗ -algebra A , wewrite A × α,ω Z l for the twisted crossed product, which is the universal C ∗ -algebragenerated by unitary multipliers { u n ∈ M ( A × α,ω Z l ) : n ∈ Z l } and a homomor-phism π : A → A × α,ω Z l such that u m π ( a ) u ∗ m = π ( α m ( a )) for all m, a and suchthat u m u n = ω ( m, n ) u m + n for all m, n . For further details on twisted crossedproducts, see [30].Let Λ be a row-finite k -graph with no sources. Consider a 1-cocycle φ ∈ Z (Λ , T l ). Define a 2-cocycle c φ ∈ Z (Λ × T l , T ) by(5.1) c φ (( λ, m ) , ( µ, n )) = φ ( µ ) m for composable λ, µ ∈ Λ and m, n ∈ N l = T l .There is a continuous cocycle ˜ φ ∈ Z ( G Λ , T l ) such that˜ φ ( µx, d ( µ ) − d ( ν ) , νx ) = φ ( µ ) φ ( ν ) for all x ∈ Λ ∞ and µ, ν ∈ Λ r ( x ).Let ω be a bicharacter of Z l . There is a cocycle c φ,ω ∈ Z (Λ × T l , T ) such that(5.2) c φ,ω (( λ, m ) , ( µ, n )) = φ ( µ ) m ω ( m, n ) . The next theorem is the main result of this section. It characterises simplicityof C ∗ (Λ × T l , c φ,ω ) in terms of ˜ φ and the centre Z ω of ω described in (3.4), underthe simplifying assumption that Λ is aperiodic. Theorem 5.1.
Let Λ be a row-finite k -graph with no sources. Take φ ∈ Z (Λ , T l ) ,and let ω be a bicharacter of Z l . Let ˜ φ ∈ Z ( G Λ , T ) and c φ,ω ∈ Z (Λ × T l , T ) beas above. Then (1) there is an action β of Z l on C ∗ (Λ) such that β m ( s λ ) = φ ( λ ) m s λ for all λ ∈ Λ and m ∈ Z l ; (2) there is an isomorphism ρ : C ∗ (Λ × T l , c φ,ω ) ∼ = C ∗ (Λ) × β,ω Z l such that ρ ( s ( λ,m ) ) = π ( s λ ) u m for all λ ∈ Λ and m ∈ T l ; and (3) if Λ is aperiodic, then C ∗ (Λ × T l , c φ,ω ) is simple if and only if the orbit { ( r ( γ ) , ˜ φ ( γ ) | Z ω ) : γ ∈ ( G Λ ) x } is dense in Λ ∞ × b Z ω for all x ∈ Λ ∞ .Remark . The observant reader may be surprised to note that there is nocofinality hypothesis on the preceding theorem. But a moment’s reflection showsthat it is still there, just hidden: each of the conditions that C ∗ (Λ × T l , c φ,ω ) issimple, and that each { ( r ( γ ) , ˜ φ ( γ ) | Z ω ) : γ ∈ ( G Λ ) x } is dense in Λ ∞ × b Z ω impliesthat Λ is cofinal.We prove (1) and (2) here and defer the proof of (3) to the end of the section. Proof of Theorem 5.1(1) and (2).
For part (1) observe that for each m ∈ Z l theset { φ ( λ ) m s λ : λ ∈ Λ } is a Cuntz–Krieger Λ-family, so induces a homomorphism β m : C ∗ (Λ) → C ∗ (Λ) such that β m ( s λ ) = φ ( λ ) m s λ . Clearly β = id and β m ◦ β n = β m + n , so β is an action.For part (2) we first check that { π ( s λ ) u m : ( λ, m ) ∈ Λ × T l } is a Cuntz–Krieger(Λ × T l , c φ,ω )-family. For λ ∈ Λ and m ∈ T l let t ( µ,m ) = π ( s λ ) u m . It is easy tocheck that (CK1), (CK3) and (CK4) hold because each β m fixes each s λ s ∗ λ . Tocheck (CK2) we compute t ( λ,m ) t ( µ,n ) = π ( s λ ) u m π ( s µ ) u n = π ( s λ ) u m π ( s µ ) u ∗ m u m u n = π ( s λ ) π ( β m ( s µ )) ω ( m, n ) u m + n = φ ( µ ) m ω ( m, n ) π ( s λµ ) u m + n = φ ( µ ) m ω ( m, n ) t ( λµ,m + n ) = c φ,ω (( λ, m )( µ, n )) t ( λ,m )( µ,n ) . Now the universal property of C ∗ (Λ × T l , c φ,ω ) gives a homomorphism ρ sat-isfying the desired formula. The gauge-invariant uniqueness theorem [26, Corol-lary 7.7] shows that ρ is injective. The map ρ is surjective because its imagecontains all the generators of the twisted crossed product. (cid:3) In order to prove (3) we first do some preparatory work to identify the action θ of Lemma 3.6 in terms of the cocycle ˜ φ ∈ Z ( G Λ , T ). Before that, though, acomment on the hypotheses of Theorem 5.1 is in order. Remark . Note that the aperiodicity hypothesis in Theorem 5.1 is needed inour proof and simplifies the statement, but is not a necessary condition for C ∗ (Λ × T l , c φ,ω ) to be simple. To see this, consider Λ = T , which is N regarded as a 1-graph. Put l = 1, define φ : Λ → T by φ (1) = e πiθ and take ω to be trivial. ThenΛ is cofinal, but certainly not aperiodic. We have Λ × T ∼ = T , and c φ,ω ∈ Z ( T , T )is given by c φ,ω ( m, n ) := e πiθm n , so C ∗ (Λ × T , c φ,ω ) is the rotation algebra A θ ,which is simple whenever θ is irrational.Let Γ := Λ × T l . The proof of Theorem 5.1 (3) requires quite a bit of preliminarywork. To begin preparations, observe that the projection map π : Γ → Λ given by( λ, m ) λ induces a homeomorphism π ∞ : Γ ∞ → Λ ∞ such that π ∞ ( x )( m, n ) = π ( x (( m, , ( n, G Γ ∼ = G Λ × Z l given by (cid:0) ( α, m ) x, ( d ( α ) , m ) − ( d ( β ) , n ) , ( β, n ) x (cid:1) (cid:0) ( απ ∞ ( x ) , d ( α ) − d ( β ) , βπ ∞ ( x )) , m − n (cid:1) . (5.3) IMPLICITY OF TWISTED k -GRAPH ALGEBRAS 21 Suppose that Λ is cofinal and aperiodic, and fix a subset P of Λ s ∗ s Λ such that( µ, s ( µ )) ∈ P for all µ ∈ Λ and such that { Z ( µ, ν ) : ( µ, ν ) ∈ P} is a partitionof G Λ . For g ∈ G Λ , let ( µ g , ν g ) ∈ P be the unique pair such that g ∈ Z ( µ g , ν g ).Observe that Per(Γ) = { } × Z l ⊆ Z k + l . Recall that H Γ is the quotient G Γ / I Γ ofthe ( k + l )-graph groupoid by the interior of its isotropy. Proposition 5.4.
Let Λ be a row-finite cofinal aperiodic k -graph with no sources,and let Γ := Λ × T l . Then Per(Γ) = { } × Z l ⊆ Z k + l , the identification of G Γ with G Λ × Z l described above carries I Γ to G (0)Λ × Z l , and ( g, m ) · I Γ g gives an isomorphism H Γ ∼ = G Λ . Let φ : Λ → T l be a -cocycle, and let ω be abicharacter of Z l . Let c φ,ω ∈ Z (Γ , T ) be the 2-cocycle of (5.2) . There exists acocycle σ ∈ Z ( G Γ , T ) such that σ | I Γ = 1 Λ ∞ × ω , C ∗ ( G Γ , σ ) ∼ = C ∗ (Γ , c φ,ω ) and theaction θ of G Λ on Λ ∞ × b Z ω of Lemma 3.6 satisfies (5.4) θ g ( s ( g ) , χ ) = ( r ( g ) , ˜ φ ( g ) | Z ω χ ) . Before proving Proposition 5.4, we establish two technical lemmas that will helpin the proof. The first of these shows that the passage from a k -graph cocycle to agroupoid cocycle described in (2.3) has no effect on cohomology when the k -graphin question is N l and its groupoid is Z l .Recall that for m ∈ Z l , we write m + := m ∨ m − := ( − m ) ∨
0; so m = m + − m − and m + ∧ m − = 0. Lemma 5.5.
Let ω be a bicharacter of Z l . Then ω | T l × T l belongs to Z ( T l , T ) .The l -graph T l has a unique infinite path x , and there is an isomorphism G T l ∼ = Z l given by ( x, m, x ) m . Let P = { ( m + , m − ) : m ∈ Z l } . Then ( λ, s ( λ )) ∈ P for all λ ∈ T l , and G T l = F ( µ,ν ) ∈P Z ( µ, ν ) . Let σ ω ∈ Z ( G T l , T ) be the 2-cocycleobtained from ω and P as in (2.3) . Then σ ω is cohomologous to ω when regardedas a cocycle on Z l .Proof. Every bicharacter of Z l is a 2-cocycle, so ω restricts to a cocycle on T l . Itis clear that G T l ∼ = Z l as claimed. We identify G T l with Z l for the rest of the proof.For m ∈ G T l , the pair ( µ m , ν m ) = ( m + , m − ) is the unique element of P suchthat m ∈ Z ( µ m , ν m ). So for m, n ∈ G T l , equation (2.3) gives σ ω ( m, n ) = ω ( m + , α ) ω ( m − , α ) ω ( n + , β ) ω ( n − , β ) ω (( m + n ) + , γ ) ω (( m + n ) − , γ )for any choice of α, β, γ ∈ T l such that m + + α = ( m + n ) + + γ , n − + β = ( m + n ) − + γ and m − + α = n + + β .We show that σ ω ( e i , e j ) = ω ( e i , e j ) for all i, j ≤ l . For this observe that for m = e i and n = e j , the elements α = e j , β = 0 and γ = 0 satisfy the aboveconditions, so σ ω ( e i , e j ) = ω ( e i , e j ) ω (0 , e j ) ω ( e j , ω (0 , ω ( e i + e j , ω (0 ,
0) = ω ( e i , e j )as claimed.Hence ( σ ω σ ∗ ω )( e i , e j ) = ( ωω ∗ )( e i , e j ) for all i, j . These are bicharacters by [29,Proposition 3.2] and so we have σ ω σ ∗ ω = ωω ∗ . Thus [29, Proposition 3.2] impliesthat σ ω is cohomologous to ω as claimed. (cid:3) Our second technical result shows how to obtain a partition Q satisfying (2.1)for the ( k + l )-graph Λ × T l from a partition P satisfying (2.1) for Λ. Lemma 5.6.
Let Λ be a row-finite k -graph with no sources. Suppose that P ⊆ Λ s ∗ s Λ satisfies ( µ, s ( µ )) ∈ P for all µ , and G Λ = F ( µ,ν ) ∈P Z ( µ, ν ) . Let Q := { (( µ, m + ) , ( ν, m − )) : ( µ, ν ) ∈ P , m ∈ Z l } . Then ( α, s ( α )) ∈ Q for all α ∈ Γ , and G Γ = F ( α,β ) ∈Q Z ( α, β ) .Proof. Consider an element α = ( µ, m ) ∈ Γ. We have m + = m and m − = 0, and( α, s ( α )) = (( µ, m ) , ( s ( µ ) , µ, s ( µ )) ∈ P , we have(( µ, m ) , ( s ( µ ) , µ, m + ) , ( s ( µ ) , m − )) ∈ Q . Let ρ : G Γ → G Λ × Z l denote the map (5.3). Then G Γ = ρ − ( G Λ × Z l ) = G ( µ,ν ) ∈P ρ − ( Z ( µ, ν ) × Z l )= G ( µ,ν ) ∈P G m ∈ Z l ρ − ( Z ( µ, ν ) × { m } )= G ( µ,ν ) ∈P G m ∈ Z l Z (( µ, m + ) , ( ν, m − )) = G ( α,β ) ∈Q Z ( α, β ) . (cid:3) Proof of Proposition 5.4.
Since π ∞ : Γ ∞ → Λ ∞ intertwines the shift maps, wehave T ( p,m ) ( x ) = T ( q,n ) ( x ) if and only if T p ( π ∞ ( x )) = T q ( π ∞ ( x )). Hence Per(Γ) =Per(Λ) × Z l = { } × Z l because Λ is aperiodic. The next two assertions arestraightforward to check using the definition of the isomorphism G Γ ∼ = G Λ × Z l .We identify G Γ with G Λ × Z l for the remainder of this proof. Choose P ⊆ Λ s ∗ s Λsuch that ( λ, s ( λ )) ∈ P for all λ and G Λ = F ( µ,ν ) ∈P Z ( µ, ν ). For g ∈ G Λ , write( µ g , ν g ) for the element of P with g ∈ Z ( µ g , ν g ). Let Q = { ( µ, m + ) , ( ν, m − ) :( µ, ν ) ∈ P , m ∈ Z l } as in Lemma 5.6. For ( g, m ) ∈ G Γ , let ˜ µ ( g,m ) := ( µ g , m + ),and ˜ ν ( g,m ) ) = ( ν g , m − ). Then (˜ µ ( g,m ) , ˜ ν ( g,m ) ) is the unique element of Q suchthat ( g, m ) ∈ Z (˜ µ ( g,m ) , ˜ ν ( g,m ) ). For (( x, , x ) , m ) ∈ I Γ , we have ˜ µ (( x, ,x ) ,m ) =( r ( x ) , m + ) and ˜ ν (( x, ,x ) ,m ) = ( r ( x ) , m − ).Let ˜ ω denote the 2-cocycle on Γ given by (( µ, m ) , ( ν, n )) ω ( m, n ), and recallthat c φ ∈ Z (Λ × T l , T ) is given by (5.1). Let σ c φ be the 2-cocycle on G Γ obtainedfrom (2.3) applied to c φ ∈ Z (Γ , T ) and Q , and let σ ˜ ω be the 2-cocycle on G Γ obtained in the same way from the cocycle (( µ, m ) , ( ν, n )) ω ( m, n ) on Γ. Usingthat c φ,ω = c φ ˜ ω and the definitions (2.3) of σ c φ , σ ˜ ω and σ c φ,ω , it is easy to seethat σ c φ,ω = σ c φ σ ˜ ω . Let σ ω ∈ Z ( G T l , T ) be the cocycle obtained from ω asin Lemma 5.5. The formulas for σ ˜ ω and σ ω show that the identification G Γ ∼ = G Λ × Z l ∼ = G Λ × G T l carries σ ˜ ω to 1 Z ( G Λ , T ) × σ ω . Thus Lemma 5.5 shows that σ c φ,ω is cohomologous to the cocycle σ ∈ Z ( G Γ , T ) given by σ (( g, m ) , ( h, n )) = σ c φ (( g, m ) , ( h, n )) ω ( m, n ) . Corollary 7.9 of [26] shows that C ∗ (Γ , c φ,ω ) ∼ = C ∗ ( G Γ , σ c φ,ω ). Proposition II.1.2of [38] says that cohomologous groupoid cocycles determine isomorphic twistedgroupoid C ∗ -algebras, and so we have C ∗ (Γ , c φ,ω ) ∼ = C ∗ ( G Γ , σ ). We have σ | I Γ = IMPLICITY OF TWISTED k -GRAPH ALGEBRAS 23 Λ ∞ × ω by construction, so it remains to calculate the action ˜ r σ of G Λ on b Z ω described in Lemma 3.6.For this, we first claim that σ c φ satisfies(5.5) σ c φ (( g, m ) , ( h, n )) = (cid:0) φ ( µ g ) φ ( µ gh ) (cid:1) m (cid:0) φ ( ν h ) φ ( ν gh ) (cid:1) n . To see this, choose α, β, γ ∈ Λ and y ∈ Λ ∞ satisfying (2.2). Then there exist m α , m β and m γ ∈ N l such that ˜ α = ( α, m α ), ˜ β = ( β, m β ) and ˜ γ = ( γ, m γ ) satisfyequations (2.2) with respect to the ˜ µ ’s and ˜ ν ’s. So σ c φ (( g, m ) , ( h, n )) = c φ (˜ µ g , ˜ α ) c φ (˜ ν g , ˜ α ) c φ (˜ µ h , ˜ β ) c φ (˜ ν h , ˜ β ) c φ (˜ µ gh , ˜ γ ) c φ (˜ ν gh , ˜ γ )= φ ( α ) m + φ ( α ) m − φ ( β ) n + φ ( β ) n − φ ( γ ) ( m + n ) + φ ( γ ) ( m + n ) − = (cid:0) φ ( α ) φ ( γ ) (cid:1) m (cid:0) φ ( β ) φ ( γ ) (cid:1) n . (5.6)We have φ ( µ g ) φ ( α ) = φ ( µ g α ) = φ ( µ gh γ ) = φ ( µ gh ) φ ( γ ), and rearranging gives φ ( α ) φ ( γ ) = φ ( µ g ) φ ( µ gh ), and similarly, φ ( β ) φ ( γ ) = φ ( ν h ) φ ( ν gh ). Substitutingthis into (5.6), we obtain (5.5).Now, fix p ∈ Z ω and ( g, n ) = (( x, m, y ) , n ) ∈ G Γ . Using (5.5) at the secondequality, we calculate: r σ ( g,n ) ( p )= σ (cid:0) ( g, n ) , (( y, , y ) , p ) (cid:1) σ (cid:0) ( g, n + p ) , ( g − , − n ) (cid:1) σ (cid:0) ( g, n ) , ( g − , − n ) (cid:1) = h(cid:0) φ ( µ g ) φ ( µ g ) (cid:1) n (cid:0) φ ( ν s ( g ) ) φ ( ν g ) (cid:1) p ih(cid:0) φ ( µ g ) φ ( µ gg − ) (cid:1) n + p (cid:0) φ ( ν g − ) φ ( ν gg − ) (cid:1) − n ih(cid:0) φ ( µ g ) φ ( µ gg − ) (cid:1) n (cid:0) φ ( ν g − ) φ ( ν gg − ) (cid:1) − n i ω ( n, p ) ω ( n + p, − n ) ω ( n, − n )= φ ( ν g ) p φ ( µ g ) n + p φ ( ν g − ) n φ ( µ g ) n φ ( ν g − ) − n ω ( n, p ) ω ( p, n )= (cid:0) φ ( µ g ) φ ( ν g ) (cid:1) p ( ωω ∗ )( n, p ) . Since p ∈ Z ω , we have ( ωω ∗ )( n, p ) = 1, and so r σ ( g,n ) ( p ) = ˜ φ ( g ) p . Since ( g, n ) g induces an isomorphism H Γ ∼ = G Λ , we deduce that ˜ r σg ( p ) = ˜ φ ( g ) p for g ∈ G Λ and p ∈ Z ω . So (5.4) follows from the definition of θ . (cid:3) Proof of part (3) of Theorem 5.1.
As observed in Remark 5.2, both conditions ap-pearing in statement (3) imply that Λ is cofinal, so we may assume that thisis the case. The cocycle σ of Proposition 5.4 satisfies the hypotheses of Theo-rem 3.4. So Theorem 3.4 combined with Proposition 5.4 says that C ∗ (Γ , c φ,ω ) issimple if and only if the action θ of G Λ on Λ ∞ × Z ω described in (5.4) is mini-mal. The condition described in the statement of Theorem 5.1(3) is precisely thestatement that G Λ · ( x, x,
1) under θ , is dense for all x . Since G Λ · ( x, z ) = { ( y, zw ) : ( y, w ) ∈ G Λ · ( x, } , the result follows. (cid:3) As a special case of Theorem 5.1, we obtain the following.
Corollary 5.7.
Let E be a strongly connected graph which is not a simple cycle.Let φ be a function from E to T . There is an action β : Z → Aut( C ∗ ( E )) such that β n ( s e ) = φ ( e ) n s e for all e ∈ E and n ∈ Z . If there is a cycle µ = µ · · · µ n such that Q i φ ( µ i ) = e πiθ for some θ Q , then C ∗ ( E ) × β Z is simple.Proof. We will apply Theorem 5.1 to the 1-graph E ∗ and the extension of φ to a1-cocycle φ : E ∗ → T . Fix x ∈ E ∞ . By Theorem 5.1, it suffices to show that { ( r ( γ ) , ˜ φ ( γ )) : γ ∈ G E , s ( γ ) = x } = E ∞ × T . Choose a basic open set Z ( η ) of E ∞ and z ∈ T . It suffices to show that there areelements γ n with s ( γ n ) = x and r ( γ n ) ∈ Z ( η ) such that ˜ φ ( γ n ) → z .Since E is strongly connected, there exist α ∈ s ( η ) E ∗ r ( µ ) and β ∈ r ( µ ) E ∗ r ( x ).The elements γ n := ( ηαµ n βx, d ( ηαµ n β ) , x ) ∈ G E satisfy s ( γ n ) = x , r ( γ n ) ∈ Z ( η ),and ˜ φ ( γ n ) := e πinθ ˜ φ ( γ ). Since θ is irrational, these values are dense in T . (cid:3) We conclude the section by giving a version of Theorem 5.1 whose statementdoes not require groupoid technology. Let Λ be a row-finite k -graph with nosources, and take φ ∈ Z (Λ , T l ) and ω a bicharacter of Z l . Regard each φ ( λ ) as acharacter of Z l so that φ ( λ ) | Z ω is a character of Z ω . Then Λ acts on Λ ∞ × b Z ω bypartial homeomorphisms { ϑ λ : λ ∈ Λ } where each dom( ϑ λ ) = Z ( s ( λ )) × b Z ω , and ϑ λ ( x, χ ) = ( λx, φ ( λ ) | Z ω χ ) for ( x, χ ) ∈ Z ( s ( λ )) × b Z ω . Define a relation ∼ on Λ ∞ × b Z ω by( x, χ ) ∼ ( x ′ , χ ′ ) iff there exist ( y, ρ ) ∈ Λ ∞ × b Z ω and λ, µ ∈ Λ such that s ( λ ) = s ( µ ) = r ( y ) , ϑ λ ( y, ρ ) = ( x, χ ) and ϑ µ ( y, ρ ) = ( x ′ , χ ′ ) . Then ∼ is an equivalence relation; indeed, it is the equivalence relation inducedby ϑ . We write [ x, χ ] for the equivalence class of ( x, χ ) under ∼ . Corollary 5.8.
Let Λ be an aperiodic row-finite k -graph with no sources and take φ ∈ Z (Λ , T l ) . Let c φ,ω ∈ Z (Λ × T l , T ) be as defined in (5.2) . Then C ∗ (Λ × T l , φ c,ω ) is simple if and only if [ x, is dense in Λ ∞ × b Z ω for all x ∈ Λ ∞ .Proof. It is routine to check that [ x,
1] is equal to the set { ( r ( λ ) , ˜ φ ( γ ) | Z ω : γ ∈ ( G Λ ) x } of Theorem 5.1 part (3). (cid:3) Examples
First we discuss an example of a nondegenerate 2-cocycle on Z pulled backover the degree map to a cocycle on a cofinal 2-graph for which the twisted C ∗ -algebra is not simple. This shows that the hypothesis of [45, Theorem 7.1] that c | Per(Λ) is nondegenerate cannot be relaxed to the weaker assumption that c isnondegenerate as a cocycle on Z k . Example . Let θ ∈ R \ Q , and define a 1-cocycle φ on Ω by φ ( m, n ) = e πi ( n − m ) θ . As in Section 5, define c φ ∈ Z (Ω × T , T ) by c φ (( α, m ) , ( β, n )) = φ ( β ) m . Then c φ = ̟ ◦ d where d : Ω × T → N is the degree map, and ̟ ∈ Z ( Z , T ) is the cocycle ̟ ( m, n ) = e πiθm n . Since θ is irrational, the an-tisymmetric bicharacter associated to ̟ is nondegenerate. Note that Per(Λ) ∼ = Z and so the restriction of ̟ to Per(Λ) is degenerate. IMPLICITY OF TWISTED k -GRAPH ALGEBRAS 25 We have G Ω ∼ = N × N as an equivalence relation. So G (0)Ω × T ∼ = N × T , andunder this identification, the action θ described in Proposition 5.4 boils down to θ ( m,n ) ( n, z ) = ( m, e πiθ ( n − m ) z ) . This is not minimal because each orbit intersects each { n } × T in a singleton. Sothe resulting twisted C ∗ -algebra is not simple.We present a second example showing that the same pulled-back cocycle as usedin Example 6.1 can yield a simple C ∗ -algebra if connectivity in the underlying 1-graph Λ is slightly more complicated than in Ω . Example . Let Λ be a strongly connected 1-graph with at least one edge andsuppose that Λ is not a simple cycle. Then Λ is cofinal and aperiodic (see, forexample, [23]) and contains a nontrivial cycle λ . A simple example is Λ = B thebouquet of 2 loops, so that C ∗ (Λ) = O .Take θ ∈ R \ Q , and define a 1-cocycle φ on Λ by φ ( λ ) = e πid ( λ ) θ . As inthe preceding section, construct c φ ∈ Z (Λ × T , T ) by c φ (( α, m ) , ( β, n )) = φ ( β ) m .Since θ ∈ R \ Q , the paths µ = λ and ν = r ( λ ) satisfy r ( µ ) = r ( ν ) and s ( µ ) = s ( ν ),and φ ( µ ) φ ( ν ) = e πid ( λ ) θ where d ( λ ) θ is irrational. So Corollary 5.7 shows that C ∗ (Λ × T , c φ ) is simple.Our next example shows that it is possible for the bicharacter ω of Per(Λ)determined by c as in Proposition 3.1 to be degenerate and still have C ∗ (Λ , c )simple. Example . Let Λ be the 1-graph with vertex v and edges { e, f } , i.e. the bouquet B of two loops. Fix θ ∈ R \ Q and define φ : Λ → T by φ ( e ) = 1 and φ ( f ) = e πiθ .As pointed out in [10, Page 917] one can check that the action β = β φ of Z on C ∗ (Λ) as given in part (1) of Theorem 5.1 is outer. Hence by [18, Theorem 3.1] C ∗ (Λ) × β Z is simple (and purely infinite by [19, Lemma 10]). Since Per(Λ × T ) ∼ = Z the bicharacter ω of Proposition 3.1 is degenerate. Note that we could also deducesimplicity of C ∗ (Λ) × β Z from Corollary 5.8.The next example illustrates that the interaction between the action θ , thegroup Per(Λ) and the subgroup Z ω can be fairly complicated. Example . Consider the 4-graph Λ = B × T , where B is the bouquet oftwo loops as in the preceding example, and T denotes N regarded as a 3-graph.Choose irrational numbers θ and ρ . Choose any φ : B → T such that φ ( e ) = 1and φ ( f ) = e πiθ , and let c φ ∈ Z ( B × T , T ) be as in (5.1). Define a bicharacter ω of Z by ω ( p, q ) = e πiρq p . Let c φ,ω be the cocycle c φ,ω (( α, m, p ) , ( β, n, q )) = c φ (( α, m ) , ( β, n )) ω ( p, q ) of (5.2). We have Per(Λ) = { } × Z ⊂ Z . It is routine tocheck using Lemma 3.5 that Z ω = { }× Z ×{ } ⊆ Per(Λ), so is a proper nontrivialsubgroup of Per(Λ). The argument of Corollary 5.7 shows that { ( r ( γ ) , ˜ φ ( γ ) | Z ω ) : γ ∈ ( G B ) x } is dense in B ∞ × b Z ω for all x ∈ B ∞ . So Theorem 5.1(3) shows that C ∗ (Λ , c φ,ω ) is simple. Note that { ( r ( γ ) , ˜ φ ( γ )) : γ ∈ ( G B ) x } may not be dense in B ∞ × Per(Λ) b for any x ; for example if φ ( e ) = φ ( f ) = 1.Finally, we present an example which is not directly related to our simplicitytheorems here, but is of independent interest. Corollary 8.2 of [26] implies that if C ∗ (Λ) is simple, then each C ∗ (Λ , c ) is simple. Moreover, Theorem 5.4 of [25]shows that C ∗ (Λ) and C ∗ (Λ , c ) very frequently (perhaps always) have the same K -theory. From this and the Kirchberg-Phillips theorem [17, 35] we deduce thatif C ∗ (Λ) is purely infinite and simple, then all its twisted C ∗ -algebras for cocyclesthat lift to R -valued cocycles coincide. On the other hand, the twisted C ∗ -algebrasof T , for example, are rotation algebras and so different choices of cocycle yielddifferent twisted C ∗ -algebras; but in this instance the untwisted algebra is notsimple. We were led to ask whether there exists a k -graph whose twisted C ∗ -algebras (including the untwisted one) are all simple but are not all identical. Wepresent a 3-graph with this property. Example . Consider the 3-coloured graph with vertices { v n : n = 1 , , . . . } ,blue edges { e n,i : n ∈ N \ { } , i ∈ Z /n Z } , red edges { f n : n ∈ N \ { }} and greenedges { g n : n ∈ N \ { }} , with r ( e n,i ) = r ( f n ) = s ( f n ) = r ( g n ) = s ( g n ) = v n and s ( e n,i ) = v n +1 for all n, i . The graph can be drawn as follows. (For those withmonochrome printers: the solid edges are blue, the dashed edges are red, and theedges drawn in a dash-dot-dash pattern are green.) v v v . . . v n v n +1 . . . e , e , e , e n, e n,n − ... f f f f n f n +1 g g g g n g n +1 Specify commuting squares by f n e n,i = e n,i +1 f n +1 , g n e n,i = e n,i + n g n +1 and f n g n = g n f n (addition in the subscripts of the e n,i takes place in the cyclic group Z /n Z ). It is easy to check that this is a complete and associative collectionof commuting squares as in [15], and so determines a 3-graph Λ, in which each d ( e n,i ) = (1 , , d ( f n ) = (0 , ,
0) and d ( g n ) = (0 , , µ, ν ∈ Λ satisfy(6.1) r ( µ ) = r ( ν ) , s ( µ ) = s ( ν ) and d ( µ ) ∧ d ( ν ) = 0and if µα Λ ∩ να Λ = ∅ for all α ∈ s ( µ )Λ then µ = ν = r ( µ ). Fix µ, ν satis-fying (6.1). Then µ = f pm and ν = g qm (or vice versa) for some p, q ∈ N . Let n = max { p, q } so that m + n > p, q , and let α = e m, e m +1 , · · · e m + n, . The fac-torisation property implies that µα has the form e m,i m e m +1 ,i m +1 · · · e m + n,p f pm + n +1 and να has the form e m,j m e m +1 ,j m +1 · · · e m + n,q ( m + n ) g qm + n +1 . So µα Λ ∩ να Λ = ∅ forces p = q ( m + n ) in Z / ( m + n ) Z . Since m + n ≥ p, q , this forces p = q = 0 sothat µ = ν = v m as required. So Λ is aperiodic as claimed.It now follows from [26, Corollary 8.2] that C ∗ (Λ , c ) is simple for every c ∈ Z (Λ , T ). Let θ ∈ [0 ,
1) and define c θ ∈ Z (Λ , T ) by c θ ( µ, ν ) = e πiθd ( µ ) d ( ν ) .We show that C ∗ (Λ , c θ ) and C ∗ (Λ , c ρ ) are nonisomorphic whenever θ and ρ arerationally independent . In an earlier version of the paper, we proved only that C ∗ (Λ , θ ) and C ∗ (Λ) are nonisomorphicwhen θ is irrational. We thank the anonymous referee for suggesting that we expand our analysisto encompass the relationship between these algebras for different irrational values of θ . We IMPLICITY OF TWISTED k -GRAPH ALGEBRAS 27 For each n = 1 , , . . . the Cuntz-Krieger relations give s f n s ∗ f n = s ∗ f n s f n = s g n s ∗ g n = s ∗ g n s g n = p v n , so s f n and s g n are unitaries in C ∗ ( { s f n , s g n } ). Thedefinition of c θ gives s g n s f n = e πiθ s f n s g n . This is the defining relation for therotation algebra A θ and so C ∗ ( { s f n , s g n } ) ∼ = A θ . We observe that we can expressthe corner p v C ∗ (Λ , c θ ) p v as the direct limit of the C ∗ -algebras p v C ∗ (Λ n , c θ ) p v ,where Λ n is the locally-convex 3-graph { v , . . . , v n } Λ { v , . . . , v n } . Each of thesesubalgebras is canonically isomorphic to M q n ( A θ ), where q n = Q n − i =1 i (note q = q = 1), so its K -group is isomorphic to Z .We claim that if θ is irrational, then the map on K induced by the inclusionmap p v C ∗ (Λ n , c θ ) p v ֒ → p v C ∗ (Λ n +1 , c θ ) p v , is given by multiplication by n . This is clear by direct computation using theCuntz–Krieger relation for the class [ p v ] of the identity, and therefore for the classof a minimal projection in M q n ( C ⊆ M q n ( A θ ). The other generating K -classis that of the matrix p n ∈ A θ ⊆ M q n ( A θ ) with (1 , M q n +1 ( A θ ) carries p n +1 to θ/q n +1 , while its restriction to the image of M q n ( A θ ) must agree with the traceon M q n ( A θ ) so carries the image of p n to θ/q n ; so [ ι ( p n )] = ( q n +1 /q n )[ p n +1 ] = n [ p n +1 ] .We deduce from the continuity of K -theory as a functor into the categoryof ordered abelian groups (see for example [43, Theorem 6.3.2]) that when θ isirrational, K ( C ∗ (Λ , c θ )) is isomorphic as an ordered abelian group to Q + θ Q with the order inherited from R , and hence isomorphic as a group to Q . Sinceeach c θ is of exponential form, Theorem 5.4 of [25] (see also [14]) shows that K ( C ∗ (Λ , c θ )) ∼ = Q for all θ .Since the tracial-state space of each p v C ∗ (Λ n ) p v is compact and nonemptyand p v C ∗ (Λ n ) p v are nested, compactness shows that p v C ∗ (Λ , c θ ) p v admits atrace τ (or one can directly construct a trace using the approach of [32, Propo-sition 3.8]). The functional τ ∗ on K ( p v C ∗ (Λ , c θ ) p v ) induced by any trace τ has range S τ ∗ | K ( p v C ∗ (Λ n ,c θ ) p v ) . Since each p v C ∗ (Λ n , c θ ) p v ∼ = M q n ( A θ ),uniqueness of the map on K ( A θ ) induced by a trace on A θ (see [9, Lemma2.3]) implies that every trace on p v C ∗ (Λ , c θ ) p v induces the same functional τ ∗ : K ( p v C ∗ (Λ n , c θ ) p v ) → R , and that this τ ∗ has range S n ( Z + θ Z ) /q n = Q + θ Q .It follows that the Morita-equivalence classes (and so in particular the isomor-phism classes) of C ∗ (Λ , c θ ) and C ∗ (Λ , c ρ ) are distinct for rationally independent θ and ρ . In particular if θ is irrational then C ∗ (Λ , c θ ) and C ∗ (Λ) = C ∗ (Λ , c ) arenot isomorphic. References [1] C. Anantharaman-Delaroche,
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E-mail address : [email protected] (D.Pask and A. Sims) School of Mathematics and Applied Statistics, University ofWollongong, NSW 2522, AUSTRALIA
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