The simplicity of the C*-algebras associated to arbitrary labeled spaces
aa r X i v : . [ m a t h . OA ] J a n THE SIMPLICITY OF THE C ∗ -ALGEBRAS ASSOCIATED TOARBITRARY LABELED SPACES EUN JI KANG
Abstract.
In this paper, we consider the simplicity of the C ∗ -algebra associatedto an arbitrary weakly left-resolving labeled space ( E, L , E ), where E is the small-est non-degenerate accommodating set. We classify all gauge-invariant ideals of C ∗ ( E, L , E ) and characterize minimality of ( E, L , E ) in terms of ideal structure of C ∗ ( E, L , E ). Using these results, we prove that C ∗ ( E, L , E ) is simple if and onlyif ( E, L , E ) is strongly cofinal and satisfies Condition (L), and for any A ∈ E \ {∅} and B ∈ E , there is C ∈ E reg such that B \ C ∈ H ( A ), and if and only if ( E, L , E )is minimal and satisfies Condition (L), and if and only if ( E, L , E ) is minimal andsatisfies Condition (K). Introductions
A class of C ∗ -algebras C ∗ ( E ) associated with directed graphs E was introduced in[9, 19] as a generalization of the Cuntz-Krieger algebras and there has been variousgeneralizations of graph C ∗ -algebras. The C ∗ -algebras associated to ultragraphs[20], higher-rank graphs [16], subshifts, labeled spaces [3], Boolean dynamical sys-tems [7] are those generalizations, and generalized Boolean dynamical systems wasintroduced in [6] to unify C ∗ -algebras of labeled spaces and C ∗ -algebras of Booleandynamical systems. Among others, we focus on the C ∗ -algebras associated to ar-bitrary weakly left-resolving normal labeled spaces. Throughout the paper, by alabeled space we always mean a weakly left-resolving normal labeled space.The ideal structure of C ∗ -algebras of set-finite, receiver set-finite labeled spaces( E, L , B ) with E having no sinks is now well understood. It is known in [11, Theo-rem 5.2] that the gauge-invariant ideals of C ∗ ( E, L , B ) are in one-to-one correspon-dence with the hereditary saturated subsets of B . We first generalize this resultsto C ∗ -algebras of arbitrary labeled spaces. We show that there is a one-to-one cor-respondences between gauge-invariant ideals of C ∗ ( E, L , B ) and pairs ( H , S ) where H is a hereditary saturated ideal of B and S is an ideal of { A ∈ B : r ( A, α ) ∈H for all but finitely many α } such that H ∪ B reg ⊆ S (Theorem 3.8). A quotientlabeled space was also introduced in [11, Definition 3.2] to realize the quotient al-gebra C ∗ ( E, L , B ) /I by a gauge-invariant ideal I as a C ∗ -algebras of a quotientlabeled space. But, a quotient labeled space is not a labeled space in general, but aBoolean dynamical system. So, in this paper we realize the quotient of C ∗ ( E, L , B ) Mathematics Subject Classification.
Key words and phrases.
Labeled graph C ∗ -algebras, simple C ∗ -algebras, gauge-invariant ideals,generalized Boolean dynamical systems.This research was supported by Basic Science Research Program through the National ResearchFoundation of Korea funded by the Ministry of Education (NRF-2020R1I1A1A01072970). by I ( H , S ) as a C ∗ -algebra of a relative generalized Boolean dynamical system in-stead of newly defining C ∗ -algebras of relative quotient labeled spaces of arbitrarylabeled spaces. Precisely, we show that the quotient of C ∗ ( E, L , B ) by the ideal I ( H , S ) is isomorphic to the C ∗ -algebra of relative generalized Boolean dynamicalsystem ( B / H , A , θ, [ I r ( α ) ]; [ S ]) (Proposition 3.7). These will be easily done by view-ing labeled graph C ∗ -algebras as C ∗ -algebras of generalized Boolean dynamicalsystems and applying results of [6].The second goal of the paper is to investigate the question of when C ∗ ( E, L , E )is simple, where ( E, L , E ) is an arbitrary weakly left-resolving labeled space with E is the smallest non-degenerate accommodating set. For an arbitrary graph E , werecall that E is said to satisfy Condition ( L ) if every loop has as exit, and is saidto be cofinal if every vertex connects to every infinite path. Then the simplicity of C ∗ ( E ) is characterized as follows. Theorem 1.1. ( [8, Corollary 2.15] ) Let E be a directed graph. Then the followingare equivalent. (1) C ∗ ( E ) is simple. (2) The following properties hold: (a) E is cofinal, (b) E satisfies Condition ( L ) , and (c) for w ∈ E and v ∈ E sing , there is a path α ∈ E ∗ such that s ( α ) = w and r ( α ) = v . (3) E satisfies Condition (L) and E has no proper hereditary saturated subsets. (4) E satisfies Condition (K) and E has no proper hereditary saturated subsets. Many authors paid a great deal of attention to extend this result to the C ∗ -algebrasassociated to set-finite and receiver set-finite labeled spaces ( E, L , E ) with E havingno sinks or sources. In [4, Definition 6.1] Bates and Pask introduced a notion ofcofinality appropriate for labeled spaces. In [10, Definition 3.1], a notion of strongcofinality of labeled spaces was given to modify minor mistake of results in [4]. Thenagain a modified version of strong cofinality of a labeled space was introduced in [15,Definition 2.10]. As an analogue of Condition (L) of usual directed graph, the notionof a disagreeable labeled space was introduced in [4, Definition 5.1]. On the otherhand, the notion of cycle was introduced in [7, Definition 9.5] to define condition( L ) for a labeled space (more generally for Boolean dynamical systems) which canbe regarded as another condition analogues to Condition (L) for usual directedgraphs. It then is known in [14, Proposition 3.7] that if ( E, L , B ) is disagreeable,then ( E, L , B ) satisfies Condition (L). But, the converse is not true, in general ([15,Proposition 3.2]). Based on these concepts, it is eventually known in [15, Theorem3.17] that for a set-finite and receiver set-finite labeled space ( E, L , E ) with E havingno sinks or sources, C ∗ ( E, L , E ) is simple if and only if ( E, L , E ) is strongly cofinal inthe sense of [15, Definition 2.10] and disagreeable. It is also prove in [15, Theorem3.17] that for a labeled space whose Boolean dynamical system satisfies a sort ofdomain condition, C ∗ ( E, L , E ) is simple if and only if ( E, L , E ) satisfies Condition(L) and there are no nonempty hereditary saturated subsets of E .We generalize these results to labeled graph C ∗ -algebras associated to arbitrarylabeled spaces. We first give an example that shows why we need to change the HE SIMPLICITY OF THE C ∗ -ALGEBRAS ASSOCIATED TO ARBITRARY LABELED SPACES3 definition of strong cofinality given in [10, Definition 3.1] to [15, Definition 2.10].We next prove that ( E, L , E ) is strongly cofinal and for any A ∈ E \ {∅} and B ∈ E ,there is C ∈ E reg such that B \ C ∈ H ( A ) if and only if ( E, L , E ) is minimal, inthe sense that {∅} and E are the only hereditary saturated subsets of E . We alsoidentify the minimal condition with the property that an ideal containing one vertexprojection must be the whole C ∗ -algebra (Theorem 4.6). It will be also shown thatif ( E, L , E ) is minimal, then ( E, L , E ) is disagreeable if and only if ( E, L , E ) satisfiesCondition (L) (Lemma 4.16). As a result, we have our main results as follows. It isa generalization of Theorem 1.1. Theorem 1.2. (Theorem 4.17) Let ( E, L , E ) be a labeled space. Then the followingare equivalent. (1) C ∗ ( E, L , E ) is simple. (2) ( E, L , E ) is minimal and satisfies Condition (L). (3) ( E, L , E ) is minimal and satisfies Condition (K). (4) The following properties hold: (a) ( E, L , E ) is strongly cofinal, (b) ( E, L , E ) satisfies Condition (L), and (c) for any A ∈ E \{∅} and B ∈ E , there is C ∈ E reg such that B \ C ∈ H ( A ) . (5) The following properties hold: (a) ( E, L , E ) is strongly cofinal, (b) ( E, L , E ) is disagreeable, and (c) for any A ∈ E \{∅} and B ∈ E , there is C ∈ E reg such that B \ C ∈ H ( A ) . As a corollary, we show for a set-finite labeled space ( E, L , E ) with E having nosinks, that C ∗ ( E, L , E ) is simple if and only if ( E, L , E ) is strongly cofinal and isdisagreeable, if and only if ( E, L , E ) is strongly cofinal and satisfies Condition (L),if and only if ( E, L , E ) is minimal and satisfies Condition (L), and if and only if( E, L , E ) is minimal and satisfies Condition (K). This generalizes [15, Theorem 3.7].This paper is organized as follows. In Section 2 we review basic definitions andterminologies needed for the rest of the paper. In Section 3 we classify the gauge-invariant ideals in the C ∗ -algebras of arbitrary labeled spaces and describe thequotients as C ∗ -algebras of relative generalized Boolean dynamical systems. InSection 4 we examine strong cofinality, minimality and disagreeablity for an arbi-trary labeled space, and prove simplicity results for C ∗ -algebras of arbitrary labeledspaces. 2. Preliminary
Directed graphs. A directed graph E = ( E , E , r, s ) consists of two count-able sets of vertices E and edges E , and the range, source maps r , s : E → E .A path of length n is a sequence λ = λ · · · λ n of edges such that r ( λ i ) = s ( λ i +1 )for 1 ≤ i ≤ n −
1. We write | λ | = n for the length of λ and the vertices in E areregarded as finite paths of length zero. By E n we mean the set of all paths of length n . The maps r, s naturally extend to the set E ≥ := ∪ n ≥ E n of all finite paths,where r ( v ) = s ( v ) = v for v ∈ E . We denote by E ∞ the set of all infinite paths x = λ λ · · · , λ i ∈ E with r ( λ i ) = s ( λ i +1 ) for i ≥
1, and define s ( x ) := s ( λ ). Weals use notation like E ≤ n and E ≥ n which should have their obvious meaning. E. J. KANG
A vertex v ∈ E is called a source if | r − ( v ) | = 0 and v is called a sink if | s − ( v ) | = 0, and v is called an infinite-emitter if | s − ( v ) | = ∞ . We define E sink tobe the set of all sinks in E . We let E reg = { v ∈ E : 0 < | s − ( v ) | < ∞} and let E sing = E \ E reg .A finite path λ = λ · · · λ | λ | ∈ E ≥ with r ( λ ) = s ( λ ) is called a loop , and an exit of a loop λ is a path δ ∈ E ≥ such that | δ | ≤ | λ | , s ( δ ) = s ( λ ) , and δ = λ · · · λ | δ | .A graph E is said to satisfy Condition (L) if every loop has an exit and E is saidto satisfy Condition (K) if every vertex v ∈ E lies on no loops, or if there are twoloops α and β such that s ( α ) = s ( β ) = v and neither α nor β is an initial path ofthe other.2.2. Labeled spaces. A labeled graph ( E, L ) over a countable alphabet A consistsof a directed graph E and a labeling map L : E → A . We assume that the map L isonto. By A ∗ and A ∞ , we denote respectively the sets of all finite words and infinitewords in symbols of A . To each finite path λ = λ · · · λ n ∈ E n , there correspondsa finite labeled path L ( λ ) := L ( λ ) · · · L ( λ n ) ∈ L ( E n ) ⊂ A ∗ , and similarly aninfinite labeled path L ( x ) := L ( λ ) L ( λ ) · · · ∈ L ( E ∞ ) ⊂ A ∞ to each infinite path x = λ λ · · · ∈ E ∞ . We use notation L ∗ ( E ) := L ( E ≥ ), where E ≥ = ∪ n ≥ E n . Wedenote the subpath α i · · · α j of α = α α · · · α | α | ∈ L ( E ≥ ) by α [ i,j ] for 1 ≤ i ≤ j ≤| α | . A subpath of the form α [1 ,j ] is called an initial path of α . The range r ( α ) of alabeled path α ∈ L ∗ ( E ) is a subset of E defined by r ( α ) = { r ( λ ) : λ ∈ E ≥ , L ( λ ) = α } . The relative range of α ∈ L ∗ ( E ) with respect to A ⊂ E is defined to be r ( A, α ) = { r ( λ ) : λ ∈ E ≥ , L ( λ ) = α, s ( λ ) ∈ A } . Let
B ⊆ E be a collection of subsets of E . We say B is closed under relativeranges for ( E, L ) if r ( A, α ) ∈ B for all A ∈ B and α ∈ L ∗ ( E ). We call B an accommodating set for ( E, L ) if it satisfies(i) r ( α ) ∈ B for all α ∈ L ∗ ( E ),(ii) it is closed under relative ranges,(iii) it is closed under finite intersections and unions.If, in addition, B is closed under relative complements, then B is said to be non-degenerate . The triple ( E, L , B ) is called a labeled space when B is accommodatingfor ( E, L ). Moreover, if B is non-degenerate, then ( E, L , B ) is called normal as in[1]. By E , we denote the smallest non-degenerate accommodating set for a labeledgraph ( E, L ).A labeled space ( E, L , B ) is said to be weakly left-resolving if it satisfies r ( A, α ) ∩ r ( B, α ) = r ( A ∩ B, α )for all
A, B ∈ B and α ∈ L ∗ ( E ). A labeled graph ( E, L ) is left-resolving if L : r − ( v ) → A is injective for each v ∈ E . Left-resolving labeled spaces are weaklyleft-resolving. Assumptions.
In this paper, ( E, L , B ) is always weakly left-resolving normal and L : E → A is onto. HE SIMPLICITY OF THE C ∗ -ALGEBRAS ASSOCIATED TO ARBITRARY LABELED SPACES5 For
A, B ⊂ E and n ≥
1, let AE n = { λ ∈ E n : s ( λ ) ∈ A } , E n B = { λ ∈ E n : r ( λ ) ∈ B } . A labeled space ( E, L , B ) is said to be set-finite ( receiver set-finite , respectively) iffor every A ∈ B and n ≥ L ( AE n ) ( L ( E n A ), respectively) is finite. Wealso say that A ∈ B is regular if 0 < |L ( BE ) | < ∞ for any ∅ 6 = B ∈ B with B ⊆ A .If A ∈ B is not regular, then it is called a singular set. We write B reg for the set ofall regular sets. Note that if E has no sinks, then ( E, L , B ) is set-finite if and onlyif B = B reg . A set A ∈ B is called minimal (in B ) if A ∩ B is either A or ∅ for all B ∈ B .Let ( E, L ) be a labeled graph and let Ω ( E ) be the set of all vertices that arenot sources, and let l ≥
1. The relation ∼ l on Ω ( E ), given by v ∼ l w if and onlyof L ( E ≤ l v ) = L ( E ≤ l w ), is an equivalence relation, and the equivalence class [ v ] l of v ∈ Ω ( E ) is called a generalized vertex . If k > l , then [ v ] k ⊂ [ v ] l is obviousand [ v ] l = ∪ mi =1 [ v i ] l +1 for some vertices v , . . . , v m ∈ [ v ] l ([4, Proposition 2.4]). Thegeneralized vertices of labeled graphs play the role of vertices in usual graphs.2.3. C ∗ -algebras of labeled spaces. We review the definition of the C ∗ -algebrasassociated to labeled spaces from [1]. Definition 2.1. ([1, Definition 2.1]) Let ( E, L , B ) be a labeled space. A represen-tation of ( E, L , B ) is a family of projections { p A : A ∈ B} and partial isometries { s α : α ∈ A} such that for A, B ∈ B and α, β ∈ A ,(i) p ∅ = 0, p A ∩ B = p A p B , and p A ∪ B = p A + p B − p A ∩ B ,(ii) p A s α = s α p r ( A,α ) ,(iii) s ∗ α s α = p r ( α ) and s ∗ α s β = 0 unless α = β ,(iv) p A = P α ∈L ( AE ) s α p r ( A,α ) s ∗ α for A ∈ B reg . Remark . Let ( E, L , B ) be a labeled space.(1) For any A ∈ B , the condition |L ( AE ) | < ∞ and A ∩ B = ∅ for all B ∈ B satisfying B ⊆ E sink is equivalent to A ∈ B reg . Thus, the condition (iv) in Definition 2.1 isequivalent to (iv) in [1, Definition 2.1].(2) If E has no sinks, the condition (iv) in Definition 2.1 is equivalent to p A = X α ∈L ( AE ) s α p r ( A,α ) s ∗ α for A ∈ B with |L ( AE ) | < ∞ . Given a labeled space ( E, L , B ), it is known in [1, Theorem 3.8] that there ex-ists a C ∗ -algebra C ∗ ( E, L , B ) generated by a universal representation { s α , p A } of( E, L , B ). We call C ∗ ( E, L , B ) the labeled graph C ∗ -algebra of a labeled space( E, L , B ) and simply write C ∗ ( E, L , B ) = C ∗ ( s α , p A ) to indicate the generators.Note that s α = 0 and p A = 0 for α ∈ A and A ∈ B , A = ∅ . Remark . Let ( E, L , B ) be a labeled space. For notational convenience, we use asymbol ǫ such that r ( ǫ ) = E , r ( A, ǫ ) = A for all A ⊂ E , and α = ǫα = αǫ for all α ∈ L ( E ≥ ), and write L ( E ) := L ( E ≥ ) ∪ { ǫ } . For ε ∈ L ( E ), let s ε denote theunit of the multiplier algebra of C ∗ ( E, L , B ). E. J. KANG (1) We have the following equality( s α p A s ∗ β )( s γ p B s ∗ δ ) = s αγ ′ p r ( A,γ ′ ) ∩ B s ∗ δ , if γ = βγ ′ s α p A ∩ r ( B,β ′ ) s ∗ δβ ′ , if β = γβ ′ s α p A ∩ B s ∗ δ , if β = γ , otherwise,for α, β, γ, δ ∈ L ( E ) and A, B ∈ B (see [3, Lemma 4.4]). Since s α p A s ∗ β = 0if and only if A ∩ r ( α ) ∩ r ( β ) = ∅ , it follows that C ∗ ( E, L , B ) = span { s α p A s ∗ β : α, β ∈ L ( E ) and A ⊆ r ( α ) ∩ r ( β ) } . (1)(2) By universal property of C ∗ ( E, L , B ) = C ∗ ( s α , p A ), there is a strongly con-tinuous action γ : T → Aut ( C ∗ ( E, L , B )), called the gauge action , suchthat γ z ( s α ) = zs α and γ z ( p A ) = p A for α ∈ A and A ∈ B .The notion of cycle was introduced ([7, Definition 9.5]) to define condition ( L )for a labeled space ( E, L , B ) (more generally for Boolean dynamical systems) whichis an analogue to Condition (L) for usual directed graphs. Definition 2.4. ([7, Definition 9.5]) Let ( E, L , B ) be a labeled space.(1) A pair ( α, A ) with α ∈ L ∗ ( E ) and ∅ 6 = A ∈ B is a cycle if B = r ( B, α ) forevery B ∈ B with B ⊆ A .(2) A cycle ( α, A ) has an exit if for there is a t ≤ | α | and a B ∈ B such that ∅ 6 = B ⊆ r ( A, α [1 ,t ] ) and L ( BE ) = { α t +1 } (where α | α | +1 := α ).(3) A cycle ( α, A ) has no exits if for all t ≤ | α | and all ∅ 6 = B ⊆ r ( A, α [1 ,t ] ), wehave B ∈ B reg with L ( BE ) = { α t +1 } (where α | α | +1 := α ).(4) A labeled space ( E, L , B ) satisfies Condition ( L ) if every cycle has an exit. Theorem 2.5. (The Cuntz-Krieger Uniqueness theorem [3, Theorem 5.5], [7, The-orem 9.9])
Let { t a , q A } be a representation of a labeled space ( E, L , B ) such that q A = 0 for all nonempty A ∈ B . If ( E, L , B ) satisfies condition ( L ) , then the canon-ical homomorphism φ : C ∗ ( E, L , B ) = C ∗ ( s a , p A ) → C ∗ ( t a , q A ) such that φ ( s a ) = t a and φ ( p A ) = q A is an isomorphism. Generalized Boolean dynamical systems and their C ∗ -algebras. Fordetails of the following, we refer the reader to [5, 6, 7].Let B be a Boolean algebra. A non-empty subset I of B is called an ideal [7,Definition 2.4] if(i) if A, B ∈ I , then A ∪ B ∈ I ,(ii) if A ∈ I and B ∈ B , then A ∩ B ∈ I .An ideal I of a Boolean algebra B is a Boolean subalgebra. For A ∈ B , the idealgenerated by A is defined by I A := { B ∈ B : B ⊆ A } . A Boolean dynamical system is a triple ( B , L , θ ) where B is a Boolean algebra, L is a set, and { θ α } α ∈L is a set of actions on B such that for α = α · · · α n ∈ L ∗ \ {∅} ,the action θ α : B → B is defined as θ α := θ α n ◦ · · · ◦ θ α . We also define θ ∅ := Id. HE SIMPLICITY OF THE C ∗ -ALGEBRAS ASSOCIATED TO ARBITRARY LABELED SPACES7 For B ∈ B , we define∆ ( B , L ,θ ) B := { α ∈ L : θ α ( B ) = ∅} and λ ( B , L ,θ ) B := | ∆ ( B , L ,θ ) B | . We will often just write ∆ B and λ B instead of ∆ ( B , L ,θ ) B and λ ( B , L ,θ ) B . We say that A ∈ B is regular ([7, Definition 3.5]) if for any ∅ 6 = B ∈ I A , we have 0 < λ B < ∞ . If A ∈ B is not regular, then it is called a singular set. We write B ( B , L ,θ ) reg or just B reg for the set of all regular sets. Notice that ∅ ∈ B reg .Let R α := R ( B , L ,θ ) α = { A ∈ B : A ⊆ θ α ( B ) for some B ∈ B} for each α ∈ L .A generalized Boolean dynamical system is a quadruple ( B , L , θ, I α ) where ( B , L , θ )is a Boolean dynamical system and {I α : α ∈ L} is a family of ideals in B suchthat R α ⊆ I α for each α ∈ L . A relative generalized Boolean dynamical system is apentamerous ( B , L , θ, I α ; J ) where ( B , L , θ, I α ) is a generalized Boolean dynamicalsystem and J is an ideal of B reg ([6, Definition 3.2]). Definition 2.6. ([6, Definition 3.2]) Let ( B , L , θ, I α ; J ) be a relative generalizedBoolean dynamical system. A ( B , L , θ, I α ; J ) -representation is a family of projec-tions { P A : A ∈ B} and a family of partial isometries { S α,B : α ∈ L , B ∈ I α } suchthat for A, A ′ ∈ B , α, α ′ ∈ L , B ∈ I α and B ′ ∈ I α ′ ,(i) P ∅ = 0, P A ∩ A ′ = P A P A ′ , and P A ∪ A ′ = P A + P A ′ − P A ∩ A ′ ;(ii) P A S α,B = S α,B P θ α ( A ) ;(iii) S ∗ α,B S α ′ ,B ′ = δ α,α ′ P B ∩ B ′ ;(iv) P A = P α ∈ ∆ A S α,θ α ( A ) S ∗ α,θ α ( A ) for all A ∈ J .Given a ( B , L , θ, I α ; J )-representation { P A , S α,B } in a C ∗ -algebra A , we denoteby C ∗ ( P A , S α,B ) the C ∗ -subalgebra of A generated by { P A , S α,B } . It is shown in [6]that there exists a universal ( B , L , θ, I α ; J )-representation { p A , s α,B : A ∈ B , α ∈L and B ∈ I α } in a C ∗ -algebra. We write C ∗ ( B , L , θ, I α ; J ) for C ∗ ( p A , s α,B ) andcall it the C ∗ -algebra of ( B , L , θ, I α ; J ).By a Cuntz–Krieger representation of ( B , L , θ, I α ) we mean a ( B , L , θ, I α ; B reg )-representation. We write C ∗ ( B , L , θ, I α ) for C ∗ ( B , L , θ, I α ; B reg ) and call it the C ∗ -algebra of ( B , L , θ, I α ). When ( B , L , θ ) is a Boolean dynamical system, then wewrite C ∗ ( B , L , θ ) for C ∗ ( B , L , θ, R α ) and call it the C ∗ -algebra of ( B , L , θ ).2.4.1. Viewing labeled graph C ∗ -algebras as C ∗ -algebras of generalized Boolean dy-namical systems. We view labeled graph C ∗ -algebras as C ∗ -algebras of generalizedBoolean dynamical systems. Let ( E, L , B ) be a labeled space where L : E → A isonto and put C ∗ ( E, L , B ) = C ∗ ( p A , s α ). Then B is a Boolean algebra and for each α ∈ A , the map θ α : B → B defined by θ α ( A ) := r ( A, α ) is an action on B ([7,Example 11.1]). Put R α = { A ∈ B : A ⊆ r ( B, α ) for some B ∈ B} and let I r ( α ) = { A ∈ B : A ⊆ r ( α ) } . It is clear that R α ⊆ I r ( α ) for each α ∈ L . Then ( B , A , θ, I r ( α ) ) is a generalizedBoolean dynamical system. We call it a generalized Boolean dynamical systemassociated to ( E, L , B ). It then is straightforward to check that { p A , s α p B : A ∈ B , α ∈ A and B ∈ I r ( α ) } E. J. KANG is a Cuntz–Krieger representation of ( B , A , θ, I r ( α ) ). Then the universal property of C ∗ ( B , A , θ, I r ( α ) ) gives a ∗ -homomorphism φ : C ∗ ( B , A , θ, I r ( α ) ) → C ∗ ( E, L , B )defined by φ ( p A ) = p A and φ ( s α,B ) = s α p B for all A ∈ B , α ∈ A and B ∈ I r ( α ) . Since s α = s α p r ( α ) , the family { p A , s α p B : A ∈ B , α ∈ A and B ∈ I r ( α ) } generates C ∗ ( E, L , B ), and hence, the map φ is onto.Applying the gauge-invariant uniqueness theorem [6, Corollary 6.2], we concludethat φ is an isomorphism.We summarize this facts in the following. Proposition 2.7. ( [6, Example 4.2] ) Let ( E, L , B ) be a labeled space, where L : E → A is onto. Then ( B , A , θ, I r ( α ) ) is a generalized Boolean dynamical systemand C ∗ ( E, L , B ) is isomorphic to C ∗ ( B , A , θ, I r ( α ) ) . Gauge-invariant ideals of C ∗ ( E, L , B )In this section, we give a complete list of the gauge-invariant ideals of C ∗ -algebrasof arbitrary labeled spaces ( E, L , B ) and describe the quotients as C ∗ -algebras ofrelative generalized Boolean dynamical systems. Most of results can be obtainedby the same arguments used in [6]. So, we omit its proof.We recall from [11, Definition 3.4] that a subset H of B is said to be hereditary if(1) if A ∈ H , then r ( A, α ) ∈ H for all α ∈ L ∗ ( E ),(2) if A, B ∈ H , then A ∪ B ∈ H ,(3) if A ∈ H and B ∈ B with B ⊂ A , then B ∈ H .A hereditary set H is said to be saturated if A ∈ H whenever A ∈ B reg satisfies r ( A, α ) ∈ H for all α ∈ L ∗ ( E ) ([6, Section 2.3]).The following lemma shows how to find a hereditary saturated subset. Lemma 3.1.
For a nonempty set A ∈ B , let H ( A ) := { B ∈ B : B ⊆ ∪ ni =1 r ( A, α i ) for some α i ∈ L ( E ) } . (2) is the smallest hereditary set that contains A , and S ( H ( A )) := { B ∈ B : there is an n ≥ such that r ( B, β ) ∈ H ( A ) for all β ∈ L n ( E ) , and r ( B, γ ) ∈ H ( A ) ⊕ B reg for all γ ∈ L ( E ) with | γ | < n } is the smallest saturated hereditary set that contains A , where H ( A ) ⊕ B reg := { C ∪ D : C ∈ H ( A ) and D ∈ B reg } . Proof.
It is straightforward to check that H ( A ) is hereditary , and it is easy to seethat if H is a hereditary set and A ∈ H , then H ( A ) ⊆ H .It is rather obvious that S ( H ( A )) is hereditary. To show that it is saturated, let B ∈ B reg satisfy r ( B, α ) ∈ S ( H ( A )) for all α ∈ L ∗ ( E ). Put L ( BE ) = { α , · · · , α n } (this is a finite set since B ∈ B reg ). Then r ( B, α i ) ∈ S ( H ( A )) for each i , andthus there is an n i ≥ r ( r ( B, α i ) , β ) ∈ S ( H ( A )) for all β ∈ L n i ( E )and r ( r ( B, α i ) , γ ) ∈ H ( A ) ⊕ B reg for all γ ∈ L ( E ) with | γ | < n i . Take n := HE SIMPLICITY OF THE C ∗ -ALGEBRAS ASSOCIATED TO ARBITRARY LABELED SPACES9 max ≤ i ≤ n { n i } . Then r ( B, β ) ∈ H ( A ) for all β ∈ L n +1 ( E ) and r ( B, γ ) ∈ H ( A ) ⊕B reg for all γ ∈ L ( E ) with | γ | < n + 1. Thus, B ∈ S ( H ( A )). It is also straightforwardto check that if S is a saturated hereditary set and A ∈ S , then S ( H ( A )) ⊆ S . (cid:3) Example 3.2.
For the labeled graph · · · ... • • • • • • / / / / / / % % % % e e b n ) n v w w v v v aa ,where v emits infinitely many labeled edges ( b n ) n ≥ , consider the labeled space( E, L , E ). It is clear that H ( r ( a )) = {∅ , { w } , { w } , { w , w }} . One sees that r ( { v } , β ) ∈ H ( r ( a )) for | β | ≥
1, but { v } / ∈ H ( r ( a )) ⊕ E reg . So, { v } / ∈ S ( H ( r ( a ))).It is also easy to see that { v i } / ∈ S ( H ( r ( a ))) for each i >
1. In fact, S ( H ( r ( a ))) = H ( r ( a )).3.1. A quotient Boolean dynamical system ( B / H , A , θ ) associated to ( E, L , B ) . Let ( E, L , B ) be a labeled space, where L : E → A is assumed to be onto. If H isa hereditary subset of B , then the relation A ∼ B ⇐⇒ A ∪ W = B ∪ W for some W ∈ H (3)defines an equivalent relation on B ([11, Proposition 3.6]). We denote the equivalentclass of A ∈ B with respect to ∼ by [ A ] (or [ A ] H if we need to specify the hereditaryset H ) and the set of all equivalent classes of B by B / H . It is easy to check that B / H is a Boolean algebra with operations defined by[ A ] ∩ [ B ] = [ A ∩ B ] , [ A ] ∪ [ B ] = [ A ∪ B ] and [ A ] \ [ B ] = [ A \ B ] . The partial order ⊆ on B / H is characterized by[ A ] ⊆ [ B ] ⇐⇒ A ⊆ B ∪ W for some W ∈ H⇐⇒ [ A ] ∩ [ B ] = [ A ] . If ,in addition, we define θ α : B / H → B / H by θ α ([ A ]) = [ r ( A, α )] for all [ A ] ∈B / H and α ∈ A , then ( B / H , A , θ ) becomes a Boolean dynamical system (see [11,Proposition 3.6]). We call it a quotient Boolean dynamical system associated to ( E, L , B ). Remark . Given a labeled space ( E, L , B ) and hereditary set H of B , there canbe a [ A ] ∈ B / H such that [ A ] = [ ∅ ], but [ r ( A, α )] = [ ∅ ] for all α ∈ A . For example,for the labeled space ( E, L , E ) in Example 3.2, we have H := H ( r ( a )) is a hereditarysaturated subset of E . It is easy to see that [ { v } ] = [ ∅ ], but [ r ( { v } , α )] = [ ∅ ] forall α ∈ A .A filter [7, Definition 2.6] ξ in a Boolean algebra B is a non-empty subset ξ ⊆ B such that F0 ∅ / ∈ ξ , F1 if A ∈ ξ and A ⊆ B , then B ∈ ξ , F2 if A, B ∈ ξ , then A ∩ B ∈ ξ . If in addition ξ satisfies F3 if A ∈ ξ and B, B ′ ∈ B with A = B ∪ B ′ , then either B ∈ ξ or B ′ ∈ ξ ,then it is called an ultrafilter [7, Definition 2.6] of B . A filter is an ultrafilter if andonly if it is a maximal element in the set of filters with respect to inclusion. Wewrite b B for the set of all ultrafilters of B . Definition 3.4. ([5, Definition 3.1 and Definition 5.1]) Let ( E, L , B ) be a labeledspace and α ∈ L ∗ ( E ) \ {∅} and η ∈ b B .(1) A pair ( α, η ) is an ultrafilter cycle if r ( A, α ) ∈ η for all A ∈ η .(2) ( E, L , B ) satisfies Condition (K) if there is no pair (( α, η ) , A ) where ( α, η )is an ultrafilter cycle and A ∈ η such that if β ∈ L ∗ \ {∅} , B ⊆ A , and r ( B, β ) ∈ η , then B ∈ η and β = α k for some k ∈ N . Lemma 3.5. ( [5, Theorem 6.3] ) Let ( E, L , B ) be a labeled space. Then ( E, L , B ) sat-isfies Condition (K) if and only if the quotient Boolean dynamical system ( B / H , A , θ ) associated to ( E, L , B ) satisfies Condition (L) for every hereditary saturated subset H of B .Proof. It follows by [5, Theorem 6.3]. (cid:3)
Gauge-invariant ideals of C ∗ ( E, L , B ) . Given a hereditary saturated subset H of B , we define an ideal B H := { A ∈ B : [ A ] ∈ ( B / H ) reg } of B . Choose an ideal S of B H (and hence an ideal of B ) such that H ∪ B reg ⊆ S .Let I ( H , S ) denote the ideal of C ∗ ( E, L , B ) := C ∗ ( p A , s α ) generated by the family ofprojections (cid:26) p A − X α ∈ ∆ [ A ] s α p r ( A,α ) s ∗ α : A ∈ S (cid:27) , where ∆ [ A ] := { α ∈ A : [ r ( A, α )] = [ ∅ ] } . Then the ideal I ( H , S ) is gauge-invariant([6, Lemma 7.1]) and I ( H , S ) = span { s α ( p A − p A, H ) s ∗ β : A ∈ S and α, β ∈ L ∗ ( E ) } , (4)where we put p A, H := P α ∈ ∆ [ A ] s α p r ( A,α ) s ∗ α .We shall prove in Theorem 3.7 that every gauge-invariant ideals of C ∗ ( E, L , B )is of the form I ( H , S ) for a hereditary saturated set H and an ideal S of B H with H ⊆ S and B reg ⊆ S . We first observe the following. Lemma 3.6. ( [6, Lemma 7.2] ) Let I be a nonzero ideal in C ∗ ( E, L , B ) . (1) The set H I := { A ∈ B : p A ∈ I } is a hereditary saturated subset of B . (2) The set S I := (cid:26) A ∈ B H I : p A − X α ∈ ∆ [ A ] s α p r ( A,α ) s ∗ α ∈ I (cid:27) is an ideal of B H I (and hence an ideal of B ) with H I ⊆ S I and B reg ⊆ S I .Proof. It follows by Proposition 2.7 and [6, Lemma 7.2]. (cid:3)
HE SIMPLICITY OF THE C ∗ -ALGEBRAS ASSOCIATED TO ARBITRARY LABELED SPACES11 A quotient labeled space was introduced in [11] to study the ideal structure of C ∗ -algebras of a set-finite and receiver set-finite labeled space ( E, L , B ) with E having no sinks. Given a hereditary saturated subset H of B , let I H be the ideal of C ∗ ( E, L , B ) generated by the projections { p A : A ∈ H} . Then the quotient algebra C ∗ ( E, L , B ) /I H by the gauge-invariant ideal I H as a C ∗ -algebras C ∗ ( E, L , B / H ) ofa quotient labeled space ( E, L , B / H ) ([11, Theorem 5.2]). The quotient labeledspace ( E, L , B / H ) is nothing but a Boolean dynamical system ( B / H , A , θ ). So,to generalize this result to C ∗ -algebras of arbitrary labeled spaces, we use relativegeneralized Boolean dynamical systems rather than we newly define relative quotientlabeled spaces of arbitrary labeled spaces. Proposition 3.7. ( [6, Proposition 7.3] ) Let ( E, L , B ) be a labeled space. Supposethat I is an ideal of C ∗ ( E, L , B ) . There is then a surjective ∗ -homomorphism φ I : C ∗ ( B / H I , A , θ, [ I r ( α ) ]; [ S I ]) → C ∗ ( E, L , B ) /I such that φ I ( p [ A ] ) = p A + I and φ I ( s α, [ B ] ) = s α p B + I, where [ I r ( α ) ] := { [ A ] ∈ B / H I : A ∈ I r ( α ) } and [ S I ] := { [ A ] ∈ B / H I : A ∈ S I } .Moreover, the following are equivalent. (1) I is gauge-invariant. (2) The map φ I is an isomorphism. (3) I = I ( H I , S I ) .Proof. It follows by Proposition 2.7 and [6, Proposition 7.3]. (cid:3)
We further say that the map ( H , S ) I ( H , S ) is a lattice isomorphism. The setof pairs ( H , S ), where H is a hereditary saturated subset of B and S is an idealof B H with H ∪ B reg ⊆ S is a lattice with respect to the order relation defined by( H , S ) ≤ ( H , S ) ⇐⇒ ( H ⊆ H and S ⊆ S ). The set of gauge-invariantideals of C ∗ ( E, L , B ) is a lattice with the order given by set inclusion. Theorem 3.8. ( [6, Theorem 7.4] ) Let ( E, L , B ) be a labeled space. Then the map ( H , S ) I ( H , S ) is a lattice isomorphism between the lattice of all pairs ( H , S ) , where H is a hereditary saturated subset of B and S is an ideal of B H with H ∪ B reg ⊆ S ,and the lattice of all gauge-invariant ideals of C ∗ ( E, L , B ) .Proof. It follows by Proposition 2.7 and [6, Theorem 7.4]. (cid:3)
Example 3.9.
Let ( E, L ) be the following labeled graph • • ; ; c c $ $ : : > > v wa a ...( c n ) n ≥ ,where v emits infinitely many labeled edges ( c n ) n ≥ . Then E = {∅ , { v } , { w } , { v, w }} , E reg = {∅ , { w }} , and H ( { w } ) = {∅ , { w }} . Since r ( { v } , a n ) / ∈ H ( { w } ) for all n ≥ { v } / ∈ S ( H ( { w } )). Thus, S ( H ( { w } )) = {∅ , { w }} . Put H := S ( H ( { w } )) = {∅ , { w }} . Then E / H = { [ ∅ ] , [ { v } ] } = ( E / H ) reg and E H = { A ∈ E : [ A ] ∈ ( E / H ) reg } = {∅ , { v } , { w } , { v, w }} = E . (5)Let I ( H , E H ) denote the only one gauge-invariant ideal of C ∗ ( E, L , E ) := C ∗ ( p A , s α )generated by the family of projections (cid:26) p A − X α ∈ ∆ [ A ] s α p r ( A,α ) s ∗ α : A ∈ E H (cid:27) . Note that [ E H ] = { [ ∅ ] , [ { v } ] } = ( E / H ) reg . Then, by Proposition 3.7, we have thatthat C ∗ ( E, L , E ) /I ( H , E H ) ∼ = C ∗ ( E / H , A , θ, [ I r ( α ) ]) . Since C ∗ ( E / H , A , θ, [ I r ( α ) ]) is generated by { p [ { v } ] , s a, [ { v } ] } satisfying s ∗ a, [ { v } ] s a, [ { v } ] = p [ { v } ] = s a, [ { v } ] s ∗ a, [ { v } ] , we conclude that C ∗ ( E, L , E ) /I ( H , E H ) is isomorphic to C ( T ).4. Simple labeled graph C ∗ -algebras In this section, we introduce strong cofinality, minimality and disagreeablity foran arbitrary labeled space ( E, L , E ), and state our main result, Theorem 4.17.4.1. Strong cofinality and minimality.
We first recall the notion of strong co-finality given [10, Definition 3.1]. A set-finite and receiver set-finite labeled space( E, L , E ) with E having no sinks or sources was called strongly cofinal if for all x ∈ L ( E ∞ ), [ v ] l ∈ E and w ∈ s ( x ), there are N ≥ λ , · · · , λ m ∈ L ∗ ( E ) such that r ([ w ] , x [1 ,N ] ) ⊂ ∪ mi =1 r ([ v ] l , λ i ) . It is shown in [10, Theorem 3.16] that if ( E, L , E ) is strongly cofinal in the abovesence and is disagreeable, then C ∗ ( E, L , E ) is simple. But, the following exampleshows that there is a strongly cofinal labeled space in the sense of [10, Definition3.1] and is disagreeable, but its associated C ∗ -algebra is not simple. Example 4.1.
Consider the following labeled graph ( E, L ): · · ·· · · • • • • • • • • ••••••••••• / / / / / / / / / / / / / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) o o o o o o o o o o o o o o o o v E, L , E ) is set-finite, receiver set-finite and left-resolving. For every vertex v ∈ E , there is l > α ∈ L l ( E ) such that r ( α ) = [ v ] l = { v } ∈ E . Thus, onesees that E = {∪ ni =1 r ( α i ) : α i ∈ L ∗ ( E ) and n ∈ N } . It then easy to see that ( E, L , E )is strongly cofinal in the sense of [10, Definition 3.1].On the other hand, consider the smallest hereditary saturated set H containing r (1). One can see that r (0) / ∈ H . So, H is a proper hereditary saturated subset HE SIMPLICITY OF THE C ∗ -ALGEBRAS ASSOCIATED TO ARBITRARY LABELED SPACES13 of E . Thus, by [11, Theorem 5.2], there is a nontrivial ideal of C ∗ ( E, L , E ), sothat C ∗ ( E, L , E ) is not simple. In fact, C ∗ ( E, L , E ) contains many ideals that isnot gauge-invariant. To see this, note that for each α ∈ L ∗ ( E ), if α = 1 α ′ or α = α ′ α ′ , then r ( α ) ∈ H . Observe also that r (0) ∼ r (0 n ) for each n ∈ N since r (0) ∪ (cid:0) r (10) ∪ · · · ∪ r (10 n − ) (cid:1) = r (0 n ) ∪ (cid:0) r (10) ∪ · · · ∪ r (10 n − ) (cid:1) for each n ∈ N . So, we have E / H = { [ r (0)] , [ ∅ ] } . One then can see that C ∗ ( E, L , E ) /I H ∼ = C ∗ ( E / H , A , θ ) ∼ = C ( T ), where A = { , } Thus, C ∗ ( E, L , E ) contains many idealsthat is not gauge-invariant.As we see in the above example, we need to modify the definition of strongcofinality given in [10, Definition 3.1]. To do that, let L ( E ∞ ) := { x ∈ A N | x [1 ,n ] ∈ L ( E n ) for all n ≥ } be the set of all right infinite sequences x such that all of its subpaths occurs as alabeled path in ( E, L ). Clearly L ( E ∞ ) ⊂ L ( E ∞ ), and in fact, L ( E ∞ ) is the closureof L ( E ∞ ) in the totally disconnected perfect space A N which has the topology witha countable basis of open-closed cylinder sets Z ( α ) := { x ∈ A N : x [1 ,n ] = α } , α ∈ A n , n ≥ L ( E ∞ ) ( L ( E ∞ ). For example,in Example 4.1, we see that 1 ∞ , ∞ / ∈ L ( E ∞ ), but 1 ∞ , ∞ ∈ L ( E ∞ ).Adopting [15, Definition 2.10], we introduce strong cofinality of an arbitrarylabeled space. Definition 4.2.
We say that a labeled space ( E, L , E ) is strongly cofinal if for eachnonempty set A ∈ E and x ∈ L ( E ∞ ), there exist N ∈ N and a finite number ofpaths λ , . . . , λ m ∈ L ∗ ( E ) such that r ( x [1 ,N ] ) ⊆ ∪ mi =1 r ( A, λ i ) . Throughout the paper if we mention strong cofinality, we mean Definition 4.2.
Example 4.3.
We continue Example 4.1. For x := 0 ∞ ∈ L ( E ∞ ) \ L ( E ∞ ) and { v } ∈ E , we see that r ( x [1 ,n ] ) = r (0 n ) * ∪ mi =1 r ( { v } , λ i ) for any paths λ , . . . , λ m ∈L ∗ ( E ) and any n ∈ N . Thus, ( E, L , E ) is not strongly cofinal. Remark . Let ( E, L , E ) be a labeled space.(1) ( E, L , E ) is strongly cofinal if and only if for all ∅ 6 = A ∈ E , B ∈ E and x ∈ L ( E ∞ ), there exists an N ≥ r ( B, x [1 ,N ] ) ∈ H ( A ) . (2) If E has no sources and ( E, L , E ) is set-finite and receiver set-finite, then( E, L , E ) is strongly cofinal if and only if for each [ v ] l ∈ E , x ∈ L ( E ∞ ) and w ∈ s ( x ), there exist an N ≥ λ , . . . , λ m ∈L ∗ ( E ) such that r ([ w ] , x [1 ,N ] ) ⊆ ∪ mi =1 r ([ v ] l , λ i ) . Proof. (1)( ⇒ ) It is clear.( ⇐ ) Let A ∈ E and x = x x · · · ∈ L ( E ∞ ). Then for r ( x ) ∈ E and x x · · · ∈L ( E ∞ ), there exist N ≥ λ , . . . , λ m ∈ L ∗ ( E ) suchthat r ( x · · · x N ) = r ( r ( x ) , x x · · · x N ) ⊆ ∪ mi =1 r ( A, λ i ) . (2) ( ⇒ ) It is clear since r ([ w ] , x [1 ,N ] ) ⊆ r ( x [1 ,N ] ) for any w ∈ s ( x ). ( ⇐ ) Let A ∈ E and x = x x · · · ∈ L ( E ∞ ). We may assume that A = [ v ] l for some v ∈ E . Since ( E, L , E ) is receiver set-finite, r ( x ) = ∪ ki =1 [ w i ] . Thus if[ w i ] ∩ s ( x ) = ∅ for i ∈ { , · · · , k } , there are N i ≥ λ i , · · · , λ im i ∈ L ∗ ( E ) suchthat r ([ w i ] , x · · · x N i ) ⊂ ∪ m i j =1 r ( A, λ ij ) . Choose N = max { N i : [ w i ] ∩ s ( x ) = ∅} . Then for all i with [ w i ] ∩ s ( x ) = ∅ , r ([ w i ] , x · · · x N ) ⊂ ∪ m i j =1 r ( A, β ij ) , where β ij = λ ij ( β ij ) ′ . Thus r ( x · · · x N ) = r ( r ( x ) , x · · · x N ) ⊂ ∪ ki =1 r ([ w i ] , x · · · x N ) ⊂ ∪ ki =1 ∪ m i j =1 r ( A, β ij ) . (cid:3) Definition 4.5.
We say that ( E, L , E ) is minimal if {∅} and E are the only hered-itary saturated subsets of E .We now describe a number of equivalent conditions of minimality. The idealstructure of C ∗ ( E, L , E ) will be used to prove theorem. Theorem 4.6.
Let ( E, L , E ) be a labeled space. The following are equivalent. (1) ( E, L , E ) is minimal. (2) S ( H ( A )) = E for every A ∈ B \ {∅} . (3) ( E, L , E ) is strongly cofinal and for any A ∈ E \ {∅} and B ∈ E , there exists C ∈ E reg such that B \ C ∈ H ( A ) . (4) The only ideal of C ∗ ( E, L , E ) containing p A for some A ∈ E\{∅} is C ∗ ( E, L , E ) . (5) The only non-zero ideal of C ∗ ( E, L , E ) which is gauge-invariant is C ∗ ( E, L , E ) .Proof. (1) = ⇒ (2): It is obvious.(2) = ⇒ (3): Choose ∅ 6 = A ∈ E , B ∈ E and x = x x · · · ∈ L ( E ∞ ). Then B ∈S ( H ( A )). It follows from the description of S ( H ( A )) givne in Lemma 3.1 that thereis an n ≥ r ( B, β ) ∈ H ( A ) for all β ∈ L n ( E ), and r ( B, γ ) ∈ H ( A ) ⊕ E reg for all γ ∈ L ( E ) with | γ | < n . If n = 0 and we let C = ∅ , then C ∈ E reg , B \ C = B ∈ H ( A ). If n >
0, then r ( B, x [1 ,n ] ) ∈ H ( A ) and there is a C ∈ E reg suchthat B \ C ∈ H ( A ). Thus, (3) holds (see Remark 4.4(1)).(3) = ⇒ (1): Choose a nonempty hereditary saturated subset H of E . Take ∅ 6 = A ∈ H . Suppose that B / ∈ H for a set B ∈ E . Then there is a C ∈ E reg suchthat B \ C ∈ H ( A )( ⊂ H ). Since B / ∈ H , we have C / ∈ H . So, there is x ∈ L ( E )such that r ( C , x ) / ∈ H since H is saturated. We can then choose C ∈ E reg suchthat r ( C , x ) \ C ∈ H ( A ). Let C := C ∩ r ( C , x ). Since r ( C , x ) / ∈ H , it followsthat C / ∈ H . Since C ∈ E reg , we deduce that there is an x ∈ L ( E ) such that r ( C , x ) / ∈ H . Continuing this process, we can construct a sequence ( C n , x n ) n ∈ N such that C n ∈ E reg \H , x n ∈ L ( E ), C n +1 ⊆ r ( C n , x n ) , and r ( C n , x n ) \ C n +1 ∈ H ( A )for each n ∈ N . Let x := x x · · · ∈ L ( E ∞ ). Then C n +1 ⊆ r ( C , x [1 ,n ] ) for each n ≥
1. Since C n +1 / ∈ H for all n ∈ N , r ( C , x [1 ,n ] ) / ∈ H for all n ∈ N . Thus, r ( x [1 ,n ] ) / ∈ H ( A ) for each n ≥
1, which contradicts to strong cofinality of ( E, L , E ).Thus, we conclude that H = E . HE SIMPLICITY OF THE C ∗ -ALGEBRAS ASSOCIATED TO ARBITRARY LABELED SPACES15 (1) = ⇒ (4): Let I be an ideal of C ∗ ( E, L , E ) such that p A ∈ I for some ∅ 6 = A ∈ E .Then H I = { A ∈ E : p A ∈ I } is a nonempty saturated hereditary subset of E (see[6, Lemma 7.2]). Then H I = E by assumption. Thus, I = C ∗ ( E, L , E ) . (4) = ⇒ (5): Let I be a nonzero gauge-invariant ideal of C ∗ ( E, L , E ). Then theset H I = { A ∈ E : p A ∈ I } is nonempty ([11]). It means that I contains a vertexprojection. So, I = C ∗ ( E, L , E ).(5) = ⇒ (1): It follows by Theorem 3.8. (cid:3) Proposition 4.7. If ( E, L , E ) is strongly cofinal and for any A ∈ E \ {∅} and B ∈ E \ E reg , we have B ∈ H ( A ) , then ( E, L , E ) is minimal.Proof. Choose a nonempty hereditary saturated subset H of E . Take ∅ 6 = A ∈ H .Note first that if B / ∈ H for a set B ∈ E , B ∈ E reg ; if B ∈ E \ E reg , then B ∈H ( A ) ⊂ H by assumption, a contradiction. So, if B / ∈ H for a set B ∈ E reg , thereis x ∈ L ( E ) such that r ( B, x ) / ∈ H since H is saturated. If r ( B, x ) ∈ E \ E reg ,then again r ( B, x ) ∈ H , a contradiction. So, r ( B, x ) ∈ E reg . Repeating thisprocess, we see that there is an infinite path x = x x · · · ∈ L ( E ∞ ) such that r ( B, x [1 ,n ] ) ∈ E reg \ H for all n ∈ N . Thus, r ( x [1 ,n ] ) / ∈ H for all n ∈ N since r ( B, x [1 ,n ] ) ⊂ r ( x [1 ,n ] ). It then contradicts to strong cofinality of ( E, L , E ). Thus,we conclude that H = E . (cid:3) The converse of Proposition 4.7 is not true.
Example 4.8.
Consider the following labeled graph ( E, L ): • • • / / / / (cid:7) (cid:7) u u ...v w α n ) n ,where w emits infinitely many labeled edges ( α n ) n ≥ . Then E = {∅ , { v } , { w } , { v, w }} and E reg = {∅ , { v }} . It is rather obvious that S ( H ( { v } )) = E and S ( H ( { v, w } )) = E .Consider H ( { w } ). Then H ( { w } ) = {∅ , { w }} and clearly { v } ∈ S ( H ( { w } )). Since r ( { v, w } , β ) ∈ H ( { w } ) for all β ∈ L ( E ) and { v, w } = { v } ∪ { w } ∈ E reg ⊕ H ( { w } ),we have { v, w } ∈ S ( H ( { w } )). Thus, S ( H ( { w } )) = E . Thus, ( E, L , E ) is mini-mal by Theorem 4.6. It is also easy to see that ( E, L , E ) is strongly cofinal. But, { v, w } / ∈ H ( { w } ). Corollary 4.9. If E has no sinks and ( E, L , E ) is set-finite, then ( E, L , E ) is min-imal if and only if ( E, L , E ) is strongly cofinal.Proof. We only need to show ”if” part: Note that E = E reg . Choose H is a nonemptyhereditary saturated subset of E . If B / ∈ H for a set B ∈ E , there is an infinite path x = x x · · · ∈ L ( E ∞ ) such that r ( B, x [1 ,n ] ) / ∈ H for all n ∈ N since H is saturated.Thus, r ( x [1 ,n ] ) / ∈ H for all n ∈ N since r ( B, x [1 ,n ] ) ⊂ r ( x [1 ,n ] ). It contradicts tostrong cofinality of ( E, L , E ). Thus, H = E . (cid:3) The following lemma is proved in [15, Lemma 3.6] for a set-finite and receiver set-finite labeled space ( E, L , E ) with E having no sinks or sources under the assumptionthat C ∗ ( E, L , E ) is simple. But, to prove it they only use minimality of ( E, L , E ) as a property of simplicity of C ∗ ( E, L , E ). So, we can weaken the assumption asfollows. The idea of proof is same with [15, Lemma 3.6]. We have only included theproof of Lemma 4.10(1) for completeness.For a path β := β · · · β | β | , let ¯ β = βββ · · · denotes the infinite repetition of β ([15, Notation 3.5]). we call a path β ∈ L ∗ ( E ) irreducible if it is not a repetition ofits proper initial path. Lemma 4.10. ( [15, Lemma 3.6] ) Let ( E, L , E ) be a minimal labeled space. If thereare a nonempty set A ∈ E and an irreducible path β ∈ L ∗ such that L ( A E n | β | ) = { β n } for all n ≥ , then the following hold. (1) There is an N ≥ such that for all n ≥ N , r ( A , ¯ β [1 ,n ] ) ⊂ ∪ n − j =1 r ( A , ¯ β [1 ,j ] ) . (2) There is an N ≥ such that for all k ≥ , r ( A , ¯ β [1 ,N + k ] ) ⊂ ∪ Nj =1 r ( A , ¯ β [1 ,j ] ) . (3) There is an N ≥ such that for all k ≥ , r ( A , β N + k ) ⊂ ∪ N j =1 r ( A , β j ) . Moreover, A = r ( A, β ) for A := ∪ N i =1 r ( A , β j ) .Proof. (1): Note first that r ( A , ¯ β [1 ,j ] ) ∩ r ( A , ¯ β [1 ,k ] ) = ∅ = ⇒ j = k (mod | β | ) . (6)Assume to the contrary that r ( A , ¯ β [1 ,n ] ) \ ∪ n − j =1 r ( A , ¯ β [1 ,j ] ) = ∅ for infinitely many n ≥
1. Then by (6), r ( A , β n )
6⊂ ∪ n − j =1 r ( A , β j ) (7)for infinitely many n ≥
1. We then claim that r ( β r ) / ∈ H ( A ) for all r ≥
1. If r ( β r ) ∈H ( A ) for some r ≥
1, then r ( β r ) ⊂ ∪ ki =1 r ( A , ¯ β [1 ,m i ] ) for some m , · · · , m k ≥ i , m i = k i | β | for some k i ≥
1. Take m := max i { k i } . Then we have r ( β r ) ⊂ ∪ mi =1 r ( A , β i ). But, then for all sufficiently large n > m | β | , r ( A , β n ) = r ( r ( A , β n − r ) , β r ) ⊂ r ( β r ) ⊂ ∪ mi =1 r ( A , β i ) ⊂ ∪ n − j =1 r ( A , ¯ β [1 ,j ] ) , which contradicts to (7). Thus, r ( β r ) / ∈ H ( A ) for all r ≥
1. It then follows that r ( β r ) / ∈ S ( H ( A )) for all r ≥
1. But, this is not the case since the minimlity of( E, L , E ) implies S ( H ( A )) = E .(2): Follows by [15, Lemma 3.6 (i)].(3): Follows by [15, Lemma 3.6 (ii)]. (cid:3) Remark . For a set-finite and receiver set-finite labeled space ( E, L , E ) with E having no sinks or sources, it is shown in [15, Theorem 3.7] that if C ∗ ( E, L , E ) issimple, then ( E, L , E ) is disagreeable. Thus, if C ∗ ( E, L , E ) is simple, the labeledspace ( E, L , E ) can not have a nonempty set A ∈ E and an irreducible path β ∈L ∗ ( E ) such that L ( AE n | β | ) = { β n } for all n ≥
1. But, a minimal labeled space can
HE SIMPLICITY OF THE C ∗ -ALGEBRAS ASSOCIATED TO ARBITRARY LABELED SPACES17 have a nonempty set A ∈ E and an irreducible path β ∈ L ∗ such that L ( AE n | β | ) = { β n } for all n ≥
1. See the following labeled graph ( E, L ): · · · · · · . • • • • • • • • • / / / / / / / / / / / / / / / / a a a a a a a a Then the smallest non-degenerate accommodating set is E = {∅ , r ( a ) } . It is easy tosee that ( E, L , E ) is minimal and L ( r ( a ) E n ) = { a n } for all n ≥ Proposition 4.12.
Let ( E, L , E ) be a minimal labeled space. If ( α, A ) is a cyclewith no exits, then A is a minimal set.Proof. Let ( α, A ) be a cycle with no exits. Say α = α · · · α n ∈ L ∗ ( E ). We showthat if B ∈ E such that ∅ 6 = B ⊆ A , then B = A . Since ( E, L , E ) is minimal,we have S ( H ( B )) = E by Theorem 4.6. Thus, A ∈ S ( H ( B )), and hence, for each1 ≤ i ≤ n , we have r ( A, α [1 ,i ] ) ⊆ B ∪ r ( B, α ) ∪ r ( B, α [1 , ) ∪ · · · ∪ r ( B, α [1 ,n − )since ( α, A ) and ( α, B ) are both cycle without exits. On the other hand, since r ( A, α [1 ,i ] ) ∩ r ( A, α [1 ,j ] ) = ∅ for i = j by [5, Lemma 6.1] and r ( B, α [1 ,j ] ) ⊆ r ( A, α [1 ,j ] )for each 1 ≤ j ≤ n , it follows that r ( A, α [1 ,i ] ) ∩ r ( B, α [1 ,j ] ) = ∅ for i = j . Thus, wehave r ( A, α [1 ,i ] ) ⊆ r ( B, α [1 ,i ] ) for each 1 ≤ i ≤ n . It then follows that A = r ( r ( A, α [1 ,i ] ) , α [ i +1 ,n ] ) ⊆ r ( r ( B, α [1 ,i ] ) , α [ i +1 ,n ] ) = B. Hence, B = A . (cid:3) Disagreeable labeled spaces.
A notion of disagreeablity of set-finite andreceiver set-finite labeled space ( E, L , E ) with E having no sinks or sources wasintroduced in [4, Definition 5.1] as another analogue notion of Condition (L) ofdirected graphs. If we briefly recall it, a labeled path α ∈ L ([ v ] l E ≥ ) is called agreeable for [ v ] l if α = βα ′ = α ′ γ for some α ′ , β, γ ∈ L ∗ ( E ) with | β | = | γ | ≤ l .Otherwise α is called disagreeable . Note that any path α agreeable for [ v ] l must beof the form α = β k β ′ for some β ∈ L ( E ≤ l ), k ≥
0, and an initial path β ′ of β . Wesay that [ v ] l is disagreeable if there is an N > n > N there is an α ∈ L ( E ≥ n ) that is disagreeable for [ v ] l . A labeled space ( E, L , E ) is disagreeable if[ v ] l is disagreeable for all v ∈ E and l ≥ Proposition 4.13. ( [14, Proposition 3.2] ) For a set-finite and receiver set-finitelabeled space ( E, L , E ) with E having no sinks, the following are equivalent: (a) ( E, L , E ) is disagreeable. (b) [ v ] l is disagreeable for all v ∈ E and l ≥ . (c) For each nonempty A ∈ E and a path β ∈ L ∗ ( E ) , there is an n ≥ suchthat L ( AE | β | n ) = { β n } . Motivated by Proposition 4.13(c), we define a notion of disagreeability of arbi-trary labeled spaces as follows.
Definition 4.14.
We say a labeled space ( E, L , E ) is disagreeable if for any nonemptyset A ∈ E and a path β ∈ L ∗ ( E ), there is an n ≥ L ( AE | β | n ) = { β n } . Definition 4.15. ([12, Definition 3.2]) Let ( E, L , E ) be a labeled space and α ∈L ∗ ( E ) and ∅ 6 = A ∈ E .(1) α is called a loop at A if A ⊆ r ( A, α ).(2) A loop ( α, A ) has an exit if one of the following holds:(i) { α [1 ,k ] : 1 ≤ k ≤ | α |} ( L ( AE ≤| α | ).(ii) A ( r ( A, α ).By [14, Proposition 3.7], one can see that if ( E, L , E ) is disagreeable, then everyloop in ( E, L , E ) has an exit, and hence, ( E, L , E ) satisfies Condition (L). It is shownin [15, Proposition 3.2] that the other implications are not true, in general. But, if( E, L , E ) is minimal, these conditions are equivalent as we see in the following. Lemma 4.16. ( [15, Proposition 3.2] ) Consider the following three conditions of alabeled space ( E, L , E ) . (1) ( E, L , E ) is disagreeable. (2) Every loop in ( E, L , E ) has an exit. (3) ( E, L , E ) satisfies Condition (L), that is, every cycle has an exit.Then we have (1) = ⇒ (2) = ⇒ (3). If, in addition, ( E, L , E ) is minimal, (1)-(3) areequivalent.Proof. We only need to show that (3) = ⇒ (1) when ( E, L , E ) is minimal: Let( E, L , E ) be minimal. Suppose that ( E, L , E ) is not disagreeable. Then there exista nonempty set A ∈ E and a path β ∈ L ∗ ( E ) such that for all n ≥ L ( A E | β | n ) = { β n } , where we assume that β is irreducible. Then by Lemma 4.10, there is an N ≥ k ≥ r ( A , β N + k ) ⊆ ∪ Nj =1 r ( A , β j ) . (8)Take A := ∪ Ni =1 r ( A , β j ). Then A = r ( A, β ). One then can show that A is aminimal set since ( E, L , E ) is minimal (see the proof of [15, Theorem 3.7]). Thus,( β, A ) is a cycle with no exits. Thus, ( E, L , E ) does not satisfy Condition (L). (cid:3) Simplicity.
We now characterize the simplicity of the C ∗ -algebra associatedto an arbitrary labeled space ( E, L , E ). Theorem 4.17.
Let ( E, L , E ) be a labeled space. Then the following are equivalent. (1) C ∗ ( E, L , E ) is simple. (2) ( E, L , E ) is minimal and satisfies Condition (L). (3) ( E, L , E ) is minimal and satisfies Condition (K). (4) The following properties hold: (a) ( E, L , E ) is strongly cofinal, (b) ( E, L , E ) satisfies Condition (L), and (c) for any A ∈ E \{∅} and B ∈ E , there is C ∈ E reg such that B \ C ∈ H ( A ) . (5) The following properties hold: (a) ( E, L , E ) is strongly cofinal, HE SIMPLICITY OF THE C ∗ -ALGEBRAS ASSOCIATED TO ARBITRARY LABELED SPACES19 (b) ( E, L , E ) is disagreeable, and (c) for any A ∈ E \{∅} and B ∈ E , there is C ∈ E reg such that B \ C ∈ H ( A ) .Proof. (1) = ⇒ (2): If C ∗ ( E, L , E ) is simple, then the only gauge-invariant ideal of C ∗ ( E, L , E ) is { } and C ∗ ( E, L , E ). Thus, ( E, L , E ) is minimal by Theorem 4.6.Suppose that ( E, L , E ) does not satisfy Condition (L). Then the labeled space hasa cycle ( α, A ) with no exits. Since ( E, L , E ) is minimal, A is a minimal set byProposition 4.12. Then, C ∗ ( E, L , E ) has a hereditary subalgebra isomorphic to M | α | ( C ( T )) by [11, Lemma 4.6]. It contradicts to C ∗ ( E, L , E ) is simple.(2) = ⇒ (1): Let I be a nonzero ideal of C ∗ ( E, L , E ). Since ( E, L , E ) satisfiesCondition (L), I contains a vertex projection p A for some ∅ 6 = A ∈ E by the Cuntz-Krieger Uniqueness Theorem 2.5. Then I = C ∗ ( E, L , E ) by Theorem 4.6.(2) ⇐⇒ (3): Follows by Lemma 3.5.(2) ⇐⇒ (4): Follows by Theorem 4.6.(4) ⇐⇒ (5): Follows by Lemma 4.16. (cid:3) As a corollary, we have the following simplicity results of labeled graph C ∗ -algebras associated to set-finite labeled spaces with no sinks. It is an improvementon [15, Theorem 3.7]. Corollary 4.18. If E has no sinks and ( E, L , E ) is set-finite, then the followingare equivalent. (1) C ∗ ( E, L , E ) is simple. (2) ( E, L , E ) is minimal and satisfies Condition (L). (3) ( E, L , E ) is minimal and satisfies Condition (K). (4) ( E, L , E ) is strongly cofinal and disagreeable. (5) ( E, L , E ) is strongly cofinal and satisfies Condition (L). (6) ( E, L , E ) is strongly cofinal and satisfies Condition (K).Proof. It follows by Corollary 4.9 and Theorem 4.17. (cid:3)
Acknowledgements.
The author wishes to many thank Toke Meier Carlsen forpointing out her mistake and having a helpful conversation.
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