-ALGEBRAS ASSOCIATED WITH

CONSTRUCTION AND PURE INFINITENESS OF C

-ALGEBRAS ASSOCIATED WITH

LAMBDA-GRAPH SYSTEMS

KENGO MATSUMOTO

Abstract

Aλ-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. In [16] the author has introduced aC^∗-algebraOᑦ

associated with aλ-graph systemᑦby using groupoid method as a generalization of the Cuntz- Krieger algebras. In this paper, we concretely construct theC^∗-algebraOᑦby using both creation operators and projections on a sub Fock Hilbert space associated withᑦ. We also introduce a new irreducible condition onᑦunder which theC^∗-algebraOᑦbecomes simple and purely infinite.

0. Introduction

For a finite set, a subshift(, σ )is a topological dynamics defined by a closed shift-invariant subsetof the compact set^Zof all bi-infinite sequences ofwith shift transformationσdefined byσ ((xi)i∈Z)=(xi+1)i∈Z.The author has introduced the notions of symbolic matrix system andλ-graph system as presentations of subshifts ([15]). They are generalized notions of symbolic matrix andλ-graph (=labeled graph) for sofic subshifts. We henceforth denote byZ₊and byNthe set of all nonnegative integers and the set of all positive integers respectively. A symbolic matrix system(M, I)overconsists of two sequences of rectangular matrices(Ml,l+1, Il,l+1),l ∈Z₊. The matricesMl,l+1

have their entries in formal sums ofand the matricesIl,l+1have their entries in{0,1}. They satisfy the following commutation relations

Il,l+1Ml+1,l+2=Ml,l+1Il+1,l+2, l ∈Z₊.

It is required that each row ofIl,l+1has at least one 1 and each column ofIl,l+1

has exactly one 1. Aλ-graph system ᑦ = (V, E, λ, ι)over consists of a vertex setV =V0∪V1∪V2∪ · · ·, an edge setE=E0,1∪E1,2∪E2,3∪ · · ·, a labeling mapλ : E → and a surjective map ι (= ιl,l+1) : Vl+1 → Vl

for eachl ∈ Z₊. It naturally arises from a symbolic matrix system(M, I). The labeled edges from a vertexv_i^l ∈Vl to a vertexv_j^l+1∈Vl+1are given by

Received August 16, 2004.

the(i, j)-componentMl,l+1(i, j)ofMl,l+1. The mapι(=ιl,l+1) is defined by ιl,l+1(v_j^l+1) = v^l_i precisely ifIl,l+1(i, j) = 1. The symbolic matrix systems and the λ-graph systems are the same objects and give rise to subshifts by gathering label sequences appearing in the labeled Bratteli diagrams of theλ- graph systems. Conversely we have a canonical method to construct a symbolic matrix system and aλ-graph system from an arbitrary subshift [15].

In [16], the author has constructed C^∗-algebras fromλ-graph systems as groupoidC^∗-algebras by using continuous graphs in the sense of Deaconu (cf. [5], [19]) and studied their structure. Letᑦ = (V, E, λ, ι)be a λ-graph system over. Let{v1^l, . . . , v_m(l)^l }be the vertex set Vl. TheC^∗-algebra Oᑦ

is generated by partial isometries Sα corresponding to the symbolsα ∈ and projectionsE_i^l corresponding to the verticesv_i^l ∈ Vl, i = 1, . . . , m(l), l∈Z₊.It is realized as a universal uniqueC^∗-algebra subject to certain operator relations amongSα, α ∈ andE_i^l,i =1, . . . , m(l),l ∈Z₊encoded by the structure of ᑦ. A condition onᑦ, called condition (I), has been introduced ([16]). Irreducibility and aperiodicity forᑦhave been also defined so that if ᑦsatisfies condition (I) and is irreducible, theC^∗-algebraOᑦis shown to be simple. It is also proved that if in particularᑦis aperiodic,Oᑦis simple and purely infinite ([16, Theorem 4.7 and Proposition 4.9]).

In this paper, we will first introduce a new construction of theC^∗-algebras Oᑦ. We will construct a sub Fock Hilbert space associated with a λ graph system_ᑦand define creation operators and sequence of projections. We will then show thatOᑦis canonically isomorphic to the quotientC^∗-algebra of the C^∗-algebra generated by the creation operators and the projections by an ideal (Theorem 2.6). This construction is a generalization of a construction of Cuntz- Krieger algebras [3] by [7], [8] andC^∗-algebras associated with subshifts [14].

We will next introduce a new irreducible condition and new condition (I) on ᑦsuch thatOᑦ becomes simple and purely infinite (Theorem 3.9). The new conditions are calledλ-irreducible condition andλ-condition (I) respectively.

In the previously proved result [16, Theorem 4.7 and Proposition 4.9], we needed aperiodicity condition onᑦforOᑦto be simple and purely infinite. It is well-known that the Cuntz-Krieger algebraOAis simple and purely infinite if the matrixAis irreducible with condition (I). Since theC^∗-algebrasOᑦare a generalization of the Cuntz-Krieger algebrasOA, the aperiodicity condition on ᑦ is too strong such that Oᑦ becomes simple and purely infinite. From this point of view, theλ-irreducible condition with λ-condition (I) on ᑦis an exact generalization of the irreducible condition with condition (I) on the nonnegative matricesA.

The author would like to thank the referee who named the termsλ-irredu- cible andλ-condition (I) instead of the originally used terms (new) irreducible and (new) condition (I), and for his useful comments.

1. Review ofC^∗-algebras associated with λ-graph systems

For aλ-graph systemᑦ= (V, E, λ, ι)over, the vertex setsVl,l ∈Z₊and the edge setsEl,l+1,l ∈Z₊are finite disjoint sets. An edgeeinEl,l+1has its source vertexs(e)inVland its terminal vertext(e)inVl+1. Every vertex inV has outgoing edges and every vertex inV, exceptV0, has incoming edges. The label of an edgee ∈Emeansλ(e) ∈ . It is then required that there exists an edge inEl,l+1with labelαand its terminal isv∈Vl+1if and only if there exists an edge inEl−1,lwith labelαand its terminal isι(v)∈Vl. Foru∈Vl−1

andv∈Vl+1, we put

E^ι(u, v)= {e∈El,l+1|t(e)=v, ι(s(e))=u}, Eι(u, v)= {e∈El−1,l|s(e)=u, t(e)=ι(v)}.

Then there exists a bijective correspondence betweenE^ι(u, v)andEι(u, v)that preserves labels for every pair(u, v) ∈ Vl−1×Vl+1. This property is called the local property of theλ-graph system. A finite sequence(e1, e2, . . . , en)of edges such thatt(ei) =s(ei+1),i = 1,2, . . . , n−1 is called a path. We put i =and define

⁺_ᑦ =

(λ(e1), λ(e2), . . .)∈

i∈N

i |ei ∈Ei−1,i, t(ei)=s(ei+1), i∈N

and

ᑦ=

(αi)i∈Z ∈

i∈Z

i |(αi, αi+1, . . .)∈⁺_ᑦ, i∈Z

.

Thenᑦis a subshift overcalled the subshift presented byᑦ. A finite se- quenceµ=(µ1, . . . , µk)ofµj ∈that appears inᑦis called an admissible word ofᑦof length|µ| =k. Denote by^k_ᑦthe set of all admissible words of lengthkofᑦand put^∗_ᑦ= ∪^∞_k=0^k_ᑦwhere⁰_ᑦdenotes the empty word∅.

We briefly review theC^∗-algebra Oᑦ associated with λ-graph system_ᑦ, that has been originally constructed in [16] to be a groupoidC^∗-algebra of a groupoid of a continuous graph obtained byᑦ(cf. [5], [6], [19]).

Let ᑦ = (V, E, λ, ι) be a left-resolving λ-graph system over , that is, fore, e ∈E,λ(e) = λ(e),t(e)= t(e)impliese =e. The vertex setVl is denoted by{v1^l, . . . , v_m(l)^l }. Define the transition matricesAl,l+1, Il,l+1of_ᑦby setting fori=1,2, . . . , m(l),j =1,2, . . . , m(l+1),α ∈,

Al,l+1(i, α, j)=

1 ifs(e)=v_i^l,λ(e)=α,t(e)=v_j^l+1for somee∈El,l+1, 0 otherwise,

Il,l+1(i, j)=

1 ifιl,l+1(v_j^l+1)=v^l_i, 0 otherwise.

TheC^∗-algebraOᑦis realized as the universal unitalC^∗-algebra generated by partial isometriesSα,α ∈ and projectionsE^l_i,i = 1,2, . . . , m(l),l ∈ Z₊ subject to the following operator relations called(ᑦ):

α∈

SαS_α^∗=1, (1.1)

m(l) i=1

E^l_i =1, E^l_i =

m(l+1) j=1

Il,l+1(i, j)E_j^l+1, (1.2)

SβS_β^∗E^l_i =E^l_iSβS_β^∗, (1.3)

S_β^∗E_i^lSβ =

m(l+1) j=1

Al,l+1(i, β, j)E_j^l+1, (1.4)

forβ ∈, i=1,2, . . . , m(l),l ∈Z₊.

For a vertexv_i^l ∈Vl, we denote by⁺(v_i^l)the set ⁺(v^l_i)=

(λ(e1), λ(e2), . . .)∈⁺_ᑦ |s(e1)=v_i^l, t(ej)=s(ej+1), j ∈N of all infinite label sequences inᑦstarting atv_i^l. We say thatᑦsatisfies condition (I) if for each v_i^l ∈ V, the set ⁺(v_i^l) contains at least two distinct label sequences.

Theorem1.1 ([16]). Suppose that_ᑦsatisfies condition (I). LetSα,α ∈ and E^l_i, i = 1,2, . . . , m(l), l ∈ Z₊ be another family of nonzero partial isometries and nonzero projections satisfying the relations(ᑦ). Then the map Sα →Sα,E_i^l →E^l_i extends to an isomorphism fromOᑦonto theC^∗-algebra Oᑦgenerated bySα,α∈andE_i^l,i =1,2, . . . , m(l),l ∈Z₊.

Hence theC^∗-algebraOᑦunder the condition that_ᑦsatisfies condition (I) is the uniqueC^∗-algebra subject to the above relations(ᑦ). By the uniqueness ofOᑦ, the correspondenceSα →zSα,E_i^l →E_i^lforz∈T= {z∈C| |z| =1} yields an actionαᑦofTcalled the gauge action. LetF_k^lbe the finite dimensional C^∗-subalgebra ofOᑦ generated bySµE^l_iS_ν^∗,µ, ν ∈ ^k_ᑦ,i = 1,2, . . . , m(l). LetFᑦbe theC^∗-subalgebra ofOᑦgenerated by the algebrasF_k^l,k ≤l. It is an AF-algebra realized as the fixed point algebraO_ᑦ^αᑦofOᑦunderαᑦ.

Aλ-graph systemᑦis said to beirreducibleif for a vertexv^l_i ∈ Vl and a sequence(u⁰, u¹, . . .)of verticesuⁿ ∈ Vn with ιn,n+1(uⁿ⁺¹) = uⁿ,n ∈ Z₊, there exists a path starting atv_i^l and terminating at u^l+N for some N ∈ N.

ᑦis said to beaperiodic if for a vertexv_i^l ∈ Vl there exists an N ∈ Nsuch that there exist paths starting at v^l_i and terminating at all vertices of Vl+N. These properties forλ-graph systems are generalizations of the corresponding properties for finite directed graphs.

Theorem 1.2 ([16], Proposition 4.9). Suppose that aλ-graph system _ᑦ satisfies condition (I). If_ᑦis irreducible, the C^∗-algebraOᑦ is simple. If in particular_ᑦis aperiodic,Oᑦis simple and purely infinite.

In what follows, we fix a left-resolvingλ-graph systemᑦ = (V, E, λ, ι) over.

2. Fock space construction

In this section, we will construct a family of partial isometries and projections satisfying the relations(ᑦ)in a concrete way. Let#ᑦbe the projective limit

#ᑦ=

(u^l)l∈Z₊ ∈

l∈Z₊

Vl |ιl,l+1(u^l+1)=u^l, l ∈Z₊

of the systemιl,l+1 : Vl+1 → Vl,l ∈ Z₊. We endow#ᑦ with the projective limit topology from the discrete topologies onVl,l ∈Z₊so that it is a compact Hausdorff space. An elementuin#ᑦis called a vertex. LetEᑦbe the set of all triplets(u, α, w)∈#ᑦ××#ᑦsuch that there existsel,l+1∈El,l+1satisfying u^l = s(el,l+1), w^l+1 = t(el,l+1) and α = λ(el,l+1)for each l ∈ Z₊ where u=(u^l)l∈Z+,w=(w^l)l∈Z+ ∈#ᑦ. The setEᑦ⊂#ᑦ××#ᑦis a continuous graph in the sense of Deaconu ([14, Proposition 2.1]). Forw=(w^l)l∈Z+ ∈#ᑦ

andα ∈ , the local property of ᑦensures that if there exists e0,1 ∈ E0,1

satisfyingw¹= t(e0,1),α = λ(e0,1), there uniquely existel,l+1∈ El,l+1and u= (u^l)l∈Z₊ ∈ #ᑦsatisfyingu^l = s(el,l+1),w^l+1 = t(el,l+1),α = λ(el,l+1) for alll ∈Z₊. Hence for everyw∈ #ᑦ, there existα ∈ andu∈#ᑦsuch that(u, α, w)∈Eᑦ. Let us consider the finite path spaces of the graphEᑦas follows:

W_ᑦ⁰=#ᑦ, W_ᑦ¹=Eᑦ,

W_ᑦ²= {(u0, α1, u1, α2, u2)|(u0, α1, u1), (u1, α2, u2)∈Eᑦ}, . . .

W_ᑦ^k = {(u0, α1, u1, α2, . . . , αk, uk)|(ui−1, αi, ui)∈Eᑦ, i=1,2, . . . , k}, . . .

We assign to a finite pathη∈W_ᑦ^kthe vectoreη. For eachk∈Z₊, letᑠ^k_ᑦbe the Hilbert space spanned by the complete orthonomal basis{eη |η ∈W_ᑦ^k}. The Hilbert space_ᑢᑦis defined by their direct sums

ᑢ_ᑦ = ⊕^∞_k=0ᑠ^k_ᑦ.

We define creation operatorsTβ forβ ∈ and projectionsP_i^l forv^l_i ∈V on ᑢᑦby setting

Tβe(u0,α1,u1,α2,...,αk,uk) =





e(u−1,β,u0,α1,u1,α2,...,αk,uk) if there existsu−1∈#ᑦ

such that(u−1,β,u⁰)∈Eᑦ,

0 otherwise,

P_i^le(u0,α1,u1,α2,...,αk,uk) =





e(u⁰,α¹,u¹,α²,...,αk,uk) ifu^l0=v^l_i, where u0=(u^l0)l∈Z+∈#ᑦ,

0 otherwise.

Note that the vertexu−1 ∈ #ᑦ satisfying(u−1, β, u0) ∈ Eᑦis unique forβ andu0if it exists, because_ᑦis left-resolving. It is direct to see that

T_β^∗e(u0,α1,u1,α2,...,αk,uk)=

e(u1,α2,...,αk,uk) ifk ≥1 andα1=β,

0 otherwise.

Lemma2.1. Forβ ∈

(i) TβT_β^∗is the projection onto the subspace spanned by the vectorseηsuch thatη=(u0, α1, u1, α2, . . . , αk, uk)∈W_ᑦ^k,α1=β,k∈N,

(ii) T_β^∗Tβis the projection onto the subspace spanned by the vectorseξsuch thatξ =(u0, α1, u1, α2, . . . , αk, uk)∈W_ᑦ^k,k ∈Z₊,(u−1, β, u0)∈Eᑦ

for someu−1∈#ᑦ.

LetP0denote the projection onᑢᑦonto the subspaceᑠ⁰_ᑦ. It is immediate to see thatP0Tβ =0 forβ∈andP0P_i^l =P_i^lP0forv^l_i ∈V. We then have

Lemma2.2.

α∈

TαT_α^∗+P0=1, (2.1)

m(l) i=1

P_i^l =1, P_i^l =

m(l+1) j=1

Il,l+1(i, j)P_j^l+1, (2.2)

TβT_β^∗P_i^l =P_i^lTβT_β^∗, (2.3)

T_β^∗P_i^lTβ =

m(l+1) j=1

Al,l+1(i, β, j)P_j^l+1, (2.4)

forβ ∈,i=1,2, . . . , m(l),l∈Z₊.

Proof. We will show the relation (2.4). Other relations are direct. Forβ ∈ ,v^l_i ∈V,(u⁰, α¹, u¹, α², . . . , αk, uk)∈W_ᑦ^k, it follows that

T_β^∗P_i^lTβe(u0,α1,u1,α2,...,αk,uk)

=





e(u⁰,α¹,u¹,α²,...,αk,uk) if(u−1, β, u⁰)∈Eᑦfor someu−1∈#ᑦ

andu^l₋1=v_i^l whereu−1=(u^l₋1)_l∈Z+ ∈#ᑦ,

0 otherwise,

=





e(u0,α1,u1,α2,...,αk,uk) ifs(e)=v^l_i,t(e)=u^l+0 ¹,λ(e)=β for somee∈El,l+1,

0 otherwise,

=

m(l+1) j=1

Al,l+1(i, β, j)P_j^l+1e(u⁰,α¹,u¹,α²,...,αk,uk).

Hence the relation (2.4) holds.

For a wordν=α1· · ·αk ∈^∗_ᑦ, we setTν =Tα1· · ·Tαk.

Lemma2.3.Every polynomial ofTα,P_i^l,α∈, i=1,2, . . . , m(l),l ∈Z₊ is a finite linear combination of elements of the formTµP_i^lT_ν^∗forµ, ν ∈^∗_ᑦ, i=1,2, . . . , m(l),l ∈Z₊.

Proof. It follows that by (2.3) and (2.4) P_i^lTα =TαT_α^∗P_i^lTα =

m(l+1) j=1

Al,l+1(i, α, j)TαP_j^l+1

and hence

T_α^∗P_i^l =

m(l+1) j=1

Al,l+1(i, α, j)P_j^l+1T_α^∗.

The assertion is immediately seen by these equations.

Let Tᑦ be the C^∗-algebra on _ᑢᑦ generated byTα, P_i^l, P0, α ∈ , i = 1,2, . . . , m(l),l ∈ Z₊andI the closed two-sided ideal ofTᑦgenerated by P0.

Lemma2.4. I is the closure of the algebra of all finite linear combinations of elements of the formTµP_i^lP0T_ν^∗forµ, ν ∈^∗_ᑦ,i =1,2, . . . , m(l),l ∈Z₊. Proof. Since P0Tβ = 0, one sees TµP_i^lT_ν^∗P0 = P0TµP_i^lT_ν^∗ = 0. As the algebra Tᑦ is generated by elements of the form TµP_i^lT_ν^∗ and P0, by

using the relationP0P_i^l =P_i^lP0,Tᑦis the closure of the algebra of all linear combinations of elements of the formsTµP_i^lP⁰T_ν^∗ andTµP_i^lT_ν^∗. SinceI = TᑦP0Tᑦ, one concludes thatI is the closure of the algebra of all finite linear combinations of elements of the formTµP_i^lP0T_ν^∗.

Lemma2.5. Tβ, P_i^l ∈I.

Proof. SupposeTβ ∈I. By Lemma 2.4, there exists a finite linear combin- ationX =

µ,ν,i,lcµ,ν,i,lTµP_i^lP0T_ν^∗ofTµP_i^lP0T_ν^∗,µ, ν ∈^∗_ᑦ,i= 1,2, . . . , m(l), l ∈ Z₊ such that X−Tβ < ¹₂. LetK denote the maximum length of the wordsν that appear in the element

µ,ν,i,lcµ,ν,i,lTµP_i^lP0T_ν^∗. Take a finite pathξ = (u0, α1, u1, α2, . . . , αK+1, uK+1)∈W_ᑦ^K+1such that there ex- ists a vertexu−1 ∈ #ᑦ satisfying(u−1, β, u⁰) ∈ Eᑦ. We haveXeξ = 0 and Tβeξ =e(u−1,β,u⁰,...,αK+1,uK+1)so that

(X−Tβ)eξ = e(u−1,β,u0,...,αK+1,uK+1) =1, a contradiction.

Suppose nextP_i^l∈I. There exists similarly an elementY =

µ,ν,i,lcµ,ν,i,l

TµP_i^lP0T_ν^∗such thatY−P_i^l< ¹₂. Take a finite pathη=(u0, α1, u1, α2, . . . , αK+1, uK+1) ∈W_ᑦ^K+1such thatu^l0 = v_i^l, whereu0 = (u^l0)_l∈Z+ ∈ #ᑦso that Y eη=0 andP_i^leη =eηa contradiction.

Definition. LetOᑦbe the quotientC^∗-algebraTᑦ/IofTᑦby the idealI, and the operatorsSαandE_i^lthe quotient images ofTαandP_i^linOᑦrespectively.

By Lemma 2.5, the elementsSα andE_i^l are not zeros for eachα ∈ and v^l_i ∈V, and satisfy the relations(ᑦ)by Lemma 2.2. Thus by Theorem 1.1 we obtain

Theorem2.6. Suppose that_ᑦsatisfies condition (I). Then theC^∗-algebra Oᑦ is canonically isomorphic to theC^∗-algebraOᑦassociated withλ-graph system_ᑦ.

Define a unitary representationU of the circle groupTon the Hilbert space ᑢᑦ by Uzeη = z^keη forη ∈ W_ᑦ^k. It is easy to see that the automorphisms Ad(Uz), z ∈ T on the algebra of all bounded linear operators on _ᑢᑦ leave invariant globally both the algebrasTᑦandI. They give rise to an action on theC^∗-algebraOᑦthat is the gauge actionαᑦonOᑦ.

This construction of theC^∗-algebraOᑦ is inspired by the construction of theC^∗-algebras of HilbertC^∗-bimodules by [18] and [10] (cf. [9]). Our con- structoin can work for the construction of theC^∗-algebras of general continu- ous graphs of Deaconu [5].

3. λ-irreducibility and pure infiniteness

As in Section 1, it has been proved in [16] that ifᑦis aperiodic, theC^∗-algebra Oᑦbecomes simple and purely infinite. The aperiodic condition onᑦhowever is too strong such that the algebraOᑦ is simple and purely infinite. In fact, the Cuntz-Krieger algebraOAis simple and purely infinite if the matrixAis irreducible with condition (I). In this section, we introduce a new irreducible condition along with a new condition (I) on ᑦunder which the C^∗-algebra Oᑦ is simple and purely infinite. The new conditions are calledλ-irreducible condition andλ-condition (I) respectively. They are exact generalization of the corresponding conditions on a finite square matrixAwith entries in{0,1}.

Definition. Aλ-graph systemᑦisλ-irreducibleif for an ordered pair of verticesv_i^l, v_j^l ∈Vl, there exists a numberLl(i, j)∈Nsuch that for a vertex v^l+L_h ^l(i,j) ∈ Vl+Ll(i,j) withι^Ll(i,j)(v^l+L_h ^l(i,j)) = v^l_i, there exists a path γ in ᑦ such that

s(γ )=v_j^l, t(γ )=v^l+L_h ^l(i,j),

whereι^Ll(i,j)means theLl(i, j)-times compositions ofι, ands(γ ), t(γ )denote the source vertex, the terminal vertex ofγ respectively. It is obvious that ifᑦis λ-irreducible, then it is irreducible in the sense of Section 1. LetGbe a finite directed graph andᑦ_Gthe associatedλ-graph system defined in [16, Section 7].

It is then immediate thatGis irreducible if and only ifᑦ_Gisλ-irreducible.

The following lemma is direct from the local property ofλ-graph system.

Lemma3.1. Suppose that aλ-graph system_ᑦisλ-irreducible. For a vertex v^l_i ∈ Vl, letLbe the numberLl(i, i)as in the definition ofλ-irreducible for the pair(v^l_i, v^l_i).

(i) For a numberk ∈Nand a vertexv_j^l+kL ∈ Vl+kLwithι^kL(v_j^l+kL)= v_i^l, there exists a pathπin_ᑦsuch thats(π)=v_i^landt(π)=v_j^l+kL. (ii) If every pathπin_ᑦof lengthLwiths(π)=v_i^lmust satisfyι^L(t(π))=v_i^l,

then every pathγ in_ᑦof lengthkLfor somek ∈Nwiths(γ )=v_i^lmust satisfyι^kL(t(γ ))=v_i^l.

We will introduceλ-condition (I).

Definition. Aλ-graph systemᑦis said to satisfyλ-condition(I) if for a vertexv_i^l ∈Vlthere exist two distinct pathsγ1, γ2in_ᑦsuch that

s(γ1)=s(γ2)=v_i^l, t(γ1)=t(γ2), λ(γ1)=λ(γ2).

It is obvious that ifᑦsatisfiesλ-condition (I), it satisfies condition (I) in the sense of Section 1. One immediately sees that the adjacency matrix of a finite

directed graphGsatisfies condition (I) in the sense of Cuntz-Krieger [3] if and only ifᑦ_Gsatisfiesλ-condition (I).

Let Al,l+1, Il,l+1 be the transition matrices of ᑦ as in Section 1. Define the matricesAl,l+k, Il,l+k fork ∈ Nby setting fori = 1,2, . . . , m(l),j = 1,2, . . . , m(l+k),µ∈^k_ᑦ,

Al,l+k(i, µ, j)=





1 ifs(γ )=v_i^l,λ(γ )=µ,t(γ )=v_j^l+k for some pathγ inᑦ,

0 otherwise, Il,l+k(i, j)=

1 ifι^k(v_j^l+k)=v_i^l, 0 otherwise,

whereλ(γ )=λ(γ1)· · ·λ(γk)forγ =(γ1, . . . , γk),γi ∈E, 1≤i ≤k. Lemma3.2. Suppose that_ᑦisλ-irreducible and satisfiesλ-condition (I).

For a vertexv_i^l ∈ Vl, letLbe the number as in Lemma 3.1. Then one of the following two conditions holds:

(1) There exist a wordη∈^L_ᑦand a vertexv_j^l+L∈Vl+Lsuch thatAl,l+L(i,η,j)

=1,Il,l+L(i, j)=0.

(2) There existsk ∈Nsuch thatIl,l+kL(i, h)=1impliesAl,l+kL(i, µ, h)= 1for someµ∈ ^kL_ᑦ , and there existsh∈ {1, . . . , m(l+L)}such that

µ∈^kL_ᑦ Al,l+kL(i, µ, h)≥2.

Proof. Suppose that the condition (1) does not hold. Asᑦisλ-irreducible, it satisfies the assumption of Lemma 3.1(ii). By theλ-condition (I), we may take a numberk ∈Nand a vertexv_h^l+kL ∈Vl+kL and two distinct pathsγ1, γ2

inᑦsuch that

s(γ1)=s(γ2)=v_i^l, t(γ1)=t(γ2)=v_h^l+kL, λ(γ1)=λ(γ2).

Hence we haveAl,l+kL(i, γ1, h)=Al,l+kL(i, γ2, h)=1 so that

µ∈^kL_ᑦ

Al,l+kL(i, µ, h)≥2

and the condition (2) holds.

Proposition3.3. Assume that_ᑦisλ-irreducible and satisfiesλ-condition (I). For the projection E_i^l in theC^∗-algebraOᑦ corresponding to the vertex v^l_i ∈Vl, there exists a numberL∈Nsuch that for every vertexv_h^l+L∈Vl+L

withι^L(v_h^l+L)=v_i^l, there exists an admissible wordµ(h)in^L_ᑦ such that Sµ(h)E^l+L_h S_µ(h)^∗ =0 and

m(l+L)

h=1

Il,l+L(i, h)Sµ(h)E_h^l+LS_µ(h)^∗ < E_i^l.

Proof. Forv^l_i ∈Vl,letLbe the number as in Lemma 3.1. One of the two conditions (1) and (2) in the preceding lemma holds. Suppose that (1) holds.

Asᑦisλ-irreducible, for a vertexv^l+L_h ∈Vl+Lwithι^L(v^l+L_h )=v_i^l, there exists a pathγ (h) inᑦ of lengthL such thats(γ (h)) = v^l_i, t(γ (h)) = v^l+L_h . Put µ(h)=λ(γ (h))∈^L_ᑦso thatSµ(h)E_h^l+LS_µ(h)^∗ =0. By the condition (1), there exists a wordη∈^L_ᑦ such thatAl,l+L(i, η, j) =1,Il,l+L(i, j)=0 for some j =1, . . . , m(l+L). Hence one has

m(l+L)

h=1

Il,l+L(i, h)Sµ(h)E_h^l+LS_µ(h)^∗ +SηE_j^l+LS_η^∗

≤

m(l+L)

h=1

ν∈^L_ᑦ

Al,l+L(i, ν, h)SνE_h^l+LS_ν^∗.

NowAl,l+L(i, η, j)=1 so thatSηE_h^l+LS_η^∗ =0. By (1.1), (1.3) and (1.4), the equality

(3.1)

m(l+L)

h=1

ν∈^L_ᑦ

Al,l+L(i, ν, h)SνE_h^l+LS_ν^∗=E_i^l

holds so that _m(l+L)

h=1

Il,l+L(i, h)Sµ(h)E_h^l+LS_µ(h)^∗ < E^l_i.

We next assume that the condition (2) holds. There exists k ∈ Nsuch that Il,l+kL(i, h) = 1 impliesAl,l+kL(i, µ, h) = 1 for someµ ∈ ^kL_ᑦ , and there existsh= 1, . . . , m(l+L)such that

µ∈^kL_ᑦ Al,l+kL(i, µ, h) ≥2. By (3.1) we obtain _m(l+kL)

h=1

Il,l+kL(i, h)Sµ(h)E_h^l+kLS_µ(h)^∗ < E_i^l. TakeLaskLso that we get the desired assertion.

Let N_h^l+n,n be the number of paths γ in ᑦstarting at a vertex in Vl and terminating atv_h^l+n. Asᑦis left-resolving, it is the number of admissible words

µinᑦof lengthnsuch thatSµE^l+n_h S_µ^∗ =0. It satisfies the equality N_h^l+n,nE_h^l+n=

µ∈ⁿ_ᑦ

S_µ^∗Sµ

E^l+n_h .

By the local property ofλ-graph system, we haveN_h^l+n,n=N_k^l+n,nifιⁿ(v^l+n_h )= ιⁿ(v_k^l+n). For a vertexv_h^l+n∈Vl+n, define a projectionP_h^l+n,nby setting

P_h^l+n,n= 1 N_h^l+n,n

µ,ν∈ⁿ_ᑦ

SµE_h^l+nS_ν^∗.

Lemma3.4. Take µ∈ ⁿ_ᑦ satisfyingSµE_h^l+nS_µ^∗ =0. Then there exists a partial isometryU_h,µ^l+ninOᑦsuch that

U_h,µ^l+nU_h,µ^l+n∗=U_h,µ^l+n∗U_h,µ^l+n=

ν∈ⁿ_ᑦ

SνE_h^l+nS_ν^∗, U_h,µ^l+nP_h^l+n,nU_h,µ^l+n∗=SµE_h^l+nS_µ^∗.

Proof. The elementsSξE_h^l+nS_η^∗,ξ, η∈ⁿ_ᑦform a matrix units of theC^∗- subalgebra ofOᑦgenerated bySξE_h^l+nS_η^∗,ξ, η∈ⁿ_ᑦthat is isomorphic to the full matrix algebra of sizeN_h^l+n,n. AsP_h^l+n,n is a projection of rank one in the subalgebra, one can find a desired partial isometry by elementary linear algebra.

The following lemma is straightforward.

Lemma3.5. PutVL= √ ¹

N_h^l+L,L

µ∈^L_ᑦSµE^l+L_h . Then we have V_L^∗VL=1, VLE_i^lV_L^∗=

m(l+L)

h=1

Il,l+L(i, h)P_h^l+L,L.

Proposition3.6. Assume that_ᑦisλ-irreducible and satisfiesλ-condition (I). Then the projectionE^l_i forv_i^l ∈V is an infinite projection inOᑦ.

Proof. Suppose that the number m(l)of the vertex setVl is one for all l ∈Z₊. Then we haveE_i^l = 1. Sinceᑦsatisfiesλ-condition (I), the alphabet is not singleton. Now 1=

α∈SαS_α^∗andAl,l+1(i, α, j)=1 for alli, α, j. Hence we see by the relations(ᑦ),

S_α^∗Sα =

α∈

SαS_α^∗=1.