Documenta Mathematica Gegr˜undet 1996 durch die Deutsche Mathematiker-Vereinigung

Documenta Mathematica

Gegr¨ undet 1996 durch die Deutsche Mathematiker-Vereinigung

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Orthogonal coloring and rigid embedding of an extended map, see page 48

Band 7

2002

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Band 7, 2002 Kengo Matsumoto

C^∗-Algebras Associated

with Presentations of Subshifts 1–30

Gordon Heier

Effective Freeness of Adjoint Line Bundles 31–42 Ezra Miller

Planar Graphs as Minimal Resolutions of

Trivariate Monomial Ideals 43–90

J. Brodzki, R. Plymen

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of Bivariant Chern Classes 133–142

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Homology Stability for Unitary Groups 143–166 Raymond Brummelhuis, Heinz Siedentop, and Edgardo Stockmeyer

The Ground State Energy

of Relativistic One-Electron Atoms

According to Jansen and Heß 167–182

Frans Oort and Thomas Zink Families of p-Divisible Groups

with Constant Newton Polygon 183–201

Paul Balmer, Stefan Gille, Ivan Panin and Charles Walter The Gersten Conjecture for Witt Groups

in the Equicharacteristic Case 203–217

Martin Olbrich

L²-Invariants of Locally Symmetric Spaces 219–237 Cornelia Busch

The Farrell Cohomology ofSP(p−1,Z) 239–254 Guihua Gong

On the Classification

of Simple Inductive Limit C^∗-Algebras, I:

The Reduction Theorem 255–461

Winfried Bruns and Joseph Gubeladze

Unimodular Covers of Multiples of Polytopes 463–480

Rost Projectors and Steenrod Operations 481–493 Tam´as Hausel and Bernd Sturmfels

Toric Hyperk¨ahler Varieties 495–534

Bernhard Keller and Amnon Neeman The Connection Between May’s Axioms for a Triangulated Tensor Product and Happel’s Description

of the Derived Category of the QuiverD4 535–560 Anthony D. Blaom

Reconstruction Phases for

Hamiltonian Systems on Cotangent Bundles 561–604 Marc A. Rieffel

Group C^∗-Algebras as Compact

Quantum Metric Spaces 605–651

Georg Schumacher

Asymptotics of Complete K¨ahler-Einstein Metrics – Negativity of the

Holomorphic Sectional Curvature 653–658

C^∗

-Algebras Associated with Presentations of Subshifts

Kengo Matsumoto

Received: May 28, 2001 Revised: March 4, 2002 Communicated by Joachim Cuntz

Abstract. Aλ-graph system is a labeled Bratteli diagram with an up- ward shift except the top vertices. We construct a continuous graph in the sense of V. Deaconu from a λ-graph system. It yields a Renault’s groupoid C^∗-algebra by following Deaconu’s construction. The class of theseC^∗-algebras generalize the class ofC^∗-algebras associated with sub- shifts and hence the class of Cuntz-Krieger algebras. They are unital, nuclear, uniqueC^∗-algebras subject to operator relations encoded in the structure of theλ-graph systems among generating partial isometries and projections. If theλ-graph systems are irreducible (resp. aperiodic), they are simple (resp. simple and purely infinite). K-theory formulae of these C^∗-algebras are presented so that we know an example of a simple and purely infinite C^∗-algebra in the class of these C^∗-algebras that is not stably isomorphic to any Cuntz-Krieger algebra.

2000 Mathematics Subject Classification: Primary 46L35, Secondary 37B10.

Keywords and Phrases: C^∗-algebras, subshifts, groupoids, Cuntz-Krieger algebras

1. Introduction

In [CK], J. Cuntz-W. Krieger have presented a class ofC^∗-algebras associated to finite square matrices with entries in {0,1}. The C^∗-algebras are simple if the matrices satisfy condition (I) and irreducible. They are also purely infinite if the matrices are aperiodic. There are many directions to general- ize the Cuntz-Krieger algebras (cf. [An],[De],[De2],[EL],[KPRR],[KPW],[Pi], [Pu],[T], etc.). The Cuntz-Krieger algebras have close relationships to topo- logical Markov shifts by Cuntz-Krieger’s observation in [CK]. Let Σ be a fi- nite set, and let σ be the shift on the infinite product space Σ^Z defined by

σ((xn)_n∈Z) = (xn+1)_n∈Z,(xn)_n∈Z ∈ Σ^Z. For a closed σ-invariant subset Λ of Σ^Z, the topological dynamical system Λ with σ is called a subshift. The topological Markov shifts form a class of subshifts. In [Ma], the author has generalized the class of Cuntz-Krieger algebras to a class ofC^∗-algebras associ- ated with subshifts. He has formulated several topological conjugacy invariants for subshifts by using the K-theory for theseC^∗-algebras ([Ma5]). He has also introduced presentations of subshifts, that are named symbolic matrix system andλ-graph system ([Ma5]). They are generalized notions of symbolic matrix andλ-graph (= labeled graph) for sofic subshifts respectively.

We henceforth denote byZ⁺andNthe set of all nonnegative integers and the set of all positive integers respectively. A symbolic matrix system (M, I) over a finite set Σ consists of two sequences of rectangular matrices (M^l,l+1, Il,l+1), l∈ Z⁺. The matrices M^l,l+1 have their entries in the formal sums of Σ and the matricesIl,l+1 have their entries in{0,1}. They satisfy the following relations (1.1) Il,l+1M^l+1,l+2=M^l,l+1Il+1,l+2, l∈Z⁺.

It is assumed for Il,l+1 that forithere existsj such that the (i, j)-component Il,l+1(i, j) = 1 and that forjthere uniquely existsisuch thatIl,l+1(i, j) = 1. A λ-graph systemL= (V, E, λ, ι) consists of a vertex set V =V0∪V1∪V2∪ · · ·, an edge set E = E0,1∪E1,2∪E2,3∪ · · ·, a labeling map λ : E → Σ and a surjective map ιl,l+1 : Vl+1 → Vl for each l ∈Z+. It naturally arises from a symbolic matrix system. For a symbolic matrix system (M, I), a labeled edge from a vertex v^l_i ∈Vl to a vertex v^l+1_j ∈Vl+1 is given by a symbol appearing in the (i, j)-component M^l,l+1(i, j) of the matrix M^l,l+1. The matrix Il,l+1

defines a surjection ιl,l+1 from Vl+1 to Vl for each l ∈ Z⁺. The symbolic matrix systems and the λ-graph systems are the same objects. They give rise to subshifts by looking the set of all label sequences appearing in the labeled Bratteli diagram (V, E, λ). A canonical method to construct a symbolic matrix system and a λ-graph system from an arbitrary subshift has been introduced in [Ma5]. The obtained symbolic matrix system and the λ-graph system are said to be canonical for the subshift. For a symbolic matrix system (M, I), let Al,l+1be the nonnegative rectangular matrix obtained from M^l,l+1 by setting all the symbols equal to 1 for each l ∈Z⁺. The resulting pair (A, I) satisfies the following relations from (1.1)

(1.2) Il,l+1Al+1,l+2=Al,l+1Il+1,l+2, l∈Z+. We call (A, I) the nonnegative matrix system for (M, I).

In the present paper, we introduce C^∗-algebras fromλ-graph systems. If a λ- graph system is the canonicalλ-graph system for a subshift Λ, theC^∗-algebra coincides with theC^∗-algebraO^Λassociated with the subshift. Hence the class of theC^∗-algebras in this paper generalize the class of Cuntz-Krieger algebras.

Let L= (V, E, λ, ι) be aλ-graph system over alphabet Σ. We first construct a continuous graph fromLin the sense of V. Deaconu ([D2],[De3],[De4]). We

then define theC^∗-algebraO^L associated withL as the Renault’sC^∗-algebra of a groupoid constructed from the continuous graph. For an edge e∈El,l+1, we denote bys(e)∈Vlandt(e)∈Vl+1its source vertex and its terminal vertex respectively. Let Λ^l be the set of all words of lengthl of symbols appearing in the labeled Bratteli diagram of L.We put Λ^∗ =∪^∞l=0Λ^l where Λ⁰ denotes the empty word. Let {v^l₁, . . . ,v^l_m(l)} be the vertex set Vl. We denote by Γ⁻_l (v^l_i) the set of all words in Λ^l presented by paths starting at a vertex of V0 and terminating at the vertexv^l_i.Lis said to be left-resolving if there are no distinct edges with the same label and the same terminal vertex. L is said to be predecessor-separated if Γ⁻_l (v^l_i) 6= Γ⁻_l (v^l_j) for distinct i, j and for all l ∈ N. Assume that L is left-resolving and satisfies condition (I), a mild condition generalizing Cuntz-Krieger’s condition (I). We then prove:

Theorem A (Theorem 3.6 and Theorem 4.3). Suppose that aλ-graph sys- temLsatisfies condition (I). Then theC^∗-algebraO^L is the universal concrete unique C^∗-algebra generated by partial isometries Sα, α ∈ Σ and projections E_i^l, i= 1,2, . . . , m(l), l∈Z⁺ satisfying the following operator relations:

X

α∈Σ

SαS_α^∗ = 1, (1.3)

m(l)X

i=1

E_i^l= 1, E_i^l=

m(l+1)

X

j=1

Il,l+1(i, j)E^l+1_j , (1.4)

SαS_α^∗E^l_i=E_i^lSαS_α^∗, (1.5)

S_α^∗E_i^lSα=

m(l+1)

X

j=1

Al,l+1(i, α, j)E_j^l+1, (1.6)

fori= 1,2, . . . , m(l), l∈Z⁺, α∈Σ, where Al,l+1(i, α, j) =

½ 1 if s(e) =v^l_i, λ(e) =α, t(e) =v^l+1_j for somee∈El,l+1, 0 otherwise,

Il,l+1(i, j) =

½ 1 if ιl,l+1(v^l+1_j ) =v^l_i, 0 otherwise

fori= 1,2, . . . , m(l), j= 1,2, . . . , m(l+ 1), α∈Σ.

IfLis predecessor-separated, the following relations:

E^l_i= Y

µ,ν∈Λ^l µ∈Γ⁻_l(v^l_i),ν6∈Γ⁻_l(v^l_i)

S_µ^∗Sµ(1−S^∗_νSν), l∈N, (1.7)

E_i⁰=X

α∈Σ m(1)X

j=1

A0,1(i, α, j)SαE_j¹S_α^∗

hold for i = 1,2, . . . , m(l), where Sµ = Sµ1· · ·Sµk for µ = (µ1, . . . , µk), µ1, . . . , µk ∈ Σ. In this case, O^L is generated by only the partial isometriesSα, α∈Σ.

IfLcomes from a finite directed graphG, the algebraO^Lbecomes the Cuntz- Krieger algebra O^AG associated to its adjacency matrix AG with entries in {0,1}.

We generalize irreducibility and aperiodicity for finite directed graphs to λ- graph systems. Then simplicity arguments of the Cuntz algebras in [C], the Cuntz-Krieger algebras in [CK] and theC^∗-algebras associated with subshifts in [Ma] are generalized to ourC^∗-algebrasO^L so that we have

Theorem B (Theorem 4.7 and Proposition 4.9). If L satisfies condi- tion (I) and is irreducible, the C^∗-algebra O^L is simple. In particular if L is aperiodic,O^L is simple and purely infinite.

There exists an actionαLof the torus groupT={z∈C| |z|= 1}on the alge- bra O^L that is called the gauge action. It satisfies αLz(Sα) = zSα, α ∈ Σ for z ∈ T. The fixed point subalgebra O^LαL of O^L under αL is an AF- algebra F^L, that is stably isomorphic to the crossed product O^Lo^αL T. Let (A, I) = (Al,l+1, Il,l+1)l∈Z+ be the nonnegative matrix system for the symbolic matrix system corresponding to the λ-graph system L. In [Ma5], its dimen- sion group (∆(A,I),∆⁺_(A,I), δ(A,I)),its Bowen-Franks groupsBFⁱ(A, I), i= 0,1 and its K-groups Ki(A, I), i= 0,1 have been formulated. They are related to topological conjugacy invariants of subshifts. The following K-theory formu- lae are generalizations of the K-theory formulae for the Cuntz-Krieger alge- bras and the C^∗-algebras associated with subshifts ([Ma2],[Ma4],[Ma5],[Ma6], cf.[C2],[C3],[CK]).

Theorem C (Proposition 5.3, Theorem 5.5 and Theorem 5.9).

(K0(F^L), K0(F^L)+,αcL∗)∼= (∆(A,I),∆⁺_(A,I), δ(A,I)), Ki(O^L)∼=Ki(A, I), i= 0,1, Extⁱ⁺¹(O^L)∼=BFⁱ(A, I), i= 0,1 whereαcL denotes the dual action of the gauge action αL onO^L.

We know that the C^∗-algebra O^L is nuclear and satisfies the Universal Coef- ficient Theorem (UCT) in the sense of Rosenberg and Schochet (Proposition 5.7)([RS], cf. [Bro2]). Hence, if L is aperiodic,O^L is a unital, separable, nu- clear, purely infinite, simple C^∗-algebra satisfying the UCT, that lives in a classifiable class by K-theory of E. Kirchberg [Kir] and N. C. Phillips [Ph]. By Rørdam’s result [Rø;Proposition 6.7], one sees that O^L is isomorphic to the C^∗-algebra of an inductive limit of a sequenceB1→B2→B3→ · · · of simple Cuntz-Krieger algebras (Corollary 5.8).

We finally present an example of a λ-graph system for which the associated C^∗-algebra is not stably isomorphic to any Cuntz-Krieger algebraO^Aand any Cuntz-algebraOⁿ forn= 2,3, . . . ,∞. The example is aλ-graph systemL(S) constructed from a certain Shannon graphS (cf.[KM]). We obtain

Theorem D (Theorem 7.7). TheC^∗-algebraOL(S) is unital, simple, purely infinite, nuclear and generated by five partial isometries with mutually orthog- onal ranges. Its K-groups are

K0(O^L(S)) = 0, K1(O^L(S)) =Z.

In [Ma7], among other things, relationships between ideals of O^L and sub λ- graph systems ofLare studied so that the class ofC^∗-algebras associated with λ-graph systems is closed under quotients by its ideals.

Acknowledgments: The author would like to thank Yasuo Watatani for his suggestions on groupoidC^∗-algebras andC^∗-algebras of HilbertC^∗-bimodules.

The author also would like to thank the referee for his valuable suggestions and comments for the presentation of this paper.

2. Continuous graphs constructed from λ-graph systems We will construct Deaconu’s continuous graphs from λ-graph systems. They yield Renault’s r-discrete groupoidC^∗-algebras by Deaconu ([De],[De2],[De3]).

Following V. Deaconu in [De3], by a continuous graph we mean a closed subset E ofV ×Σ× V where V is a compact metric space and Σ is a finite set. If in particularV is zero-dimensional, that is, the set of all clopen sets form a basis of the open sets, we sayE to be zero-dimensional or Stonean.

Let L= (V, E, λ, ι) be aλ-graph system over Σ with vertex set V =∪^l∈Z⁺Vl

and edge setE=∪^l∈Z⁺El,l+1that is labeled with symbols in Σ byλ:E→Σ, and that is supplied with surjective maps ι(= ιl,l+1) : Vl+1 → Vl for l ∈Z⁺. Here the vertex sets Vl, l∈Z⁺ are finite disjoint sets. Also El,l+1, l∈Z⁺ are finite disjoint sets. An edgeein El,l+1 has its source vertexs(e) in Vland its terminal vertext(e) inVl+1respectively. Every vertex inV has a successor and every vertex in Vl forl ∈ Nhas a predecessor. It is then required that there exists an edge in El,l+1 with labelα and its terminal isv ∈Vl+1 if and only if there exists an edge inE_l−1,l with labelαand its terminal is ι(v)∈Vl. For u∈V_l−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 each pair of vertices u, v. We call this property the local property of L. Let ΩL be the projective limit of the system ιl,l+1 : Vl+1 → Vl, l∈Z⁺,that is defined by

ΩL={(v^l)_l∈Z+ ∈ Y

l∈Z⁺

Vl |ιl,l+1(v^l+1) =v^l, l∈Z+}.

We endow ΩL with the projective limit topology so that it is a compact Haus- dorff space. An elementvin ΩLis called anι-orbit or also a vertex. LetELbe

the set of all triplets (u, α, v)∈ΩL×Σ×ΩL such that for eachl ∈Z⁺, there existsel,l+1∈El,l+1 satisfying

u^l=s(el,l+1), v^l+1=t(el,l+1) and α=λ(el,l+1) whereu= (u^l)l∈Z⁺, v= (v^l)l∈Z⁺∈ΩL.

Proposition 2.1. The setEL⊂ΩL×Σ×ΩLis a zero-dimensional continuous graph.

Proof. It suffices to show thatEL is closed. For (u, β, v)∈ΩL×Σ×ΩL with (u, β, v) 6∈ EL, one finds l ∈ N such that there does not exist any edge e in El,l+1 withs(e) =u^l, t(e) =v^l+1 andλ(e) =β. Put

Uu^l ={(wⁱ)i∈Z⁺∈ΩL|w^l=u^l}, Uv^l+1 ={(wⁱ)i∈Z⁺∈ΩL|w^l+1=v^l+1}. They are open sets in ΩL. Hence U_u^l× {β} ×U_v^l+1 is an open neighborhood of (u, β, v) that does not intersect withEL so thatEL is closed. ¤

We denote by{v^l₁, . . . ,v^l_m(l)} the vertex setVl.Put forα∈Σ, i= 1, . . . , m(1) U_i¹(α) ={(u, α, v)∈EL|v¹=v¹_i wherev= (v^l)_l∈Z+∈ΩL}.

ThenU_i¹(α) is a clopen set inEL such that

∪α∈Σ∪^m(1)i=1 U_i¹(α) =EL, U_i¹(α)∩U_j¹(β) =∅ if (i, α)6= (j, β).

Put t(u, α, v) = v for (u, α, v) ∈ EL. Suppose that L is left-resolving. It is easy to see that if U_i¹(α) 6= ∅, the restriction of t to U_i¹(α) is a homeomor- phism onto U_v¹

i ={(v^l)_l∈Z+ ∈ ΩL | v¹ =v¹_i}. Hence t : EL → ΩL is a local homeomorphism.

Following Deaconu [De3], we consider the setXL of all one-sided paths ofEL: XL={(αi, ui)^∞_i=1∈

Y∞ i=1

(Σ×ΩL)|(ui, αi+1, ui+1)∈EL for alli∈N and (u0, α1, u1)∈EL for someu0∈ΩL}. The set XL has the relative topology from the infinite product topology of Σ×ΩL. It is a zero-dimensional compact Hausdorff space. The shift map σ: (αi, ui)^∞_i=1∈XL →(αi+1, ui+1)^∞_i=1∈XL is continuous. Forv= (v^l)l∈Z+∈ ΩL and α ∈ Σ, the local property of L ensures that if there exists e0,1 ∈ E0,1 satisfying v¹ = t(e0,1), α = λ(e0,1), there exist el,l+1 ∈ El,l+1 and u = (u^l)l∈Z⁺ ∈ΩL satisfying u^l =s(el,l+1), v^l+1 =t(el,l+1), α =λ(el,l+1) for each l∈Z⁺.Hence ifLis left-resolving, for anyx= (αi, vi)^∞_i=1∈XL, there uniquely exists v0∈ΩL such that (v0, α1, v1)∈EL. Denote byv(x)0 the unique vertex v0 forx∈XL.

Lemma 2.2. For aλ-graph system L, consider the following conditions (i) Lis left-resolving.

(ii) EL is left-resolving, that is, for (u, α, v),(u⁰, α, v⁰)∈EL,the condition v=v⁰ impliesu=u⁰.

(iii) σis a local homeomorphism on XL. Then we have

(i)⇔(ii)⇒(iii).

Proof. The implications (i)⇔(ii) are direct. We will see that (ii)⇒(iii).Sup- pose thatLis left-resolving. Let{γ1, . . . , γm}= Σ be the list of the alphabet.

Put

XL(k) ={(αi, vi)^∞_i=1∈XL|α1=γk}

that is a clopen set of XL. Since the familyXL(k), k = 1, . . . , m is a disjoint covering ofXL and the restriction ofσto each of themσ|^XL(k):XL(k)→XL

is a homeomorphism, the continuous surjectionσis a local homeomorphism on XL. ¤

Remark. We will remark that a continuous graph coming from a left-resolving, predecessor-separatedλ-graph system is characterized as in the following way.

LetE ⊂ V ×Σ× Vbe a continuous graph. Following [KM], we define thel-past context ofv∈ V as follows:

Γ⁻_l (v) ={(α1, . . . , αl)∈Σ^l | ∃v0, v1, . . . , vl−1∈ V;

(v_i−1, αi, vi)∈ E, i= 1,2, . . . , l−1,(v_l−1, αl, v)∈ E}. We sayE to be predecessor-separated if for two verticesu, v ∈ V, there exists l ∈ N such that Γ⁻_l (u) 6= Γ⁻_l (v). The following proposition can be directly proved by using an idea of [KM]. Its result will not be used in our further discussions so that we omit its proof.

Proposition 2.3. Let E ⊂ V ×Σ× V be a zero-dimensional continuous graph such thatEis left-resolving, predecessor-separated. If the mapt:E → Vdefined byt(u, α, v) =vis a surjective open map, there exists aλ-graph systemL^E over Σ and a homeomorphismΦ from V onto ΩL^E such that the map Φ×id×Φ: V ×Σ× V →ΩL^E ×Σ×ΩL^E satisfies(Φ×id×Φ)(E) =EL^E.

3. The C^∗-algebraO^L.

In what follows we assume L to be left-resolving. Following V. Deaconu [De2],[De3],[De4], one may construct a locally compact r-discrete groupoid from a local homeomorphismσonXL as in the following way (cf. [An],[Re]). Set

GL={(x, n, y)∈XL×Z×XL | ∃k, l≥0; σ^k(x) =σ^l(y), n=k−l}. The range map and the domain map are defined by

r(x, n, y) =x, d(x, n, y) =y.

The multiplication and the inverse operation are defined by

(x, n, y)(y, m, z) = (x, n+m, z), (x, n, y)⁻¹= (y,−n, x).

The unit spaceG⁰_L is defined to be the spaceXL={(x,0, x)∈GL| x∈XL}. A basis of the open sets forGL is given by

Z(U, V, k, l) ={(x, k−l,(σ^l|^V)⁻¹◦(σ^k(x))∈GL |x∈U}

where U, V are open sets ofXL, and k, l∈Nare such that σ^k|^U andσ^l|^V are homeomorphisms with the same open range. Hence we see

Z(U, V, k, l) ={(x, k−l, y)∈GL |x∈U, y∈V, σ^k(x) =σ^l(y)}. The groupoid C^∗-algebra C^∗(GL) for the groupoid GL is defined as in the following way ([Re], cf. [An],[De2],[De3],[De4]). Let Cc(GL) be the set of all continuous functions onGL with compact support that has a natural product structure of∗-algebra given by

(f∗g)(s) = X

t∈GL, r(t)=r(s)

f(t)g(t⁻¹s) = X

t1,t2∈GL, s=t1t2

f(t1)g(t2),

f^∗(s) =f(s⁻¹), f, g∈Cc(GL), s∈GL.

Let C0(G⁰_L) be the C^∗-algebra of all continuous functions onG⁰_L that vanish at infinity. The algebraCc(GL) is aC0(G⁰_L)-module, endowed with aC0(G⁰_L)- valued inner product by

(ξf)(x, n, y) =ξ(x, n, y)f(y), ξ∈Cc(GL), f ∈C0(G⁰_L), (x, n, y)∈GL,

< ξ, η >(y) = X

x,n (x,n,y)∈GL

ξ(x, n, y)η(x, n, y), ξ, η∈Cc(GL), y∈XL.

Let us denote by l²(GL) the completion of the inner productC0(G⁰_L)-module Cc(GL). It is a Hilbert C^∗-right module over the commutative C^∗-algebra C0(G⁰_L). We denote by B(l²(GL)) the C^∗-algebra of all bounded adjointable C0(G⁰_L)-module maps on l²(GL). Let π be the ∗-homomorphism of Cc(GL) into B(l²(GL)) defined by π(f)ξ=f ∗ξ for f, ξ ∈Cc(GL). Then the closure ofπ(Cc(GL)) in B(l²(GL)) is called the (reduced)C^∗-algebra of the groupoid GL, that we denote byC^∗(GL).

Definition. The C^∗-algebraO^L associated with λ-graph system L is defined to be theC^∗-algebraC^∗(GL)of the groupoid GL.

We will study the algebraic structure of the C^∗-algebra O^L. Recall that Λ^k denotes the set of all words of Σ^k that appear inL. Forx= (αn, un)^∞_n=1∈XL, we put λ(x)n = αn ∈ Σ, v(x)n = un ∈ ΩL respectively. The ι-orbit v(x)n

is written as v(x)n = (v(x)^l_n)_l∈Z+ ∈ ΩL = lim←−Vl. Now L is left-resolving so that there uniquely exists v(x)0 ∈ΩL satisfying (v(x)0, α1, u1) ∈EL. Set for µ= (µ1, . . . , µk)∈Λ^k,

U(µ) ={(x, k, y)∈GL|σ^k(x) =y, λ(x)1=µ1, . . . , λ(x)k =µk} and forv^l_i∈Vl,

U(v^l_i) ={(x,0, x)∈GL |v(x)^l₀=v^l_i}

wherev(x)0= (v(x)^l₀)l∈Z+∈ΩL.They are clopen sets ofGL. We set Sµ =π(χU(µ)), E^l_i=π(χ_U(v^l

i)) in π(Cc(GL))

where χF ∈ Cc(GL) denotes the characteristic function of a clopen set F on the spaceGL.Then it is straightforward to see the following lemmas.

Lemma 3.1.

(i) Sµ is a partial isometry satisfying Sµ = Sµ1· · ·Sµk, where µ = (µ1, . . . , µk)∈Λ^k.

(ii) P

µ∈Λ^kSµS_µ^∗= 1 fork∈N.We in particular have

(3.1) X

α∈Σ

SαS_α^∗ = 1.

(iii) E_i^l is a projection such that (3.2)

m(l)X

i=1

E_i^l= 1, E_i^l=

m(l+1)

X

j=1

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

where Il,l+1 is the matrix defined in Theorem A in Section 1, corre- sponding to the mapιl,l+1:Vl+1→Vl.

Take µ= (µ1, . . . , µk)∈Λ^k, ν = (ν1, . . . , νk⁰)∈Λ^k0 and v^l_i ∈Vl withk, k⁰≤l such that there exist paths ξ, η in L satisfyingλ(ξ) =µ, λ(η) = ν andt(ξ) = t(η) =v^l_i.We set

U(µ,v^l_i, ν) ={(x, k−k⁰, y)∈GL|σ^k(x) =σ^k0(y), v(x)^l_k=v(y)^l_k0 =v^l_i, λ(x)1=µ1, . . . , λ(x)k =µk, λ(y)1=ν1, . . . , λ(y)k⁰ =νk⁰ }. The setsU(µ,v^l_i, ν), µ∈Λ^k, ν∈Λ^k0, i= 1, . . . , m(l) are clopen sets and gener- ate the topology ofGL.

Lemma 3.2.

SµE_i^lS_ν^∗=π(χ_U(µ,v^l

i,ν))∈π(Cc(GL)).

Hence theC^∗-algebraO^L is generated bySα, α∈ΣandE^l_i, i= 1, . . . , m(l), l∈ Z⁺.

The generatorsSα, E_i^lsatisfy the following operator relations, that are straight- forwardly checked.

Lemma 3.3.

SαS_α^∗E^l_i=E_i^lSαS_α^∗, (3.3)

S^∗_αE_i^lSα=

m(l+1)

X

j=1

Al,l+1(i, α, j)E^l+1_j , (3.4)

forα∈Σ, i= 1,2, . . . , m(l), l∈Z⁺,whereAl,l+1(i, α, j)is defined in Theorem A in Section 1.

The four operator relations (3.1),(3.2),(3.3),(3.4) are called the relations (L).

LetA^l, l∈Z⁺be theC^∗-subalgebra ofO^Lgenerated by the projectionsE_i^l, i= 1, . . . , m(l), that is,

A^l=CE₁^l⊕ · · · ⊕CE_m(l)^l .

The projections S_α^∗Sα, α ∈ Σ and S_µ^∗Sµ, µ ∈ Λ^k, k ≤ l belong to A^l, l ∈ N by (3.4) and the first relation of (3.2). Let A^L be the C^∗-subalgebra of O^L generated by all the projections E_i^l, i = 1, . . . , m(l), l ∈ Z⁺. By the second relation of (3.2), the algebraA^lis naturally embedded inA^l+1 so thatA^Lis a commutative AF-algebra. We note that there exists an isomorphism between A^l and C(Vl) for eachl ∈Z+ that is compatible with the embeddings A^l ,→ A^l+1andI_l,l+1^t (=ι^∗_l,l+1) :C(Vl),→C(Vl+1).Hence there exists an isomorphism betweenAL andC(ΩL). Letk, l be natural numbers withk≤l. We set

D^L=TheC^∗-subalgebra ofO^L generated bySµaS_µ^∗, µ∈Λ^∗, a∈ A^L. Fk^l =TheC^∗-subalgebra ofO^L generated bySµaS_ν^∗, µ, ν∈Λ^k, a∈ A^l. Fk^∞=TheC^∗-subalgebra ofO^L generated bySµaS_ν^∗, µ, ν∈Λ^k, a∈ A^L.

F^L=TheC^∗-subalgebra ofO^L generated bySµaS_ν^∗, µ, ν∈Λ^∗,

|µ|=|ν|, a∈ A^L. The algebraD^L is isomorphic to C(XL). It is obvious that the algebra Fk^l is finite dimensional and there exists an embeddingιl,l+1 :Fk^l ,→ Fk^l+1 through the preceding embeddingA^l ,→ A^l+1. Define a homomorphismc : (x, n, y)∈ GL→n∈Z.We denote byFL the subgroupoidc⁻¹(0) ofGL. LetC^∗(FL) be its groupoidC^∗-algebra. It is also immediate that the algebraF^Lis isomorphic toC^∗(FL). By (3.1),(3.3),(3.4), the relations:

E_i^l=X

α∈Σ m(l+1)

X

j=1

Al,l+1(i, α, j)SαE_j^l+1S_α^∗, i= 1,2, . . . , m(l) hold. They yield

SµE_i^lS_ν^∗=X

α∈Σ m(l+1)

X

j=1

Al,l+1(i, α, j)SµαE_j^l+1S^∗_να forµ, ν ∈Λ^k, that give rise to an embedding Fk^l ,→ Fk+1^l+1. It induces an embedding ofFk^∞

intoFk+1^∞ that we denote byλk,k+1.

Proposition 3.4.

(i) Fk^∞ is an AF-algebra defined by the inductive limit of the embeddings ιl,l+1:Fk^l ,→ Fk^l+1, l∈N.

(ii) F^L is an AF-algebra defined by the inductive limit of the embeddings λk,k+1:Fk^∞,→ Fk+1^∞ , k∈Z⁺.

Let Uz, z ∈T ={z ∈ C| |z| = 1} be an action of T to the unitary group of B(l²(GL)) defined by

(Uzξ)(x, n, y) =zⁿξ(x, n, y) for ξ∈l²(GL),(x, n, y)∈GL.

The actionAd(Uz) onB(l²(GL)) leavesO^L globally invariant. It gives rise to an action on O^L. We denote it by αL and call it the gauge action. LetEL

be the expectation from O^L onto the fixed point subalgebra O^LαL under αL

defined by

(3.5) EL(X) =

Z

z∈TαLz(X)dz, X ∈ O^L.

Let P^L be the ∗-algebra generated algebraically by Sα, α ∈ Σ and E_i^l, i = 1, . . . , m(l), l ∈ Z⁺. For µ = (µ1, . . . , µk) ∈ Λ^k, it follows that by (3.3), E_i^lSµ1· · ·Sµk =Sµ1S^∗_µ1E_i^lSµ1· · ·Sµk. As S_µ^∗₁E_i^lSµ1 is a linear combination of E_j^l+1, j= 1, . . . , m(l+1) by (3.4), one seesS_µ^∗₁E^l_iSµ1Sµ2 =Sµ2S_µ^∗₂S_µ^∗₁E_i^lSµ1Sµ2

and inductively

(3.6) E_i^lSµ=SµS_µ^∗E_i^lSµ, E^l_iSµS_µ^∗ =SµS^∗_µE_i^l.

By the relations (3.6), each element X∈ P^L is expressed as a finite sum X = X

|ν|≥1

X₋νS_ν^∗+X0+ X

|µ|≥1

SµXµ for some X₋ν, X0, Xµ∈ F^L.

Then the following lemma is routine.

Lemma 3.5. The fixed point subalgebra O^LαL of O^L under αL is the AF- algebra F^L.

We can now prove a universal property ofO^L.

Theorem 3.6. The C^∗-algebraO^L is the universal C^∗-algebra subject to the relations (L).

Proof. Let O^[L] be the universal C^∗-algebra generated by partial isometries sα, α ∈ Σ and projections e^l_i, i = 1, . . . , m(l), l ∈ Z⁺ subject to the op- erator relations (L). This means that O^[L] is generated by sα, α ∈ Σ and e^l_i, i = 1, . . . , m(l), l ∈ Z+, that have only operator relations (L). The C^∗- norm of O^[L] is given by the universal C^∗-norm. Let us denote by F[k]^[l],F^[L]

the similarly defined subalgebras of O[L] to Fk^l,F^L respectively. The algebra F[k]^[l] as well asFk^l is a finite dimensional algebra. Since sµe^l_is^∗_ν6= 0 if and only ifSµE^l_iS_ν^∗ 6= 0,the correspondence sµe^l_is^∗_ν →SµE_i^lS_ν^∗,|µ|=|ν|=k≤l yields an isomorphism fromF[k]^[l] to Fk^l. It induces an isomorphism fromF^[L] to F^L. By the universality, for z ∈C,|z| = 1 the correspondence sα →zsα, α ∈Σ, e^l_i → e^l_i, i = 1, . . . , m(l), l ∈ Z⁺ gives rise to an action of the torus group T on O[L], which we denote by α[L]. Let E[L] be the expectation from O[L]

onto the fixed point subalgebra O[L]α[L] under α[L] similarly defined to (3.5).

The algebra O[L]α_[L]

is nothing but the algebra F[L]. By the universality of O^[L], the correspondence sα → Sα, α ∈ Σ, e^l_i → E_i^l, i = 1, . . . , m(l), l ∈ Z⁺ extends to a surjective homomorphism from O^[L] to O^L, which we denote by πL. The restriction ofπLtoF^[L] is the preceding isomorphism. As we see that EL◦πL=πL◦E[L] andE[L] is faithful, we conclude thatπL is isomorphic by a similar argument to [CK; 2.12. Proposition]. ¤

4. Uniqueness and simplicity

We will prove thatO^Lis the uniqueC^∗-algebra subject to the operator relations (L) under a mild condition onL,called (I). The condition (I) is a generalization of condition (I) for a finite square matrix with entries in{0,1}defined by Cuntz- Krieger in [CK] and condition (I) for a subshift defined in [Ma4]. A related condition for a HilbertC^∗-bimodule has been introduced by Kajiwara-Pinzari- Watatani in [KPW]. For an infinite directed graph, such a condition is defined by Kumjian-Pask-Raeburn-Renault in [KPRR]. For a vertexv^l_i∈Vl, let Γ⁺(v^l_i) be the set of all label sequences in Lstarting atv^l_i. That is,

Γ⁺(v^l_i) ={(α1, α2, . . . ,)∈Σ^N | ∃en,n+1∈En,n+1 forn=l, l+ 1, . . .; v^l_i=s(el,l+1), t(en,n+1) =s(en+1,n+2), λ(en,n+1) =αn−l+1}. Definition. Aλ-graph systemLsatisfies condition (I) if for eachv^l_i∈V,the setΓ⁺(v^l_i)contains at least two distinct sequences.

Forv^l_i∈VlsetF_i^l={x∈XL |v(x)^l₀=v^l_i} wherev(x)0= (v(x)^l₀)_l∈Z+∈ΩL= lim←−Vl is the uniqueι-orbit forx∈XL such that (v(x)0, λ(x)1, v(x)1)∈EL as in the preceding section. By a similar discussion to [Ma4; Section 5] (cf.[CK;

2.6.Lemma]), we know that ifLsatisfies (I), forl, k∈Nwithl≥k, there exists y^l_i∈F_i^lfor each i= 1,2, . . . , m(l) such that

σ^m(y_i^l)6=y_j^l for all 1≤i, j≤m(l), 1≤m≤k.

By the same manner as the proof of [Ma4;Lemma 5.3], we obtain

Lemma 4.1. Suppose that L satisfies condition (I). Then for l, k ∈ N with l≥k, there exists a projectionq_k^l ∈ D^L such that

(i) q_k^la6= 0for all nonzeroa∈ A^l,

(ii) q_k^lφ^m_L(q_k^l) = 0for all m= 1,2, . . . , k,whereφ^m_L(X) =P

µ∈Λ^mSµXS_µ^∗.

Now we put Q^l_k = φ^k_L(q^l_k) a projection in D^L. Note that each element of D^L commutes with elements of A^L. As we see Sµφ^j_L(X) = φ^j+|µ|_L (X)Sµ for X ∈ D^L, j ∈Z+, µ∈Λ^∗, a similar argument to [CK;2.9.Proposition] leads to the following lemma.

Lemma 4.2.

(i) The correspondence: X ∈ Fk^l −→ Q^l_kXQ^l_k ∈ Q^l_kFk^lQ^l_k extends to an isomorphism fromFk^l ontoQ^l_kFk^lQ^l_k.

(ii) Q^l_kX−XQ^l_k →0, kQ^l_kXk → kXkask, l→ ∞ forX ∈ F^L. (iii) Q^l_kSµQ^l_k, Q^l_kS_µ^∗Q^l_k →0as k, l→ ∞ forµ∈Λ^∗.

We then prove the uniqueness of the algebraO^L subject to the relations (L).

Theorem 4.3. Suppose that L satisfies condition (I). Let Sbα, α ∈ Σ and Eb_i^l, i= 1,2, . . . , m(l), l∈Z⁺be another family of nonzero partial isometries and nonzero projections satisfying the relations(L). Then the mapSα→Sbα, α∈Σ, E_i^l→Eb_i^l, i= 1, . . . , m(l), l∈Z⁺ extends to an isomorphism fromO^L onto the C^∗-algebraOb^L generated by Sbα, α∈ΣandEb_i^l, i= 1, . . . , m(l), l∈Z⁺.

Proof. We may define C^∗-subalgebras Db^L,Fbk^l,Fb^L of Ob^L by using the ele- mentsSbµEb_i^lSb_ν^∗by the same manners as the constructions of theC^∗-subalgebras D^L,Fk^l,F^L of O^L respectively. As in the proof of Theorem 3.6, the map SµE_i^lS_ν^∗ ∈ Fk^l → SbµEb_i^lSb_ν^∗ ∈Fbk^l,|µ| =|ν| =k ≤l extends to an isomorphism from the AF-algebraF^Lonto the AF-algebraFb^L.By Theorem 3.6, the algebra O^L has a universal property subject to the relations (L) so that there exists a surjective homomorphism ˆπ from O^L onto Ob^L satisfying ˆπ(Sα) = Sbα and ˆ

π(E_i^l) =Eb_i^l.The restriction of ˆπto F^L is the preceding isomorphism onto Fb^L. NowLsatisfies (I). LetQ^l_kbe the sequence of projections as in Lemma 4.2. We putQb^l_k= ˆπ(Q^l_k)∈DbLthat has the corresponding properties to Lemma 4.2 for the algebra Fb^L. LetPb^L be the ∗-algebra generated algebraically by Sbα, α∈Σ and Eb^l_i, i= 1, . . . , m(l), l∈Z⁺. By the relations (L), each element X ∈Pb^L is expressed as a finite sum

X = X

|ν|≥1

X_−νSb_ν^∗+X0+ X

|µ|≥1

SbµXµ for some X_−ν, X0, Xµ∈Fb^L. By a similar argument to [CK;2.9.Proposition], it follows that the map X ∈ Pb^L→X0∈Fb^L extends to an expectationEbL from Ob^LontoFb^L,that satisfies EbL◦πˆ= ˆπ◦EL.As ELis faithful, we conclude that ˆπis isomorphic. ¤ Remark. Letexbe a vector assigned tox∈XL. LetH_L be the Hilbert space spanned by the vectors ex, x ∈XL such that the vectorsex, x∈ XL form its complete orthonormal basis. For x= (αi, vi)^∞_i=1 ∈XL, take v0=v(x)0∈ΩL. For a symbolβ ∈Σ, if there exists a vertexv₋₁∈ΩL such that (v₋₁, β, v0)∈ EL,we defineβx∈XL, by puttingα0=β, as

βx= (αi−1, vi−1)^∞_i=1∈XL.

Put

Γ⁻₁(x) ={γ∈Σ|(v₋₁, γ, v(x)0)∈ELfor some v₋₁∈ΩL}. We define the creation operatorsSeβ, β∈Σ onH_L by

Seβex=

½eβx ifβ ∈Γ⁻₁(x), 0 ifβ 6∈Γ⁻₁(x).

Proposition 4.4. Suppose that Lsatisfies condition (I). If Lis predecessor- separated, O^L is isomorphic to the C^∗-algebraC^∗(Seβ, β ∈Σ)generated by the partial isometries Seβ, β∈Σon the Hilbert space HL.

Proof. Suppose thatL is predecessor-separated. Define a sequence of projec- tionsEe_i^l, i= 1, . . . , m(l), l∈N and Ee_i⁰, i= 1, . . . , m(0) by using the formulae (1.7) from the partial isometries Seβ, β ∈ Σ. It is straightforward to see that Ee_i^l, i = 1, . . . , m(l), l ∈ Z+ are nonzero. The partial isometries Seβ and the projectionsEe_i^l satisfy the relations (L). ¤

Let Λ be a subshift andL^Λits canonical λ-graph system, that is left-resolving and predecessor-separated. It is easy to see that Λ satisfies condition (I) in the sense of [Ma4] if and only if L^Λ satisfies condition (I).

Corollary 4.5(cf.[Ma],[CaM]). The C^∗-algebra OL^Λ associated with λ- graph system L^Λ is canonically isomorphic to the C^∗-algebra O^Λ associated with subshift Λ.

We next refer simplicity and purely infiniteness of the algebraO^L. We introduce the notions of irreducibility and aperiodicity forλ-graph system

Definition.

(i) Aλ-graph systemL is said to be irreducible if for a vertex v∈Vl and x= (x1, x2, . . .)∈ ΩL = lim

←−Vl, there exists a path in L starting atv and terminating atxl+N for someN ∈N.

(ii) Aλ-graph system Lis said to be aperiodic if for a vertex v ∈Vl there exists an N ∈ N such that there exist paths in L starting at v and terminating at all the vertices ofVl+N.

Aperiodicity automatically implies irreducibility. Define a positive operatorλL

onA^L by

λL(X) =X

α∈Σ

S_α^∗XSα for X ∈ A^L.

We say thatλLisirreducibleif there exists no non-trivial ideal ofA^L invariant under λL, andλL isaperiodic if for a projectionE_i^l ∈ A^l there exists N ∈ N such thatλ^N_L(E_i^l)≥1. The following lemma is easy to prove (cf.[Ma4]).