Representations of Cuntz algebras associated to random walks on graphs
aa r X i v : . [ m a t h . OA ] S e p REPRESENTATIONS OF CUNTZ ALGEBRAS ASSOCIATED TO RANDOMWALKS ON GRAPHS
DORIN DUTKAY AND NICHOLAS CHRISTOFFERSEN
Abstract.
Motivated by the harmonic analysis of self-affine measures, we introduce a class ofrepresentations of the Cuntz algebra associated to random walks on graphs. The representationsare constructed using the dilation theory of row coisometries. We study these representations, theircommutant and the intertwining operators.
Contents
1. Introduction 12. A motivation from harmonic analysis 53. General results 104. An explicit construction of the Cuntz dilation associated to a random walk 125. Intertwining operators 166. Examples 22References 291.
Introduction
Definition 1.1.
Let Λ be a finite set of cardinality | Λ | = N , N ∈ N , N ≥
2. A representationof the
Cuntz algebra O N , is a family of N isometries ( S λ ) λ ∈ Λ on a Hilbert space H , such that theisometries have mutually orthogonal ranges whose sum is the entire space H . These properties areexpressed in the Cuntz relations(1.1) S ∗ λ S λ ′ = δ λλ ′ I H , ( λ, λ ′ ∈ Λ) , X λ ∈ Λ S λ S ∗ λ = I H . The Cuntz algebra was introduced in [Cun77] as a simple, purely infinite C ∗ -algebra. Therepresentation theory of Cuntz algebras is extremely rich, unclassifiable even [Gli60, Gli61]. Therepresentations of Cuntz algebras have been proved to have applications in mathematical physics[BJ02, Bur04, GN07, AK08, Kaw03, Kaw06, Kaw09, KHL09, JP11, AJLM13, AJLV16, JT20], inwavelets [BEJ00, Jor04, Jor06b, DJ08, DJ07b, Jor06a, Jor01], harmonic analysis [DHJ09, DJ07a,DJ12], and in fractal geometry [DJ06, DJ11].We will introduce here a class of representations of the Cuntz algebra, associated to a randomwalk. The construction will follow two major steps: from a random walk, one constructs a row Mathematics Subject Classification.
Key words and phrases.
Cuntz algebras, random walks. coisometry; then, from the row coisometry, using dilation theory, one obtains a representation ofthe Cuntz algebra.
Definition 1.2.
A family of operators ( V λ ) λ ∈ Λ on a Hilbert space K is called a row coisometry , if(1.2) X λ ∈ Λ V λ V ∗ λ = I K . It is known [Pop89, BJKW00] that every row coisometry can be dilated to a representation ofthe Cuntz algebra. More precisely
Theorem 1.3. [BJKW00, Theorem 5.1]
Let K be a Hilbert space and ( V λ ) λ ∈ Λ be a row coisometryon K . Then K can be embedded into a larger Hilbert space H = H V carrying a representation ( S λ ) λ ∈ Λ of the Cuntz algebra O N such that, if P : H → K is the projection onto K , we have (1.3) V ∗ λ = S ∗ λ P, (i.e., S ∗ λ K ⊂ K and S ∗ λ P = P S ∗ λ P = V ∗ λ ) and K is cyclic for the representation.The system ( H , ( S λ ) λ ∈ Λ , P ) is unique up to unitary equivalence, and if σ : B ( K ) → B ( K ) isdefined by (1.4) σ ( A ) = X λ ∈ Λ V λ AV ∗ λ , then the commutant of the representation { S λ } ′ λ ∈ Λ is isometrically order isomorphic to the fixedpoint set B ( K ) σ = { A ∈ B ( K ) : σ ( A ) = A } , by the map A ′ P A ′ P . More generally, if ( W λ ) λ ∈ Λ is another row coisometry on the space K ′ , and ( T λ ) λ ∈ Λ is the corresponding Cuntz dilation, thenthere exists an isometric linear isomorphism between the intertwiners U : H V → H W , i.e., operatorssatisfying (1.5) U S λ = T λ U, and operators V ∈ B ( K , K ′ ) such that (1.6) X λ ∈ Λ W λ V V ∗ λ = V, given by the map U V = P K ′ U P K . Definition 1.4.
Given a row coisometry ( V λ ) λ ∈ Λ on the Hilbert space K , we call the representation( S λ ) λ ∈ Λ of the Cuntz algebra O | Λ | in Theorem 1.3, the Cuntz dilation of the row coisometry ( V λ ) λ ∈ Λ . Definition 1.5.
Let G = ( V , E ) be a directed graph, where the set of vertices V is finite or countable.For each edge e ∈ E we assume we have a label λ ( e ) chosen from a finite set of labels Λ, | Λ | = N .We assume in addition, that given a vertex i , two different edges e = e from i have differentlabels λ ( e ) = λ ( e ) and their end vertices are different. We use the notation i λ → j to indicatethat there is an edge from i to j with label λ , and, in this case we also write i · λ = j . Thus, foreach vertex i , there is at most one edge leaving i with label λ .We also assume that for each vertex i there is at most one edge coming into the vertex i withlabel λ . UNTZ ALGEBRAS AND RANDOM WALKS 3
For each vertex i and each label λ we assume that we have an associated complex number α i,λ , α i,λ = 0 in case there is no edge from i with label λ , and we assume that(1.7) X λ ∈ Λ | α i,λ | = 1 , ( i ∈ V ) . Thus, we have a random walk on the graph G , the probability of transition from i to i · λ withlabel λ being | α i,λ | .Here is the way we define the representation of the Cuntz algebra O N .First, we define the row coisometry ( V λ ) λ ∈ Λ on K := ℓ [ V ],(1.8) V ∗ λ ( ~i ) = ( α i,λ ~j, if i λ → j , otherwise.(Here, the vertices i in V are considered as the canonical basis vectors ~i := δ i for l [ V ]).Then, we use Theorem 1.3, to construct the Cuntz dilation ( S λ ) λ ∈ Λ . We call it the representationof the Cuntz algebra associated to the random walk .The paper is structured as follows: in section 2, we present some motivation from the harmonicanalysis of self-affine measures and we show how our representations of the Cuntz algebra appearin that context. In section 3, we present some general properties of the Cuntz representationsobtained from dilations of row coisometries. Of particular interest, it is the following fact: thecommutant of the Cuntz dilation is an algebra, it has a product structure, by composition. Theisometric correspondence between the commutant and the space B ( K ) σ , implies that we get aproduct structure also on this space. We make this product structure explicit in Proposition 3.4.Since the general dilation theorem is a bit abstract, in section 4 we present an explicit constructionof the Cuntz dilation associated to a random walk on a graph. In section 5, we study the intertwiningoperators between the Cuntz dilations associated to two random walks on finite graphs. Of course,in particular, this covers the case of the commutant of the Cuntz dilation. It turns out thatthese operators are completely determined by the balanced minimal invariant sets . Here are thedefinitions. Definition 1.6.
Suppose we have two directed finite graphs ( V , E ) and ( V ′ , E ′ ), with labels fromthe same set Λ, and with weights ( α i,λ ) i ∈V ,λ ∈ Λ and ( α ′ i ′ ,λ ) i ′ ∈V ′ ,λ ∈ Λ respectively.Given two pairs of vertices ( i, i ′ ) , ( j, j ′ ) ∈ V × V ′ , we say that the transition from ( i, i ′ ) to ( j, j ′ ) is possible, with word λ = λ . . . λ n , and we write ( i, i ′ ) λ → ( j, j ′ ), if there are pairs ofvertices ( i, i ′ ) = ( i , i ′ ) , ( i , i ′ ) , . . . , ( i n , i ′ n ) = ( j, j ′ ) such that the transitions ( i k − , i ′ k − ) λ k → ( i k , i ′ k )is possible, meaning α i k − ,λ k = 0, α ′ i ′ k − ,λ k = 0, for all k = 1 , . . . , n . We use also the notations i · ( λ . . . λ n ) = ( . . . (( i · λ ) · λ ) · . . . ) · λ n ,α i,λ = α i,λ α i · λ ,λ . . . α i · λ λ ...λ n − ,λ n , and i λ → j if j = i · λ with α i,λ = 0.So, the transition ( i, i ′ ) λ → ( j, j ′ ) is possible, with λ = λ . . . λ n if i λ → j , i ′ λ → j ′ , α i,λ = 0, and α ′ i ′ ,λ = 0. DORIN DUTKAY AND NICHOLAS CHRISTOFFERSEN
A non-empty subset M of V × V ′ is called invariant , if for any ( i, i ′ ) ∈ M , and λ = λ . . . λ n , ifthe transition ( i, i ′ ) λ → ( j, j ′ ) is possible, then ( j, j ′ ) ∈ M . The invariant set M is called minimal if it has no proper invariant subsets.The orbit O ( i, i ′ ) of a pair of vertices ( i, i ′ ) is the smallest invariant set that contains ( i, i ′ ). Definition 1.7.
For each minimal invariant set M , pick a point ( i M , i ′M ) in M . For ( i, i ′ ) ∈ V ×V ′ ,define the set F ( i, i ′ ) of paths/words that arrive at one of the points ( i M , i ′M ) for the first time;that is λ = λ . . . λ n , n ≥
1, is in F ( i, i ′ ) if and only if ( i · λ, i ′ · λ ) = ( i M , i ′M ) for some minimalinvariant set M , and ( i · λ . . . λ k , i ′ · λ . . . λ k ) = ( i N , i ′N ) for all 1 ≤ k < n and all minimal invariantsets N .We say that a minimal invariant set M is balanced if(i) For all ( i, i ′ ) ∈ M , and for all λ ∈ Λ, | α i,λ | = | α ′ i ′ ,λ | .(ii) For all ( i, i ′ ) ∈ M , and all loops λ = λ . . . λ n at ( i, i ′ ), i.e., ( i, i ′ ) λ → ( i, i ′ ), one has α i,λ = α ′ i ′ ,λ .For reasons that will be apparent in section 4, we also use the notation ( i, ∅ ) := ~i = δ i , for theorthonormal basis of the space K .According to Theorem 1.3, the operators that intertwine the two Cuntz dilations associated tothe two random walks are in bijective correspondence with the operators in the space B ( K , K ′ ) σ which is the space of operators T : K → K ′ with T = X λ ∈ Λ V ′ λ T V ∗ λ =: σ ( T ) . The main result of section 5 describes the space B ( K , K ′ ) σ in terms of balanced minimal invariantsets, thus describing also the intertwining operators between the two Cuntz dilations. Theorem 1.8.
Let C = (cid:8) ( i M , i ′M ) : M balanced minimal invariant set (cid:9) , and let C [ C ] be the space of complex valued functions on C .Let B ( K , K ′ ) σ be the space of operators T : K → K ′ with (1.9) T = X λ ∈ Λ V ′ λ T V ∗ λ =: σ ( T ) . Then there is a linear isomorphism between C [ C ] and B ( K , K ′ ) σ , C [ C ] ∋ c T ∈ B ( K , K ′ ) σ , definedby (1.10) (cid:10) T ( i, ∅ ) , ( i ′ , ∅ ) (cid:11) = X λ ∈ F ( i,i ′ ) α i,λ α ′ i ′ ,λ c ( i · λ, i ′ · λ ) . The inverse of this map is B ( K , K ′ ) σ ∋ T c ∈ C [ C ] , (1.11) c ( i M , i ′M ) = (cid:10) T ( i M , ∅ ) , ( i ′M , ∅ ) (cid:11) , ( M ∈ C ) . We end the paper with some examples, in section 6.
UNTZ ALGEBRAS AND RANDOM WALKS 5 A motivation from harmonic analysis
The study of orthogonal Fourier series on fractal measures began with the paper [JP98], whereJorgensen and Pedersen proved that, for the Cantor measure µ with scale 4 and digits 0 and 2,the set of exponential functions ( e πiλx : λ = n X k =0 k l k , n ∈ N , l k ∈ { , } ) , is an orthonormal basis in L ( µ ). Such a measure, which possesses an orthonormal Fourier basis ofexponential functions is called a spectral measure . Many more examples of spectral measures havebeen constructed since, see, e.g., [Str00, DHL19]. For the classical Middle Third Cantor measure,Jorgensen and Pedersen proved that this construction is not possible. Strichartz [Str00] posedthe question, if this measure, has a frame of exponential functions. The question is still open atthe time of writing this article. In [PW17], Picioroaga and Weber, trying to construct frames ofexponential functions for Cantor measures, introduced a new idea: to use Cuntz dilations to obtainorthonormal bases in spaces larger than the L -space of the given measure and then project themto construct Parseval frames. Even though the idea did not apply to the Middle Third Cantor set, anew class of Parseval frames was constructed for certain Cantor measures. The ideas were extendedin [DR18, DR16] and we present some of them here, in the context of Cuntz representations.First, one needs to define the ground space: the Cantor measure, or more generally the self-affinemeasure. It is associated to a scale R and a digit set B . For the Middle Third Cantor set R = 3and B = { , } . For the Jorgensen-Pedersen example in [JP98], R = 4, B = { , } . Definition 2.1.
For a given integer R ≥ B with cardinality | B | =: N, we define the affine iterated function system (IFS) τ b ( x ) = R − ( x + b ) , x ∈ R , b ∈ B. The self-affinemeasure (with equal weights) is the unique probability measure µ = µ ( R, B ) satisfying(2.1) Z f dµ = 1 N X b ∈ B Z f ◦ τ b dµ, ( f ∈ C c ( R )) . This measure is supported on the attractor X B which is the unique compact set that satisfies X B = [ b ∈ B τ b ( X B ) . The set X B is also called the self-affine set associated with the IFS, and it can be described as X B = ( ∞ X k =1 R − k b k : b k ∈ B ) . One can refer to [Hut81] for a detailed exposition of the theory of iterated function systems. Wesay that µ = µ ( R, B ) satisfies the no overlap condition if µ ( τ b ( X B ) ∩ τ b ′ ( X B )) = 0 , ∀ b = b ′ ∈ B. For λ ∈ R , define e λ ( x ) = e πiλx , ( x ∈ R ) . DORIN DUTKAY AND NICHOLAS CHRISTOFFERSEN A frame for a Hilbert space H is a family { e i } i ∈ I ⊂ H such that there exist constants A, B > v ∈ H , A k v k ≤ X i ∈ I | h v , e i i | ≤ B k v k . The largest A and smallest B which satisfy these inequalities are called the frame bounds . Theframe is called a Parseval frame if both frame bounds are 1.Next, to construct the Parseval frame of exponential functions, one needs the dual set L whichacts as the starting point for the construction of the frequencies associated to the Fourier series.We make the following assumptions. Assumptions 2.1.
Suppose that there exists a finite set L ⊂ Z with 0 ∈ L, | L | =: M andnon-zero complex numbers ( α l ) l ∈ L such that the following properties are satisfied:(i) α = 1 . (ii) The matrix(2.2) T := 1 √ N (cid:16) e πiR − l · b α l (cid:17) l ∈ L,b ∈ B is an isometry, i.e., T T ∗ = I N , i.e., its columns are orthonormal, which means that(2.3) 1 N X l ∈ L | α l | e πiR − l · ( b − b ′ ) = δ b,b ′ , ( b, b ′ ∈ B ) . (iii) The measure µ ( R, B ) has no overlap.To formulate the result, we need some extra notations and definitions.
Definition 2.2.
Let(2.4) m B ( x ) = 1 N X b ∈ B e πibx , ( x ∈ R ) . Since the measure µ ( R, B ) has no overlap, we can define the map R : X B → X B , by R ( x ) = τ − b ( x ) = Rx − b, if x ∈ τ b ( X B ) , b ∈ B. A set
M ⊂ R is called invariant if for any point t ∈ M , and any l ∈ L , if α l m B ( R − ( t − l )) = 0,then g l ( t ) := R − ( t − l ) ∈ M . M is said to be non-trivial if M 6 = { } . We call a finite minimalinvariant set a min-set .Note that(2.5) X l ∈ L | α l | | m B ( g l ( t )) | = 1 ( t ∈ R d ) , (see (3.2) in [DR16, p.1615]), and therefore, we can interpret the number | α l | | m B ( g l ( t )) | asthe probability of transition from t to g l ( t ), and if this number is not zero then we say that this transition is possible in one step (with digit l ) , and we write t → g l ( t ) or t l → g l ( t ). We say thatthe transition is possible from a point t to a point t ′ if there exist t = t , t , . . . , t n = t ′ such that t = t → t → · · · → t n = t ′ . The trajectory of a point t is the set of all points t ′ (including thepoint t ) such that the transition is possible from t to t ′ .A cycle is a finite set { t , . . . , t p − } such that there exist l , . . . , l p − in L such that g l ( t ) = t , . . . , g l p − ( t p − ) = t p := t . Points in a cycle are called cycle points . UNTZ ALGEBRAS AND RANDOM WALKS 7
A cycle { t , . . . , t p − } is called extreme if | m B ( t i ) | = 1 for all i ; by the triangle inequality, since0 ∈ B , this is equivalent to t i · b ∈ Z for all b ∈ B .For k ∈ Z , we denote [ k ] := { k ′ ∈ Z : ( k ′ − k ) · R − b ∈ Z , for all b ∈ B } . The next proposition gives some information about the structure of finite, minimal sets, whichmakes it easier to find such sets in concrete examples.
Proposition 2.3. [DR18]
Let M be a non-trivial finite, minimal invariant set. Then, for everytwo points t, t ′ ∈ M the transition is possible from t to t ′ in several steps. In particular, every pointin the set M is a cycle point. The set M is contained in the interval h min( − L ) R − , max( − L ) R − i .If t is in M and if there are two possible transitions t → g l ( t ) and t → g l ( t ) , then l ≡ l (mod R ) .Every point t in M is an extreme cycle point, i.e., | m B ( t ) | = 1 and if t → g l ( t ) is a possibletransition in one step, then [ l ] ∩ L = { l ∈ L : l ≡ l (mod R ) } and (2.6) X l ∈ L,l ≡ l (mod R ) | α l | = 1 . In particular t · b ∈ Z for all b ∈ B . Definition 2.4.
Let c be an extreme cycle point in some finite minimal invariant set. A word l . . . l p − in L is called a cycle word for c if g l p − . . . g l ( c ) = c and g l k . . . g l ( c ) = c for 0 ≤ k < p − , and the transitions c → g l ( c ) → g l g l ( c ) → · · · → g l p − . . . g l ( c ) → g l p − . . . g l ( c ) = c are possible.For every finite minimal invariant set M , pick a point c ( M ) in M and define Ω( c ( M )) to be theset of finite words with digits in L that do not end in a cycle word for c ( M ), i.e., they are not ofthe form ωω where ω is a cycle word for c and ω is an arbitrary word with digits in L . Theorem 2.5. [DR18]
Suppose ( R, B, L ) and ( α l ) l ∈ L satisfy the Assumptions 1.1. Then the set ( k Y j =0 α l j e l + Rl + ··· + R k l k + R k +1 c ( M ) : l . . . l n ∈ Ω( c ( M )) , M is a min-set ) is a Parseval frame for L ( µ ( R, B )) . Here we will formulate these results, in our context, of row coisometries, Cuntz representationsand random walks on graphs. This will give us a better understanding of the structure associatedto the Fourier series on self-affine measures.
Definition 2.6.
We denote by Ω the set of all finite words with letters in Λ, including the emptyword denoted ∅ , Ω = { λ . . . λ m : λ , . . . , λ m ∈ Λ , m ≥ } , For λ = λ . . . λ m ∈ Ω, we denote by | λ | = m , the length of λ .We use the notation, for λ . . . λ m ∈ Ω, V λ λ ...λ m = V λ V λ . . . V λ m , S λ ...λ m = S λ . . . S λ m . DORIN DUTKAY AND NICHOLAS CHRISTOFFERSEN
Theorem 2.7.
Define the operators ( V l ) l ∈ L on L ( µ ( R, B )) by (2.7) V l f ( x ) = α l e l ( x ) f ( R ( x )) , ( x ∈ X B , f ∈ L ( µ ( R, B )) , l ∈ L ) . Then ( V l ) l ∈ L is a row coisometry.Let M be a min-set and let K M = span { e t : t ∈ M} . Then K M is invariant for V ∗ l , l ∈ L and (2.8) V ∗ l e t = α l m B ( g l ( t )) e g l ( t ) , ( t ∈ R ) . In addition, for all t ∈ M , m B ( g l ( t )) = 1 , if the transition t l → g l ( t ) is possible and (2.9) X l ∈ L,t l → gl ( t ) is possible | α l | = 1 . The exponential functions { e t : t ∈ M} form an orthonormal basis for K M and the spaces K M aremutually orthogonal.Let H M = span { V l ...l k e t : k ≥ , l , . . . , l k ∈ L, t ∈ M} . (2.10) V l ...l k e t = k Y j =0 α l j e l + Rl + ··· + R k l k + R k +1 t , ( k ≥ , l , . . . , l k ∈ L, t ∈ M ) . (2.11) span {H M : M min-set } = L ( µ ( R, B )) . The coisometry ( P K M V l P K M ) l ∈ L on K is isomorphic to the coisometry associated to the graphwith vertices V = M and transition weights α t,l := α l , t ∈ M , l ∈ L , through the linear map definedby C [ M ] ∋ ~t e t ∈ K M .The Cuntz dilation of this coisometry is irreducible.Proof. Let µ = µ ( R, B ). First, we compute the adjoints V ∗ l . We have, using (2.1), for f, g ∈ L ( µ ): h V l f , g i = Z α l e πilx f ( R x ) g ( x ) dµ = 1 N X b ∈ B Z α l e πilτ b ( x ) f ( R τ b ( x )) g ( τ b ( x )) dµ = * f , N X b ∈ B α l e − l ◦ τ b · g ◦ τ b + . Therefore,(2.12) V ∗ l g = 1 N X b ∈ B α l e − l ◦ τ b · g ◦ τ b , ( g ∈ L ( µ )) . Then, for f ∈ L ( µ ) and x ∈ X B , suppose x ∈ τ b ′ ( X B ), and we have, using the assumptions, X l ∈ L V l V ∗ l f ( x ) = X l ∈ L α l e πilx N X b ∈ B α l e − πil Rx − b ′ + bR f (cid:18) Rx − b ′ + bR (cid:19) UNTZ ALGEBRAS AND RANDOM WALKS 9 = X b ∈ B f (cid:18) x + b − b ′ R (cid:19) N X l ∈ L | α l | e πil b − b ′ R = X b ∈ B f (cid:18) x + b − b ′ R (cid:19) δ bb ′ = f ( x ) . This shows that ( V l ) l ∈ L is a row coisometry.We compute, for t ∈ R , V ∗ l e t ( x ) = 1 N X b ∈ B α l e − πil x + bR e πit x + bR = α l N X b ∈ B e πib · t − lR ! e πix · t − lR = α l m B ( g l ( t )) e g l ( t ) ( x ) . This implies (2.8).Now let M be a min-set. For t ∈ M , we have V ∗ l e t = α l m B ( g l ( t )) e g l ( t ) . If the transition t l → g l ( t )is not possible, that means that α l m B ( g l ( t )) = 0 so V ∗ l e t = 0. If the transition t l → g l ( t ) is possible,then g l ( t ) ∈ M so V ∗ l e t ∈ K M . Thus K M is invariant for V ∗ l .For t ∈ M , if the transition t l → g l ( t ) is possible, then g l ( t ) ∈ M , and, by Proposition 2.3, bg l ( t ) ∈ Z for all b ∈ B so m B ( g l ( t )) = 1. Also, from the same Proposition, we have thatif the transitions t l → g l ( t ) and t l → g l ( t ) are possible, then l ≡ l (mod R ). Conversely, ifthe transition t l → g l ( t ) is possible, and l ≡ l (mod R ), then m B ( g l ( t )) = N P b ∈ B e πib t − l R = N P b ∈ B e πib t − l R = m B ( g l ( t )) = 0, so the transition t l → g l ( t ) is possible (note that we assumedthat the numbers α l are all non-zero). Therefore, using again Proposition 2.3, fixing some l ∈ L such that the transition t l → g l ( t )) is possible, X l ∈ L,t l → gl ( t ) is possible | α l | | m B ( g l ( t )) | = X l ≡ l ( mod R ) | α l | = 1 . The fact that the functions e t , t ∈ M , M min-set, are mutually orthogonal is in the proof of[DR18, Theorem 1.6]. Equation (2.10) follows from a simple computation, using the fact that, byProposition 2.3, for all t ∈ M , bt ∈ Z and so t is a period for m B , i.e., m B ( x + kt ) = m B ( t ) for all x ∈ R , k ∈ Z . The relation (2.11) follows from Theorem 2.5.For t ∈ M , by (2.8), P K M V ∗ l P K M e t = α l m B ( g l ( t )) P K M e g l ( t ) = ( α l e g l ( t ) , if the transition t l → g l ( t ) is possible , , otherwise.So , the coisometry ( P K M V ∗ l P K M ) l ∈ L on K M is isometric to the given graph coisometry.To see that the associated Cuntz dilation is irreducible, we use Corollary 5.10 and show that therandom walk is connected and separating.The fact that the random walk is connected follows from the minimality of M and Proposition2.3. To check that the random walk is separating take t = t ′ in M . Let l , . . . , l n in L such thatboth transitions t l ...l n → g l n ...l ( t ) and t ′ l ...l n → g l n ...l ( t ′ ) are possible. Then g l n ...l ( t ) and g l n ...l ( t ′ ) arein M . But, the maps g l are contractions so lim n dist( g l n ...l ( t ) , g l n ...l ( t ′ )) = 0. Since M is finite,for n large enough, we get that g l n ...l ( t ) = g l n ...l ( t ′ ), but that means t = t ′ , a contradiction. Thus,for n large enough, we cannot have that both transitions t l ...l n → g l n ...l ( t ) and t ′ l ...l n → g l n ...l ( t ′ ) arepossible. So, the random walk is separating. (cid:3) General results
Proposition 3.1.
Let ( V λ ) λ ∈ Λ be a row coisometry on the Hilbert space K and let ( S λ ) λ ∈ Λ be itsCuntz dilation on the Hilbert space H . Define the subspaces (3.1) K m = span { S λ ...λ m k : k ∈ K , λ , . . . λ m ∈ Λ } , ( m ∈ N ) , K = K . (i) {K m } is an increasing sequence of subspaces and [ m ∈ N K m = H . (ii) For each m ≥ and λ ∈ Λ , S ∗ λ K m +1 ⊆ K m . (iii) For k ∈ K m , and each n ≤ m , we have that the representation k = X λ ∈ Ω , | λ | = n S λ k λ , is unique with k λ ∈ H , and moreover, k λ is given by k λ = S ∗ λ k ∈ K m − n . (iv) Let P K m be the orthogonal projection onto K m . Then P K m = X λ ,...,λ m ∈ Λ S λ ...λ m P K S ∗ λ ...λ m . Proof. (i) Let S λ ...λ m k ∈ K m , with k ∈ K . Then, using the Cuntz relations S λ ...λ m k = S λ ...λ m X λ ∈ Λ S λ S ∗ λ k = X λ ∈ Λ S λ ...λ m S λ ( S ∗ λ k ) . Since S ∗ λ k = V ∗ λ k ∈ K for each λ , it follows that S λ ...λ m k ∈ K m +1 , so K m ⊆ K m +1 . The density of the union follows from the fact that K is cyclic for the representation.(ii) Let S λ ...λ m +1 k ∈ K m +1 . Then S ∗ λ S λ ...λ m +1 k = δ λλ S λ ...λ m +1 k ∈ K m .(iii) Assume we have two such representations k = X λ ∈ Ω , | λ | = n S λ k λ = X λ ∈ Ω , | λ | = n S λ k ′ λ . Then, using the orthogonality of the ranges of the isometries0 = k k − k k = X | λ | = n k S λ ( k λ − k ′ λ ) k = X | λ | = n k k λ − k ′ λ k . Therefore k λ = k ′ λ for all | λ | = n .Further, for all n ≤ m and | λ | = n , by inducting on (ii) we have k λ = S ∗ λ k ∈ K m − n , and, usingthe Cuntz relations, k = X | λ | = n S λ S ∗ λ k = X | λ | = n S λ k λ . (iv) Denote T = X λ ...λ m S λ ...λ m P K S ∗ λ ...λ m . UNTZ ALGEBRAS AND RANDOM WALKS 11 If h ∈ K m , then, by (ii), we have that S ∗ λ ...λ m ∈ K , so P K S ∗ λ ...λ m h = S ∗ λ ...λ m h . Therefore T h = X λ ...λ m S λ ...λ m (cid:0) P K S ∗ λ ...λ m h (cid:1) = X λ ...λ m S λ ...λ m S ∗ λ ...λ m h = h. Now, if h ⊥ K m , then we have that for each k ∈ K , (cid:10) S ∗ λ ...λ m h , k (cid:11) = h h , S λ ...λ m k i = 0 . This means that P K S ∗ λ ...λ m h = 0, and therefore T h = 0 for h ⊥ K m and T h = h for h ∈ K m . Inconclusion T = P K m . (cid:3) Proposition 3.2.
With the notations in Theorem 1.3, let T ∈ B ( K ) σ and let A T be the associatedoperator on H which commutes with the representation ( S λ ) λ ∈ Λ , defined uniquely by the property T = P K A T P K . Then, for m ∈ N , (3.2) P K m A T P K m = X | λ | = m S λ T S ∗ λ . Also, P K m A T P K m ξ m →∞ → A T ξ for all ξ ∈ H .Proof. Using Proposition 3.1(iv), and the fact that A T commutes with the representation, we have P K m A T P K m = X | λ | = m X | λ ′ | = m S λ P K S ∗ λ A T S λ ′ P K S ∗ λ ′ = X | λ | = m X | λ ′ | = m S λ P K S ∗ λ S λ ′ A T P K S ∗ λ ′ = X | λ | = m S λ P K A T P K S ∗ λ = X | λ | = m S λ T S ∗ λ . The following Lemma is probably well known.
Lemma 3.3.
Let ( A n ) n ∈ N , ( B n ) n ∈ N be sequences of bounded operators on some Hilbert space H .Assume that A n → A and B n → B in the Strong Operator Topology (SOT), i.e., A n ξ → Aξ , B n ξ → Bξ , for all vectors ξ ∈ H . Then A n B n → AB in the SOT.Proof. By the Uniform Boundedness Principle, sup n k A n k =: M < ∞ . Let ξ ∈ H . We have k A n B n ξ − ABξ k ≤ k A n B n ξ − A n Bξ k + k A n Bξ − ABξ k ≤ k A n kk B n ξ − Bξ k + k ( A n − A )( Bξ ) k → . (cid:3) Since P K m → I H in the SOT, with Lemma 3.3, we obtain that P K m A T P K m → A T in SOT. (cid:3) Proposition 3.4.
With the notations as in Theorem 1.3, let
T, T ′ ∈ B ( K ) σ and let A and A ′ ,respectively, be the associated operators in the commutant of the Cuntz dilation, so T = P K AP K and T ′ = P K A ′ P K . Then AA ′ is also in the commutant of the Cuntz dilation, so T ∗ T ′ := P K AA ′ P K is an element in B ( K ) σ . Then, (3.3) ( P K m AP K m )( P K m A ′ P K m ) ξ → AA ′ ξ, ( ξ ∈ H ) , and (3.4) ( T ∗ T ′ ) ξ = lim m →∞ X | λ | = m V λ T T ′ V ∗ λ ξ, ( ξ ∈ K ) . Proof.
The limit in (3.3) follows from Proposition 3.2 and Lemma 3.3. Then, with Lemma 3.3,(3.2), for ξ ∈ K , P K AA ′ P K ξ = lim m →∞ P K P K m AP K m P K m A ′ P K m P K ξ = lim m →∞ P K X | λ | = m S λ T S ∗ λ X | λ ′ | = m S λ ′ T ′ S ∗ λ ′ P K ξ = lim m →∞ P K X | λ | = m X | λ ′ | = m S λ T S ∗ λ S λ ′ T ′ S ∗ λ ′ P K ξ = lim m →∞ P K X | λ | = m S λ T T ′ S ∗ λ P K ξ = lim m →∞ X | λ | = m P K S λ T T ′ V ∗ λ ξ = lim m →∞ X | λ | = m V λ T T ′ V ∗ λ ξ. (cid:3) An explicit construction of the Cuntz dilation associated to a random walk
Consider now, as in Definition 1.5 a directed graph G = ( V , E ), with edges labeled from a finiteset Λ, | Λ | = N . Recall, that we assume that for each vertex i , different labels λ , λ , lead, from i to different vertices, so, if i λ → j , and i λ → j , then j = j . We write j = i · λ if i λ → j . Also, weassume that for each vertex i there is at most one edge coming into the vertex i with label λ .Each edge has an associated weight | α i,λ | defined by some complex number α i,λ , α i,λ = 0 in thecase when there is no edge from i , labeled λ , and(4.1) X λ ∈ Λ | α i,λ | = 1 , ( i ∈ V ) . We recall that we define the operators ( V λ ) λ ∈ Λ on the Hilbert space K = l [ V ],(4.2) V ∗ λ ( ~i ) = ( α i,λ ~j, if i λ → j , otherwise. Proposition 4.1.
The operators ( V λ ) λ ∈ Λ form a row coisometry.Proof. A simple computation shows that(4.3) V λ ( ~j ) = ( α i,λ ~i, if i λ → j, , otherwise.Then, for i ∈ V , X λ ∈ Λ V λ V ∗ λ ~i = X λ ∈ Λ α i,λ V λ ( ~i · λ ) = X λ ∈ Λ | α i,λ | ~i = ~i. (cid:3) Since ( V λ ) λ ∈ Λ is a row coisometry, by Theorem 1.3, it has a unique Cuntz dilation. In this sectionwe will give an explicit construction of the Cuntz dilation associated to this random walk, under acertain mild assumption.We will need some notation. Recall that, for a word λ = λ λ . . . λ l , we define(4.4) α i,λ = α i,λ α i · λ ,λ . . . α i · λ · λ ··· λ l − ,λ l . Note that | α i,λ | is the probability that, starting from the vertex i , the random walk follows thelabels λ , λ , . . . , λ l . UNTZ ALGEBRAS AND RANDOM WALKS 13
For each vertex i , let Λ i be the set of labels that originate from i ,(4.5) Λ i := n λ ∈ Λ : There exists j such that i λ → j o . For each vertex j , let Λ j be the set of all labels that end in j ,(4.6) Λ j := n λ ∈ Λ : There exists i such that i λ → j o . Further, we define n i := | Λ i | and n j := | Λ j | .To construct the Cuntz dilation, first we will construct, for each vertex i , some unitary matrixthat has, as the first column, the weights ( α i,λ ) λ ∈ Λ i . Indeed, since X λ ∈ Λ i | α i,λ | = 1 , we may create the following unitary matrix (not necessarily unique)(4.7) C i = α i,λ c i,λ c i,λ · · · c n i − i,λ α i,λ c i,λ c i,λ · · · c n i − i,λ ... ... ... . . . ... α i,λ ni − c i,λ ni − c i,λ ni − · · · c n i − i,λ ni − = h c ki,λ i k =0 ,...,n i − λ ∈ Λ i , where we adopt the notation c i,λ j := α i,λ j .To define the Hilbert space of the Cuntz dilation, we will use the set Ω ∗ N , defined as the set offinite words over the alphabet { , , . . . , N − } not ending in 0, including the empty word.For a digit k ∈ { , . . . , N − } and a word w ∈ Ω ∗ N , define kw ∈ Ω ∗ N as the concatenation of k and w . We make the important convention :0 ∅ = ∅ . Additionally, we define the “inverse concatenation”, \ : Ω ∗ N × { , . . . , N − } → Ω ∗ N ∪ { null } , \ ( w, k ) := w \ k := (cid:26) w ′ , if kw ′ = w, null , otherwise.Note that ∅ \ ∅ and ∅ \ k = null for k = 0.Expanding our notation, we define λ · j := ( i, if i λ → j null , otherwise. i · λ := ( j, if i λ → j null , otherwise.We define the Hilbert space of the dilation as H = l [ V × Ω ∗ N ] = span { ( i, w ) : i vertex in V , w ∈ Ω ∗ N } . We identify K with K = span { ( i, ∅ ) : i ∈ V} . Remark 4.2.
For a fixed w ∈ Ω ∗ N , w \ k = null for exactly one digit k ∈ { , . . . , N − } . We makethe conventions( null , null ) = ( null , w ) = ( i, null ) = 0 ∈ H , δ null i = δ null w = δ nullnull = 0 , for all vertices i and all words w .Note also that δ w ′ kw = δ w ′ \ kw ,δ i ′ i = δ i ′ · λi · λ , ( λ ∈ Λ i ) , δ j ′ j = δ λ · j ′ λ · j , ( λ ∈ Λ j ) . We will make the following assumption:(4.8) X i ∈V ( N − n i ) = X j ∈V ( N − n j ) . In this case, consider the two sets (cid:8) ( j, λ ) : j ∈ V , λ ∈ Λ \ Λ j (cid:9) , { ( i, k ) : i ∈ V , n i ≤ k ≤ N − } . Note that the first set has cardinality P j ∈V ( N − n j ), and the second set has cardinality P i ∈V ( N − n i ). Therefore, under assumption (4.8), the two sets have equal cardinality, so there is a bijectionbetween them ϕ : (cid:8) ( j, λ ) : j ∈ V , λ ∈ Λ \ Λ j (cid:9) → { ( i, k ) : i ∈ V , n i ≤ k ≤ N − } . We define:(4.9) F = π ◦ ϕ, G = π ◦ ϕ, ˜ F = π ◦ ϕ − , ˜ G = π ◦ ϕ − , where π , π are the projections onto the first and second component. Remark 4.3.
Note that, when the graph is finite, the assumption (4.8) holds. Indeed, each edgeis completely and uniquely determined by its starting vertex i and a label in Λ i . Therefore thenumber of edges is P i ∈V n i . On the other hand each edge is completely and uniquely determinedby its end vertex j and a label in Λ j , therefore the number of edges is also P j ∈V n j . Since thegraph is finite, this implies (4.8). Theorem 4.4.
Suppose the assumption (4.8) holds. Define the operators ( S λ ) λ ∈ Λ on H by (4.10) S λ ( j, w ) = ( P n i − k =0 c ki,λ ( i, kw ) , if i λ → j, ( F ( j, λ ) , G ( j, λ ) w ) , otherwise.Then ( S λ ) λ ∈ Λ is the Cuntz dilation of the row coisometry ( V λ ) λ ∈ Λ in (4.3) . Moreover, the adjointoperators are (4.11) S ∗ λ ( i, k ′ w ) = P n i − k =0 c ki,λ ( j, k ′ w \ k ) , if i λ → j, ( ˜ F ( i, k ′ ) , w ) , if λ = ˜ G ( i, k ′ ) , , otherwise. UNTZ ALGEBRAS AND RANDOM WALKS 15
Proof.
First, we compute the adjoints. Let λ ∈ Λ, j, i ′ ∈ V , w, w ′ ∈ Ω ∗ N . Case 1: λ ∈ Λ j ; thendenote i λ → j . (cid:10) ( j, w ) , S ∗ λ ( i ′ , w ′ ) (cid:11) = (cid:10) S λ ( j, w ) , ( i ′ , w ′ ) (cid:11) = * n i − X k =0 c ki,λ ( i, kw ) , ( i ′ , w ′ ) + = n i − X k =0 c ki,λ δ i ′ i δ w ′ kw = n i ′ − X k =0 c ki ′ ,λ δ i ′ · λj δ w ′ \ kw = * ( j, w ) , δ i ′ · λj n i ′ − X k =0 c ki ′ ,λ ( i ′ · λ, w ′ \ k ) + . Case 2: λ Λ j . Then (cid:10) ( j, w ) , S ∗ λ ( i ′ , k ′ w ′ ) (cid:11) = (cid:10) S λ ( j, w ) , ( i ′ , k ′ w ′ ) (cid:11) = (cid:10) ( F ( j, λ ) , G ( j, λ ) w ) , ( i ′ , kw ′ ) (cid:11) = δ i ′ F ( j,λ ) δ k ′ G ( j,λ ) δ w ′ w = δ ( i ′ ,k ′ ) ϕ ( j,λ ) δ w ′ w = δ ˜ F ( i ′ ,k ′ ) j δ ˜ G ( i ′ ,k ′ ) λ δ w ′ w = D ( j, w ) , δ ˜ G ( i ′ ,k ′ ) λ ( ˜ F ( i ′ , k ′ ) , w ′ ) E . This proves (4.11).Now, we verify the Cuntz relations. Let j ∈ V , w ∈ Ω ∗ N , λ, λ ′ ∈ Λ. Case 1: λ ∈ Λ j . Then wetake i such that i λ → j . Then S ∗ λ ′ S λ ( j, w ) = S ∗ λ ′ n i − X k =0 c ki,λ ( i, kw ) = n i − X k ′ =0 n i − X k =0 c ki,λ c k ′ i,λ ′ ( i · λ ′ , kw \ k ′ )= n i − X k =0 c ki,λ c ki,λ ′ ( i · λ ′ , w ) = δ λ ′ λ ( j, w ) , where we used the orthogonality of the matrix C i in the last equality.Case 2: λ Λ j . Then, since ϕ is a bijection, S ∗ λ ′ S λ ( j, w ) = S ∗ λ ′ ( F ( j, λ ) , G ( j, λ ) w ) = δ λ ′ λ ( j, w ) . Now, we check the second Cuntz relation. Let i ∈ V , k ∈ { , . . . , N − } , w ∈ Ω ∗ N . Case 1: k ≥ n i . Then X λ ∈ Λ S λ S ∗ λ ( i, kw ) = X λ ∈ Λ i S λ S ∗ λ ( i, kw ) + X λ Λ i S λ S ∗ λ ( i, kw )= X λ ∈ Λ i S λ n i − X k ′ =0 c k ′ i,λ ( i · λ, kw \ k ′ ) + X λ Λ i S λ δ ˜ G ( i,k ) λ ( ˜ F ( i, k ) , w )= 0 + S ˜ G ( i,k ) ( ˜ F ( i, k ) , w ) = ( i, kw ) . Case 2: k < n i . Then X λ ∈ Λ S λ S ∗ λ ( i, kw ) = X λ ∈ Λ i S λ S ∗ λ ( i, kw ) + X λ Λ i S λ S ∗ λ ( i, kw )= X λ ∈ Λ i S λ n i − X k ′ =0 c k ′ i,λ ( i · λ, kw \ k ′ ) + 0 = X λ ∈ Λ i c ki,λ S λ ( i · λ, w )= X λ ∈ Λ i n i − X k ′ =0 c k ′ i,λ c ki,λ ( i, k ′ w ) = n i − X k ′ =0 δ k ′ k ( i, k ′ w ) = ( i, kw ) . Thus, the Cuntz relations are satisfied.It is clear from (4.11), that S ∗ λ ( i, ∅ ) = c i,λ ( j, ∅ ), if i λ → j , (recall the convention ∅ = 0 ∅ ), and S ∗ λ ( i, ∅ ) = 0 if λ Λ i . Therefore S ∗ λ coincides with V ∗ λ on K .It remains to prove that K is cyclic for the representation. We will prove, by induction, that, forall words w ∈ Ω ∗ N of length | w | = m and all vertices i ∈ V , ( i, w ) is in S := span (cid:8) S λ ...λ p K : λ , . . . , λ p ∈ Λ , p ≥ (cid:9) . For m = 0, we have ( i, w ) = ( i, ∅ ) ∈ K , so the assertion is clear.Assume now, that the assertion is true for m . Let kw an arbitrary word of length m + 1, and i ∈ V . Case 1: k ≥ n i . Take ( j, λ ) = ϕ − ( i, k ). Then( i, kw ) = ( F ( j, λ ) , G ( j, λ ) w ) = S λ ( j, w ) ∈ S . Case 2: k < n i . ( i, kw ) = X λ ∈ Λ i n i − X k ′ =0 c ki,λ c k ′ i,λ ( i, k ′ w ) = X λ ∈ Λ i c ki,λ S λ ( i · λ, w ) ∈ S . By induction, if follows that S = H , which means that K is cyclic for the representation. (cid:3) Intertwining operators
The main goal in this section is to prove Theorem 1.8, which describes the intertwining operatorsbetween the Cuntz dilations associated to two random walks. We begin with some properties ofinvariant sets. We assume from this point on that the graphs are finite.
Proposition 5.1.
The invariant sets have the following properties (i)
Every invariant set contains a minimal invariant subset. (ii)
For ( i, i ′ ) ∈ V × V ′ , its orbit O ( i, i ′ ) is invariant. (iii) If M is a minimal invariant set and ( i, i ′ ) ∈ M , then O ( i, i ′ ) = M ; in other words, if ( j, j ′ ) ∈ M , then there exists a possible transition ( i, i ′ ) λ → ( j, j ′ ) . (iv) For every pair of vertices ( i, i ′ ) ∈ V × V ′ , there exists a minimal invariant set M , suchthat for every pair of vertices ( j, j ′ ) ∈ M the transition ( i, i ′ ) λ → ( j, j ′ ) is possible.Proof. (i) is obvious, just take an invariant subset of the smallest cardinality.(ii) If ( j, j ′ ) ∈ O ( i, i ′ ) and the transition ( j, j ′ ) λ → ( k, k ′ ) is possible, then there is a possibletransition ( i, i ′ ) λ → ( j, j ′ ) and therefore, the transition ( i, i ′ ) λ λ → ( k, k ′ ) is also possible, so ( k, k ′ ) ∈O ( i, i ′ ).(iii) Since M is invariant, it contains O ( i, i ′ ). Since O ( i, i ′ ) is invariant and M is minimal, M = O ( i, i ′ ).(iv) Consider O ( i, i ′ ); it is an invariant set, therefore, it contains a minimal invariant set M . Anypoint ( j, j ′ ) in M is in the orbit of ( i, i ′ ), so there is a possible transition from ( i, i ′ ) to ( j, j ′ ). (cid:3) Heuristically, the next key lemma says that, for each pair of vertices ( i, i ′ ), with probabilityone, the random walk will reach a prescribed point in one of the minimal invariant sets. It isa generalization of the well known result that a finite irreducible random walk is recurrent. SeeRemark 5.4 below. UNTZ ALGEBRAS AND RANDOM WALKS 17
Lemma 5.2.
For each minimal invariant set M , pick a point ( i M , i ′M ) in M . For ( i, i ′ ) ∈ V × V ′ ,define the following set of paths/words that do not go through any of the points ( i M , i ′M ) : (5.1) A ( i, i ′ ) := (cid:8) λ = λ . . . λ n : n ≥ , ( i · λ . . . λ k , i ′ · λ . . . λ ′ k ) = ( i M , i ′M ) for all M minimal invariant and ≤ k ≤ n } . Define, for ( i, i ′ ) ∈ V × V ′ , n ∈ N , (5.2) P (( i, i ′ ); n ) := X | λ | = n,λ ∈ A ( i,i ′ ) | α i,λ || α ′ i ′ ,λ | . Then (5.3) lim n →∞ P (( i, i ′ ); n ) = 0 . Proof.
We prove first, that P (( i, i ′ ); n ) is decreasing. P (( i, i ′ ); n + 1) = X λ = λ ...λ n +1 ,λ ∈ A ( i,i ′ ) | α i,λ ...λ n +1 || α ′ i ′ ,λ ...λ n +1 |≤ X λ ′ = λ ...λ n ,λ ′ ∈ A ( i,i ′ ) X λ n +1 ∈ Λ | α i,λ ′ || α ′ i ′ ,λ ′ || α i · λ ′ ,λ n +1 || α ′ i ′ · λ ′ ,λ n +1 | . But, by the Cauchy-Schwarz inequality, X λ n +1 ∈ Λ | α i · λ ′ ,λ n +1 || α i ′ · λ ′ ,λ n +1 | ≤ X λ n +1 ∈ Λ | α i · λ ′ ,λ n +1 | / X λ n +1 ∈ Λ | α ′ i ′ · λ ′ ,λ n +1 | / = 1Therefore, we obtain further P (( i, i ′ ); n + 1) ≤ X λ ′ = λ ...λ n ,λ ′ ∈ A ( i,i ′ ) | α i,λ ′ || α ′ i ′ ,λ ′ | = P (( i, i ′ ); n ) . Next, we claim that, for each pair of vertices ( i, i ′ ), there exists n ∈ N such that P (( i, i ′ ); n ) < n ( i,i ′ ) to be the minimal one.Indeed, using Proposition 5.1 (iv) and (iii), there exists some possible transition( i, i ′ ) λ → ( i M , i ′M ), for some minimal invariant set M . Let n = | λ | . Then P (( i, i ′ ); n ) = X | λ | = n,λ ∈ A ( i,i ′ ) | α i,λ || α ′ i ′ ,λ | ≤ X | λ | = n | α i,λ || α ′ i ′ ,λ | − | α i,λ || α ′ i ′ ,λ |≤ X | λ | = n | α i,λ | / X | λ | = n | α ′ i ′ ,λ | / − | α i,λ || α ′ i ′ ,λ | = 1 − | α i,λ || α ′ i ′ ,λ | < . Now, let L be the maximum of n ( i,i ′ ) for all ( i, i ′ ) ∈ V × V ′ . Then L ≥ n ( i,i ′ ) for all ( i, i ′ ) so P (( i, i ′ ); L ) ≤ P (( i, i ′ ); n ( i,i ′ ) ) <
1. Define p := max ( i,i ′ ) P (( i, i ′ ); L ) < . We have P (( i, i ′ ); ( n + 1) L ) = X | λ | = nL, | λ | = L,λ λ ∈ A ( i,i ′ ) | α i,λ λ || α ′ i ′ ,λ λ | = X | λ | = nL,λ ∈ A ( i,i ′ ) X | λ | = L,λ ∈ A ( i · λ ,i ′ · λ ) | α i,λ || α ′ i ′ ,λ || α i · λ ,λ || α ′ i · λ ,λ | = X | λ | = nL,λ | α i,λ || α ′ i ′ ,λ | P (( i · λ , i ′ · λ ); L ) ≤ X | λ | = nL,λ ∈ A ( i,i ′ ) | α i,λ || α ′ i ′ ,λ | · p = P (( i, i ′ ); nL ) · p. Therefore, P (( i, i ′ ); nL ) →
0. Since P (( i, i ′ ); n ) is decreasing, we get that P (( i, i ′ ); n ) → (cid:3) Recall now the Definition 1.7. Given a pair of vertices ( i, i ′ ), the set F ( i, i ′ ) consists of pathswhich reach one the prescribed points ( i M , i ′M ) for the first time. The next theorem, shows thatthe matrix entries of an operator T in B ( K , K ′ ) σ are completely determined by the matrix entriescorresponding to ( i M , i ′M ). Theorem 5.3.
Let T : K → K ′ be an operator in B ( K , K ′ ) σ , so (5.4) T = X λ ∈ Λ V ′ λ T V ∗ λ . Then (5.5) (cid:10) T ( i, ∅ ) , ( i ′ , ∅ ) (cid:11) = X λ ∈ F ( i,i ′ ) α i,λ α ′ i ′ ,λ (cid:10) T ( i · λ, ∅ ) , ( i ′ · λ, ∅ ) (cid:11) . Proof.
We denote by T i,i ′ = h T ( i, ∅ ) , ( i ′ , ∅ ) i . From (5.4), we get(5.6) T i,i ′ = X λ ∈ Λ α i,λ α ′ i ′ ,λ T i · λ,i ′ · λ . Iterating (5.6), by induction, we obtain(5.7) T i,i ′ = X λ = λ ...λ n α i,λ α ′ i ′ ,λ T i · λ,i ′ · λ . We split the sum in (5.7) into the sum over the paths λ that go through one of the points ( i M , i ′M ),and the ones that do not. We have, with the notation from Lemma 5.2, T i,i ′ = X | λ | = n,λ A ( i,i ′ ) α i,λ α ′ i ′ ,λ T i · λ,i ′ · λ + X | λ | = n,λ ∈ A ( i,i ′ ) α i,λ α ′ i ′ ,λ T i · λ,i ′ · λ . Since T is bounded, using Lemma 5.2 and the triangle inequality, we get that the second sumconverges to 0 as n → ∞ .For the first sum, each λ A ( i, i ′ ) goes through one of the points ( i M , i ′M ), so we split λ intotwo parts, where it reaches one of these points for the first time, λ = βγ with β ∈ F ( i, i ′ ), | β | ≤ n ,and | γ | = n − | β | . Therefore we have: X | λ | = n,λ A ( i,i ′ ) α i,λ α ′ i ′ ,λ T i · λ,i ′ · λ = X | β |≤ n,β ∈ F ( i,i ′ ) X | γ | = n −| β | α i,βγ α ′ i ′ ,βγ T i · βγ,i ′ · βγ UNTZ ALGEBRAS AND RANDOM WALKS 19 = X | β |≤ n,β ∈ F ( i,i ′ ) α i,β α ′ i ′ ,β X | γ | = n −| β | α i · β,γ α ′ i ′ · β,γ T ( i · β ) · γ, ( i ′ · β ) · γ (5.7) = X | β |≤ n,β ∈ F ( i,i ′ ) α i,β α ′ i ′ ,β T i · β,i ′ · β . Letting n → ∞ we get (5.5). (cid:3) Remark 5.4.
If the two graphs ( V , E ) and ( V ′ , E ′ ) are the same (including the weights), then wecan always take T = I K in (5.4). Let’s see what (5.5) gives us in this case.If i = i ′ , then i · λ = i ′ · λ for any path λ = λ . . . λ n , so (5.5) is trivial in this case, with bothsides equal to zero.However, if i = i ′ , then i · λ = i ′ · λ for all paths λ , and therefore (5.5) gives us the followinginteresting relation(5.8) 1 = X λ ∈ F ( i,i ) | α i,λ | , which can be interpreted as: the probability to reach one of the points i M is one. This a well knownfact from probability: any finite irreducible Markov chain is recurrent (see e.g. [Dur10, Theorem6.6.4]).As a byproduct of the relation (5.8), we obtain an interesting relation in the dilation space. Theorem 5.5.
With the notations in Theorem 5.3, assume that the two graphs are the same.Then, for all i ∈ V , (5.9) ( i, ∅ ) = X λ ∈ F ( i,i ) α i,λ S λ ( i · λ, ∅ ) . Proof.
We begin with a general Lemma.
Lemma 5.6.
Let ( S λ ) λ ∈ Λ be some representation of a Cuntz algebra on a Hilbert space H and let h , h ∈ H and λ , λ two finite words. Then S λ h is orthogonal to S λ h unless λ is a prefix of λ or vice versa.Proof. If λ and λ are not prefixes, one for the other, then, there exists 1 ≤ k ≤ min {| λ | , | λ |} such that λ ,j = λ ,j for 1 ≤ j < k , and λ ,k = λ ,k . Then h S λ h , S λ h i = D S λ , . . . S λ ,k − S λ ,k . . . S λ , | λ | h , S λ , . . . S λ ,k − S λ ,k . . . S λ , | λ | h E = D S λ ,k . . . S λ , | λ | h , S λ ,k . . . S λ , | λ | h E = 0 . (cid:3) Using Lemma 5.6, we notice that words in F ( i, i ) cannot be prefixes of each other, since λ isin F ( i, i ) if the path reaches one of the points ( i M , i M ) for the first time. Therefore the vectors S λ ( i · λ, ∅ ), λ ∈ F ( i, i ) form an orthonormal set. We project the vector ( i, ∅ ) onto this orthonormalset, and we compute the coefficients. h ( i, ∅ ) , S λ ( i · λ ) i = h S ∗ λ ( i, ∅ ) , ( i · λ, ∅ ) i = h V ∗ λ ( i, ∅ ) , ( i · λ, ∅ ) i = h α i,λ ( i · λ, ∅ ) , ( i · λ, ∅ ) i = α i,λ . This means that the right-hand side of (5.9) is the projection of ( i, ∅ ) onto the span of ( S λ ( i · λ, ∅ )) λ ∈ F ( i,i ) . But the square of the norm of this projection is X λ ∈ F ( i,i ) | α i,λ | = 1 = k ( i, ∅ ) k , by (5.8). Thus, (5.9) follows. (cid:3) The next Lemma shows that, given an operator T in B ( K , K ′ ) σ , the matrix entries of T have tobe 0 on non-balanced minimal invariant sets, and there are important restrictions on the balancedones. Lemma 5.7.
With the notations as in Theorem 5.3, let T i,i ′ := h T ( i, ∅ ) , ( i ′ , ∅ ) i , for all ( i, i ′ ) ∈V × V ′ . We have the following two possibilities: (i) Either, M is not balanced, and then T i,i ′ = 0 for all ( i, i ′ ) ∈ M , or (ii) M is balanced and (5.10) T i · λ,i ′ · λ = T i,i ′ α i,λ α ′ i ′ ,λ , for all ( i, i ′ ) in M and all words λ for which the transition ( i, i ′ ) λ → ( j, j ′ ) is possible (forsome ( j, j ′ ) ).Proof. Let’s assume that T i,i ′ = 0 for some ( i, i ′ ) ∈ M and pick ( i, i ′ ) ∈ M so that | T i,i ′ | =max ( j,j ′ ) ∈M | T j,j ′ | . With the Schwarz inequality, we have, for all n ≥ | T i,i ′ | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | λ | = n α i,λ α ′ i ′ ,λ T i · λ,i ′ · λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X | λ | = n | α i,λ | | T i · λ,i ′ · λ | / X | λ | = n | α ′ i ′ ,λ | / ≤ | T i,i ′ | · · | T i,i ′ | . Thus, we must have equalities in all inequalities.Since we have equality, there exists some constant c = c ( i, i ′ , n ) ∈ C such that(5.11) α i,λ T i · λ,i ′ · λ = cα ′ i ′ ,λ , for all words λ with | λ | = n . Thus α ′ i ′ ,λ = 0 if α i,λ = 0, and also conversely, by symmetry, whichmeans that the transition i λ → i · λ is possible if and only if the transition i ′ λ → i ′ · λ is. Note that c cannot be 0 because of the assumptions. Since we have equality in the second inequality, it followsthat | T i · λ,i ′ · λ | = | T i,i ′ | if α i,λ = 0. By Proposition 5.1(iii), there are possible transitions from ( i, i ′ )to any other ( j, j ′ ) in M . Thus | T i,i ′ | is constant on M .We have then T i,i ′ = X | λ | = n α i,λ α ′ i ′ ,λ T i · λ,i ′ · λ = X | λ | = n c | α ′ i ′ ,λ | = c. This and (5.11) implies (5.10).From this, we get that | α i,λ || T i · λ,i ′ · λ | = | T i,i ′ || α ′ i ′ ,λ | for all | λ | = n . Since | T i,i ′ | is constant on M ,we get that | α i,λ | = | α ′ i ′ ,λ | when the transition i λ → i · λ is possible. If the transition is not possible,then | α i,λ | = | α ′ i ′ ,λ | = 0. Thus condition (i), for M to be balanced, is satisfied.Now take a point ( i M , i ′M ) in M . With (5.5), we have, T i M ,i ′M = X λ ∈ F ( i M ,i ′M ) α i M ,λ α ′ i ′M ,λ T i M · λ,i ′M · λ = X λ ∈ F ( i M ,i ′M ) α i M ,λ α ′ i ′M ,λ T i M ,i ′M . UNTZ ALGEBRAS AND RANDOM WALKS 21
Since T i M ,i ′M = 0 we get(5.12) X λ ∈ F ( i M ,i ′M ) α i M ,λ α ′ i ′M ,λ = 1 . Using the Schwarz inequality and (5.8), we get1 ≤ X λ ∈ F ( i M ,i ′M ) | α i M ,λ | / X λ ∈ F ( i M ,i ′M ) | α ′ i ′M ,λ | / = 1 . Thus, we have equality in the Schwarz inequality, so there exists a constant c ∈ C such that α i M ,λ = cα ′ i ′M ,λ for all λ ∈ F ( i M , i ′M ). Using (5.12) again, we get that c = 1. Thus α i M ,λ = α i ′M ,λ for all λ ∈ F ( i M , i ′M ). Since every loop at ( i M , i ′M ) is a concatenation of loops from F ( i M , i ′M ),and since ( i M , i ′M ) is arbitrary, it follows that condition (ii), for M to be balanced, is satisfied.Thus, M is balanced. (cid:3) The next Lemma shows that, if we can construct an operator T in B ( K , K ′ ) σ from some arbitraryprescribed matrix entries on balanced minimal invariant sets. Lemma 5.8.
For each minimal invariant set M , let c i M ,i ′M = 0 if M is not balanced and let c i M ,i ′M be in C , arbitrary, if M is balanced. Define the operator T : K → K ′ by (5.13) (cid:10) T ( i, ∅ ) , ( i ′ , ∅ ) (cid:11) = X λ ∈ F ( i,i ′ ) α i,λ α i ′ ,λ c i · λ,i ′ · λ . Then the operator T satisfies (5.4) .Proof. Let ( i, i ′ ) ∈ V × V ′ . We compute S := X λ ∈ Λ α i,λ α i ′ ,λ T i · λ ,i ′ · λ . We split the sum in two. Consider the set A of λ ∈ Λ such that ( i · λ , i ′ · λ ) = ( i M , i ′M ) for someminimal invariant set M . Then S = X λ ∈ A α i,λ α i ′ ,λ T i · λ ,i ′ · λ + X λ A α i,λ α i ′ ,λ T i · λ ,i ′ · λ But if λ ∈ A , so ( i · λ , i ′ · λ ) = ( i M , i ′M ) for some M , then, we can assume M is balanced,otherwise T i · λ ,i ′ · λ = 0, and we have: T i · λ ,i ′ · λ = T i M ,i ′M = X λ ∈ F ( i M ,i ′M ) α i M ,λ α ′ i ′M ,λ c i M · λ,i ′M · λ = X λ ∈ F ( i M ,i ′M ) α i M ,λ α i ′M ,λ c i M ,i ′M = c i M ,i ′M X λ ∈ F ( i M ,i ′M ) | α i M ,λ | = c i M ,i ′M . Thus S = X λ ∈ A α i,λ α ′ i ′ ,λ c i · λ ,i ′ · λ + X λ A α i,λ α ′ i ′ ,λ X λ ∈ F ( i · λ ,i ′ · λ ) α i · λ ,λ α i ′ · λ ,λ c i · λ λ,i ′ · λ λ = X λ ∈ A α i,λ α ′ i ′ ,λ c i · λ ,i ′ · λ + X λ A X λ ∈ F ( i · λ ,i ′ · λ ) α i,λ λ α ′ i ′ ,λ λ c i · λ λ,i ′ · λ λ = X γ ∈ F ( i,i ′ ) α i,γ α ′ i ′ ,γ c i · γ,i ′ · γ = T i,i ′ . This proves (5.4). (cid:3)
Proof of Theorem 1.8.
Lemma 5.8 shows that the map in (1.10) is well defined. Lemma 5.7 showsthat T i,i ′ has to be zero on non-balanced minimal sets, and then Theorem 5.3 shows that the mapsare inverse to each other. It is clear that the maps are linear. (cid:3) Definition 5.9.
We say that the random walk is connected if, for any pair of vertices i, j ∈ V ,there is a possible transition from i to j . We say that the random walk is separating , if, for anypair of distinct vertices i = i ′ in V , there exists n ∈ N such that, for any vertices j, j ′ in V and forany word λ of length | λ | = n , either the transition i λ → j is not possible or the transition i ′ λ → j ′ isnot possible. Corollary 5.10.
If the random walk is connected and separating, then the Cuntz dilation is irre-ducible.Proof.
We prove that there is only one balanced minimal invariant set. Let M be a balancedminimal invariant set. Suppose there is a pair ( i, i ′ ) in M with i = i ′ . Since the random walk isseparating there exists n ∈ N such that for every word λ with | λ | = n , either α i,λ = 0 or α i ′ ,λ = 0.Since the | α i,λ | are probabilities, there exists λ with | λ | = n such that α i,λ = 0. Then α i ′ ,λ = 0,which contradicts the fact that M is balanced.Thus M cannot contain pairs ( i, i ′ ) with i = i ′ . Let ( i, i ) ∈ M , since the random walk is connectedthe orbit of ( i, i ) is { ( j, j ) : j ∈ V} . By Proposition 5.1, it follows that M = { ( j, j ) : j ∈ V} .Hence, there is only one balanced minimal invariant set, which means, according to Theorem 1.8,that the commutant of the Cuntz dilation is one-dimensional so the Cuntz dilation is irreducible. (cid:3) Examples
Example 6.1.
Let G be a finite group and let Λ be a set of generators of G in the sense that eachelement in G is a product of elements of Λ. Let N = | Λ | . We consider the Cayley graph of thegroup: the vertices are the elements of G , and, for each g ∈ G , there is an edge from g to λg , forall λ ∈ Λ. We will take the numbers α g,λ = √ N , for all g ∈ G , λ ∈ Λ. This corresponds to equalprobability of transition, for all λ ∈ Λ.Then, the Hilbert space K = l ( G ). The canonical vectors are ~g = δ g , g ∈ G . For the row-coisometry, we have V ∗ λ δ g = √ N δ λg , which means that V ∗ λ = √ N L ( λ ), λ ∈ Λ, where ( L ( g )) g ∈ G isthe left regular representation of G .Next we compute the minimal invariant sets. Let M be a minimal invariant set. Take ( g , g ) ∈M . Then, M = O ( g , g ), by Proposition 5.1(iii). So, for ( x, y ) ∈ M , there exists a word λ = λ . . . λ n such that g λ → x and g λ → y , so x = λ n . . . λ g and y = λ n . . . λ g . Then x − y = g − g . Conversely, if x − y = g − g , then yg − = xg − = λ n . . . λ for some elements λ , . . . , λ n in Λ. Then x = λ n . . . λ g and y = λ n . . . λ g , which means that the transitions g λ → x , g λ → y are possible with the word λ . . . λ n . UNTZ ALGEBRAS AND RANDOM WALKS 23
Thus each minimal invariant set is of the form M = { ( x, y ) ∈ G × G : x − y = g } , for some g ∈ G . Note that the minimal invariant sets form a partition of G × G , corresponding tothe right-cosets of the diagonal subgroup in G × G .All minimal invariant sets are balanced, because the transition probabilities are equal.Next, we compute the elements in B ( K ) σ . Let T in B ( K ) σ . By Lemma 5.7(ii), T x,y is constanton each minimal invariant set. Define, for all g ∈ G , T ( g ) x,y = (cid:26) , if x − y = g, , otherwise. , ( x, y ∈ G ) . Then T ( g ) is in B ( K ) σ and every operator in B ( K ) σ is a linear combination P g ∈ G a g T ( g ).Note that T ( g ) δ a = δ ag − , a, g ∈ G . Therefore T ( g ) = R ( g ), where ( R ( g )) g ∈ G is the right regularrepresentation of G . Thus B ( K ) σ is the linear span of the right regular representation.Moving on to the Cuntz dilation, for each g ∈ G , let A ( g ) be the operator in the commutant ofthe Cuntz dilation that corresponds to T ( g ), by the map A P K AP K = T . The map is a linear ∗ isomorphism. We check that it preserves the product too.By Proposition 3.4, for g , g ∈ G , the operator A ( g ) A ( g ) corresponds to the operator T ( g ) ∗ T ( g ) which is the SOT-limit of X | λ | = n V λ T ( g ) T ( g ) V ∗ λ . But T ( g ) T ( g ) = R ( g ) R ( g ) = R ( g g ) = T ( g g ) which is also in B ( K ) σ . Therefore, X | λ | = n V λ T ( g ) T ( g ) V ∗ λ = T ( g g ) , so T ( g ) ∗ T ( g ) = T ( g ) T ( g ) = T ( g g ), which corresponds to A ( g g ). So A ( g ) A ( g ) = A ( g g ),which means that the isomorphism preserves the product too. Example 6.2.
We consider the groups Z /M Z for M ∈ N . As with the Example 6.1, we will takethe generators to be Λ = { +1 , − } , with the random walk having equal probabilities and positive α i,λ . Let V be the following graph: V = 0 1 . . . M − Via the Example 6.1, we already know that the minimal invariant sets of
V × V are in one-to-onecorrespondence with the cosets of the diagonal subgroup H = { ( i, i ) : i ∈ Z /M Z } E Z /M Z × Z /M Z . We also want to understand the intertwiners between the graphs of Z /M Z and Z /N Z . To this end,let V ′ be the labeled random walk associated to Z /N Z , with equal probabilities of transition, andpositive α ′ i ′ ,λ . We see that:i. For each ( i, i ′ ) ∈ V × V ′ , O ( i, i ′ ) is a balanced, minimal invariant set.ii. The dimension of the intertwiners is exactly gcd( M, N ). Proof. (i) A quick calculation shows that O ( i, i ′ ) = { ( i + M k, i ′ + N k ) : k ∈ N } , which is by defini-tion invariant. It is also minimal since for every two elements in the orbit ( l, l ′ ) , ( j, j ′ ) ∈ O ( i, i ′ ),the transition ( l, l ′ ) → ( j, j ′ ) is possible. Further, it follows directly from the assumption that α i,λ = α ′ i ′ ,λ is constant for all i, i ′ , λ , that O ( i, i ′ ) is also balanced.(ii) Recall that every minimal invariant set can be written as the orbit of some element. Now,consider the product group Z /M Z × Z /N Z , and the cyclic subgroup H = h (1 , i . There is a one-to-one correspondence between elements of the quotient group ( Z /M Z × Z /N Z ) /H and minimalinvariant sets (given by φ (( i, i ′ ) H ) = O ( i, i ′ )). The cardinality of H is lcm ( M, N ). Thus, thecardinality of the quotient group, and thus the number of minimal invariant sets, is MN lcm( M,N ) =gcd( M, N ). (cid:3) Example 6.3.
We illustrate our theory here with another example. Consider the following graphs: V = 0 1 234 λ λ λ λ λ λ λ V ′ = 0 1 234 5 λ λ λ λ λ λ λ λ We will again consider the case where all α i,λ , α ′ i ′ ,λ are positive, and the probabilities of transitionare split evenly between all possible transitions. α ,λ = 0 α ,λ = α ,λ = 1 √ α ,λ = α ,λ = 0 α ,λ = 1 α ,λ = α ,λ = 0 α ,λ = 1 α ,λ = 0 α ,λ = α ,λ = 1 √ α ,λ = α ,λ = 0 α ,λ = 1 α ′ ,λ = 0 α ′ ,λ = α ′ ,λ = 1 √ α ′ ,λ = α ′ ,λ = 0 α ′ ,λ = 1 α ′ ,λ = α ′ ,λ = 0 α ′ ,λ = 1 α ′ ,λ = 0 α ′ ,λ = α ′ ,λ = 1 √ α ′ ,λ = α ′ ,λ = 0 α ′ ,λ = 1 α ′ ,λ = α ′ ,λ = 0 α ′ ,λ = 1To analyze the reducibility of the Cuntz dilation of V , we consider the graph of V × V . We havethe following orbits on
V × V : O (0 ,
0) = { ( i, i ) : i ∈ V}O (1 ,
1) = { (1 , , (2 , , (3 , } = O (2 ,
2) = O (3 , O (4 ,
4) = { (4 , }O (0 ,
1) = { (0 , }O (0 ,
2) = { (0 , }O (0 ,
3) = { (0 , , (4 , , (4 , , (4 , }O (0 ,
4) = { (0 , }O (1 ,
0) = { (1 , }O (1 ,
2) = { (1 , , (2 , , (3 , } = O (2 ,
3) = O (3 , O (1 ,
3) = { (1 , , (2 , , (3 , } = O (2 ,
1) = O (3 , O (1 ,
4) = { (1 , , (2 , , (3 , } = O (2 ,
4) = O (3 , O (2 ,
0) = { (2 , }O (3 ,
0) = { (3 , , (2 , , (3 , , (1 , }O (4 ,
0) = { (4 , }O (4 ,
1) = { (4 , , (4 , , (4 , } = O (4 ,
2) = O (4 , . Which gives us the following minimal invariant sets: O (1 , O (4 , O (0 , O (0 , O (0 , O (1 , O (1 , O (1 , O (1 , O (2 , O (4 , O (4 , . Of which only the following are balanced: O (1 , O (4 , , since all the other minimal invariants do not meet condition (i) of Definition 1.7. So the commutantof the Cuntz dilation of V is two-dimensional. Similarly, we can compute the dimension of thecommutant of the Cuntz dilation of V ′ . We see that the orbits are: O (0 ,
0) = { ( i, i ) : i ∈ V}O (1 ,
1) = { (1 , , (2 , , (3 , } = O (2 ,
2) = O (3 , O (4 ,
4) = { (4 , , (5 , } = O (5 , O (0 ,
1) = { (0 , }O (0 ,
2) = { (0 , }O (0 ,
3) = { (0 , , (4 , , (5 , , (4 , , (5 , , (4 , , (5 , }O (0 ,
4) = { (0 , }O (0 ,
5) = { (0 , }O (1 ,
0) = { (1 , }O (1 ,
2) = { (1 , , (2 , , (3 , } = O (2 ,
3) = O (3 , O (1 ,
3) = { (1 , , (2 , , (3 , } = O (2 ,
1) = O (3 , O (1 ,
4) = { (1 , , (2 , , (3 , , (1 , , (2 , , (3 , } == O (2 ,
5) = O (3 ,
4) = O (1 ,
5) = O (2 ,
4) = O (3 , O (2 ,
0) = { (2 , }O (3 ,
0) = { (3 , , (2 , , (3 , , (1 , , (2 , , (3 , , (1 , }O (4 ,
0) = { (4 , }O (4 ,
1) = { (4 , , (5 , , (4 , , (5 , , (4 , , (5 , } = O (5 ,
2) = O (4 ,
3) = O (5 ,
1) = O (4 ,
2) = O (5 , O (4 ,
5) = { (4 , , (5 , } = O (5 , . These yield the following minimal invariant sets: O (1 ,
1) = { (1 , , (2 , , (3 , } = O (2 ,
2) = O (3 , O (4 ,
4) = { (4 , , (5 , } = O (5 , O (0 ,
1) = { (0 , } UNTZ ALGEBRAS AND RANDOM WALKS 27 O (0 ,
2) = { (0 , }O (0 ,
4) = { (0 , }O (0 ,
5) = { (0 , }O (1 ,
0) = { (1 , }O (1 ,
2) = { (1 , , (2 , , (3 , } = O (2 ,
3) = O (3 , O (1 ,
3) = { (1 , , (2 , , (3 , } = O (2 ,
1) = O (3 , O (1 ,
4) = { (1 , , (2 , , (3 , , (1 , , (2 , , (3 , } == O (2 ,
5) = O (3 ,
4) = O (1 ,
5) = O (2 ,
4) = O (3 , O (2 ,
0) = { (2 , }O (4 ,
0) = { (4 , }O (4 ,
1) = { (4 , , (5 , , (4 , , (5 , , (4 , , (5 , } = O (5 ,
2) = O (4 ,
3) = O (5 ,
1) = O (4 ,
2) = O (5 , O (4 ,
5) = { (4 , , (5 , } = O (5 , . Of which, the following are balanced: O (1 ,
1) = { (1 , , (2 , , (3 , } = O (2 ,
2) = O (3 , O (4 ,
4) = { (4 , , (5 , } = O (5 , O (4 ,
5) = { (4 , , (5 , } = O (5 , . So we have that the Cuntz dilation of V ′ has dimension 3. Further, for any T ∈ B σ , T has theform: T = a + b a a a b c c b , a, b, c ∈ C , where the entry T , = P λ ∈ Λ α ,λ , α ′ ,λ T · λ, · λ = ( T , + T , ).Now, we look at the intertwiners between the Cuntz dilations of V and V ′ . By the same processwe first identify the balanced, minimal, invariant sets of V × V ′ as: O (1 ,
1) = { (1 , , (2 , , (3 , } = O (2 ,
2) = O (3 , O (4 ,
4) = { (4 , , (4 , } = O (4 , . So then the space of intertwiners between the Cuntz dilations of V and V ′ has dimension 2, andany such intertwiner has the form: T = a + b a a a
00 0 0 0 b b , a, b ∈ C , where the entry T , = P λ ∈ Λ α ,λ , α ′ ,λ T · λ, · λ = ( T , + T , ). Example 6.4.
This example will show that the phase of the numbers α i,λ matters for the re-ducibility of the Cuntz dilation.We consider the following graph: V = 123 λ λ λ λ λ λ with the α i,λ defined to be: α ,λ = 1 √ , α ,λ = 1 √ α ,λ = i √ , α ,λ = 1 √ α ,λ = − √ , α ,λ = 1 √ . We see that the minimal invariant sets of
V × V are: M = { (1 , , (2 , , (3 , }M = { (1 , , (2 , , (3 , }M = { (1 , , (2 , , (3 , } . So then we look to see if these minimal invariants are balanced. For each minimal invariant set M ,take ( i M , i ′M ) to be the first one in the above set.For each minimal invariant, we clearly have condition (i) since | α i,λ | = (cid:12)(cid:12) α i ′ ,λ (cid:12)(cid:12) for all λ ∈ Λ andall i ∈ V . However, for the minimal invariants M and M , condition (ii) does not hold since forthe loops (1 , λ λ → (1 ,
2) and (1 , λ λ → (1 , α ,λ α · λ ,λ = 12 α ,λ α · λ ,λ = i α ,λ α · λ ,λ = − . Therefore, for some arbitrary T ∈ B σ , we have that T i,i ′ = 0 , ( i, i ′ ) ∈ M ∪ M T i,i = T j,j , i, j = 1 , , . So T = cI for some c ∈ C , and the Cuntz dilation of V is irreducible. UNTZ ALGEBRAS AND RANDOM WALKS 29
In contrast, we know from Example 6.2, that if all the α i,λ are equal (i.e. we consider the graphof Z / Z with equal probabilities of transition and real, positive α i,λ ), then the Cuntz dilation isreducible with dim B σ = 3. Thus the choice of phase for the α i,λ does matter. Acknowledgements.
We would like to thank professor Deguang Han for very helpful conversations.
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J.Anal. Math. , 81:209–238, 2000. [Dorin Ervin Dutkay] University of Central Florida, Department of Mathematics, 4000 CentralFlorida Blvd., P.O. Box 161364, Orlando, FL 32816-1364, U.S.A.,
E-mail address : [email protected] [Nicholas Christoffersen] Louisiana State University, Department of Mathematics, Baton Rouge,LA 70803-4918, U.S.A.,
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