New York Journal of Mathematics New York J. Math.

New York Journal of Mathematics

New York J. Math.22(2016) 277–291.

Realising the Toeplitz algebra of a higher-rank graph as a Cuntz–Krieger

algebra

Yosafat E. P. Pangalela

Abstract. For a row-finite higher-rank graph Λ, we construct a higher- rank graph TΛ such that the Toeplitz algebra of Λ is isomorphic to the Cuntz–Krieger algebra ofTΛ. We then prove that the higher-rank graphTΛ is always aperiodic and use this fact to give another proof of a uniqueness theorem for the Toeplitz algebras of higher-rank graphs.

Contents

1. Introduction 277

2. Higher-rank graphs 278

3. The k-graph TΛ 280

4. RealisingT C^∗(Λ) as a Cuntz–Krieger algebra 283

References 290

1. Introduction

Higher-rank graphs and their Cuntz–Krieger algebras were introduced by Kumjian and Pask in [5] as a generalisation of the Cuntz–Krieger algebras of directed graphs. Kumjian and Pask proved an analogue of the Cuntz–

Krieger uniqueness theorem for a family ofaperiodic higher-rank graphs [5, Theorem 4.6]. Aperiodicity is a generalisation of Condition (L) for directed graphs and comes in several forms for different kinds of higher-rank graphs (see [1, 5, 6, 10, 11, 12, 13, 14]).

The Toeplitz algebra of a directed graph is an extension of the Cuntz–

Krieger algebra in which the Cuntz–Krieger equations at vertices are re- placed by inequalities. An analogous family of Toeplitz algebras for higher- rank graph was introduced and studied by Raeburn and Sims [9]. They

Received April 29, 2015.

2010Mathematics Subject Classification. Primary 46L05.

Key words and phrases. Cuntz–Krieger algebra, Toeplitz algebra, higher-rank graph, the Cuntz–Krieger uniqueness theorem, the uniqueness theorem for Toeplitz algebras.

This research is part of the author’s Ph.D. thesis, supervised by Professor Iain Raeburn and Dr. Lisa Orloff Clark.

ISSN 1076-9803/2016

277

proved a uniqueness theorem for Toeplitz algebras [9, Theorem 8.1], gene- ralising a previous theorem for directed graphs [3, Theorem 4.1].

For a directed graphE, the Toeplitz algebra ofEis canonically isomorphic to the Cuntz–Krieger algebra of a graphT E (see [7, Theorem 3.7] and [15, Lemma 3.5]). Here we provide an analogous construction for a row-finite higher-rank graph Λ. We build a higher-rank graph TΛ, and show that the Toeplitz algebra of Λ is canonically isomorphic to the Cuntz–Krieger algebra of TΛ (Theorem 4.1). Our proof relies on the uniqueness theorem of [9]. However, it is interesting to observe that the higher-rank graph TΛ is always aperiodic. Hence our isomorphism shows that the uniqueness theorem of [9] is a consequence of the general Cuntz–Krieger uniqueness theorem of [11] (see Remark 4.3).

2. Higher-rank graphs

Let k be a positive integer. We regard N^k as an additive semigroup with identity 0. For m, n ∈ N^k, we write m∨n for their coordinate-wise maximum.

A higher-rank graph ork-graph is a pair (Λ, d) consisting of a countable small category Λ together with a functor d: Λ→N^k satisfying the factori- sation property: for every λ∈ Λ andm, n ∈ N^k with d(λ) =m+n, there are unique elements µ, ν ∈ Λ such that λ =µυ and d(µ) = m, d(ν) = n.

We then write λ(0, m) for µ and λ(m, m+n) for ν. We regard elements of Λ⁰ as vertices and elements of Λ as paths. For detailed explanation and examples, see [8, Chapter 10].

Forv∈Λ⁰ andE ⊆Λ, we define vE:={λ∈E :r(λ) =v} and m∈N^k, we write Λ^m := {λ∈Λ :d(λ) =m}.We use term edge to denote a path e∈Λ^ei where 1≤i≤k, and write

Λ¹:= [

1≤i≤k

Λ^ei

for the set of all edges. We say that Λ is row-finite if for every v ∈Λ⁰, the set vΛ^ei is finite for 1≤i ≤k. Finally, we say v ∈Λ⁰ is a source if there existsm∈N^k such thatvΛ^m=∅.

For a row-finitek-graph Λ, we shall construct ak-graphTΛ which is row- finite and always has sources. Ourk-graphTΛ is typically notlocally convex in the sense of [10, Definition 3.9] (see Remark 3.3), so the appropriate defi- nition of Cuntz–Krieger Λ-family is the one in [11]. For detailed discussion about row-finitek-graphs and their generalisations, see [16, Section 2].

From now on, we focus on a row-finite k-graph Λ. For λ, µ∈Λ, we say thatτ is aminimal common extension of λand µif

d(τ) =d(λ)∨d(µ) , τ(0, d(λ)) =λand τ(0, d(µ)) =µ.

Let MCE (λ, µ) denote the collection of all minimal common extensions of λand µ. Then we write

Λ^min(λ, µ) :=

λ⁰, µ⁰

∈Λ×Λ :λλ⁰ =µµ⁰∈MCE (λ, µ) .

A set E ⊆vΛ¹ is exhaustive if for all λ∈vΛ, there exists e∈E such that Λ^min(λ, e)6=∅.

A Toeplitz–Cuntz–Krieger Λ-family is a collection{t_λ:λ∈Λ} of partial isometries in aC^∗-algebra B satisfying:

(TCK1)

tv :v∈Λ⁰ is a collection of mutually orthogonal projections.

(TCK2) t_λt_µ=t_λµ whenever s(λ) =r(µ).

(TCK3) t^∗_λtµ=P

(λ⁰,µ⁰)∈Λ^min(λ,µ)tλ⁰t^∗_µ0 for all λ, µ∈Λ.

Remark 2.1. In [9, Lemma 9.2], Raeburn and Sims required also that

“for all m ∈ N^k\ {0}, v ∈ Λ⁰, and every set E ⊆ vΛ^m, tv ≥ P

λ∈Et_λt^∗_λ”.

However, by [11, Lemma 2.7 (iii)], this follows from (TCK1)–(TCK3), and hence our definition is basically same as that of [9].

Meanwhile, based on [11, Proposition C.3], aCuntz–Krieger Λ-family is a Toeplitz–Cuntz–Krieger Λ-family{t_λ:λ∈Λ} which satisfies

(CK) Q

e∈E(t_v−t_et^∗_e) = 0 for all v∈Λ⁰ and exhaustiveE ⊆vΛ¹.

Raeburn and Sims proved in [9, Section 4] that there is a C^∗-algebra T C^∗(Λ) generated by a universal Toeplitz–Cuntz–Krieger Λ-family

{t_λ :λ∈Λ}.

If{T_λ :λ∈Λ}is a Toeplitz–Cuntz–Krieger Λ-family in aC^∗-algebraB, we write φ_T for the homomorphism of T C^∗(Λ) intoB such that φ_T(t_λ) =T_λ forλ∈Λ. The quotient ofT C^∗(Λ) by the ideal generated by

( Y

e∈E

(tv−tet^∗_e) :v∈Λ⁰, E⊆vΛ¹ is exhaustive )

is generated by a universal family of the Cuntz–Krieger Λ-family {s_λ :λ∈Λ},

and hence we can identify it with the C^∗-algebra C^∗(Λ). For a Cuntz–

Krieger Λ-family {S_λ:λ∈Λ} in a C^∗-algebra B, we write πS for the ho- morphism ofC^∗(Λ) intoB such thatπ_S(s_λ) =S_λ forλ∈Λ. Furthermore, we have sv 6= 0 for v∈Λ⁰ [11, Proposition 2.12].

As for directed graphs, we have uniqueness theorems for the Toeplitz algebra [9, Theorem 8.1] and the Cuntz–Krieger algebra [6, Theorem 4.7].

The former does not need any hypothesis on the k-graph as stated in the following theorem.

Theorem 2.2. LetΛbe a row-finitek-graph. Let{T_λ :λ∈Λ}be a Toeplitz–

Cuntz–Krieger Λ-family in aC^∗-algebra B. Suppose that for every v∈Λ⁰,

(∗) Y

e∈vΛ¹

(Tv−TeT_e^∗)6= 0

(where this includes T_v 6= 0 if vΛ¹=∅). Suppose that φ_T :T C^∗(Λ)→B is the homomorphism such that φT (tλ) =Tλ for λ∈Λ. Then

φ_T :T C^∗(Λ)→B is injective.

Remark 2.3. Every k-graph Λ gives a product system of graphs overN^k and a Toeplitz–Cuntz–Krieger Λ-family gives a Toeplitz Λ-family of the product system [9, Lemma 9.2]. Lemma 9.3 of [9] shows that, if the Toeplitz–

Cuntz–Krieger Λ-family satisfies (∗), then the Toeplitz Λ-family satisfies the hypothesis of [9, Theorem 8.1].

Remark 2.4. In the actual hypothesis, we need to verify whether Y

1≤i≤k



T_v− X

e∈Gi

T_eT_e^∗



6= 0

for every v ∈ Λ⁰, 1 ≤ i ≤k, and finite set G_i ⊆ vΛ^ei. However, since we only consider row-finite k-graphs, then for everyv∈Λ⁰ and 1≤i≤k, the setvΛ^ei is finite. Thus for a row finitek-graph, we can simplify Lemma 9.3 of [9] as Theorem 2.2.

On the other hand, Lewin and Sims in [6, Theorem 4.7] proved that the Cuntz–Krieger uniqueness theorem only holds fork-graphs which satisfy the following aperiodicity condition: for every pair of distinct paths λ, µ ∈ Λ with s(λ) = s(µ), there exists η ∈ s(λ) Λ such that MCE (λη, µη) =∅ [6, Definition 3.1]. (For discussion about the equivalence of various aperiodicity definitions, see [6, 12, 13, 14].) Now we state the uniqueness theorem as follows:

Theorem 2.5 ([6, Theorem 4.7]). Suppose that Λis an aperiodic row-finite k-graph and {S_λ :λ∈Λ} is a Cuntz–Krieger Λ-family in a C^∗-algebra B such thatS_v 6= 0forv∈Λ⁰. Suppose that π_S :C^∗(Λ)→B is the homomor- phism such that πS(sλ) =Sλ for λ∈Λ for λ∈Λ. Then πS is an injective homomorphism.

3. The k-graph TΛ

Suppose that Λ is a row-finitek-graph. In this section, we define ak-graph TΛ; later we show that T C^∗(Λ) ∼= C^∗(TΛ) (Theorem 4.1). Interestingly, ourk-graph TΛ is always aperiodic (Proposition 3.5).

Proposition 3.1. Let Λ = (Λ, d, r, s) be a row-finite k-graph. Then define sets TΛ⁰ and TΛ as follows:

TΛ⁰ :=

α(v) :v∈Λ⁰ ∪

β(v) :vΛ¹ 6=∅ ; TΛ :={α(λ) :λ∈Λ} ∪

β(λ) :λ∈Λ, s(λ) Λ¹6=∅ .

Define functions r, s:TΛ\TΛ⁰→TΛ⁰ by

r(α(λ)) =α(r(λ)), s(α(λ)) =α(s(λ)), r(β(λ)) =α(r(λ)), s(β(λ)) =β(s(λ))

(r, s are the identity on TΛ⁰). We also define a partially defined product (τ, ω)7→τ ω from

{(τ, ω)∈TΛ×TΛ :s(τ) =r(ω)}

to TΛ, where

(α(λ), α(µ))7→α(λµ) (α(λ), β(µ))7→β(λµ) and a functiond:TΛ→N^k where

d(α(λ)) =d(β(λ)) =d(λ). Then (TΛ, d) is a k-graph.

Proof. First we claim that TΛ is a countable category. Note that TΛ is countable since Λ is countable.

Now we show that for all paths η, τ, ω in TΛ where s(η) = r(τ) and s(τ) = r(ω), we have s(τ ω) = s(ω), r(τ ω) = r(τ), and (ητ)ω =η(τ ω).

If one of τ, ω is a vertex then we are done. So assume otherwise, and we have η = α(λ), τ = α(µ), and ω is either α(ν) or β(ν) for some paths λ, µ, ν in Λ. In both cases, we always have s(λ) =r(µ), s(µ) = r(ν), and (λµ)ν =λ(µν). If ω=α(ν), we have

s(τ ω) =s(α(µ)α(ν)) =s(α(µν))

=α(s(µν)) =α(s(ν)) =s(α(ν)) =s(ω) , r(τ ω) =r(α(µ)α(ν)) =r(α(µν))

=α(r(µν)) =α(r(µ)) =r(α(µ)) =r(τ) , and

(ητ)ω= α(λ)α(µ)

α(ν) =α(λµ)α(ν) =α((λµ)ν)

=α(λ(µν)) =α(λ)α(µν) =α(λ) α(µ)α(ν)

=η(τ ω) . On the other hand, if ω=β(ν), then

s(τ ω) =s(α(µ)β(ν)) =s(β(µν))

=β(s(µν)) =β(s(ν)) =s(β(ν)) =s(ω) , r(τ ω) =r(α(µ)β(ν)) =r(β(µν))

=α(r(µν)) =α(r(µ)) =r(α(µ)) =r(τ) , and

(ητ)ω= α(λ)α(µ)

β(ν) =α(λµ)β(ν) =β((λµ)ν)

=β(λ(µν)) =α(λ)β(µν) =α(λ) α(µ)β(ν)

=η(τ ω) .

Thus,TΛ is a countable category, as claimed.

Now we show thatdis a functor. Note that bothTΛ andN^kare categories.

First take objectx∈TΛ⁰, thend(x) = 0 is an object in categoryN^k. Next take morphismsτ, ω∈TΛ withs(τ) =r(ω). Then by definition ofd,

d(τ ω) =d(τ) +d(ω) . Hence,dis a functor.

To show that d satisfies the factorisation property, take ω ∈ TΛ and m, n ∈ N^k such that d(ω) = m+n. By definition, ω is either α(λ) or β(λ) for some pathλ in Λ. In both cases, there exist paths µ, ν in Λ such that λ = µν, d(µ) = m, and d(ν) = n. Then, we have d(α(µ)) = m, d(α(ν)) =d(β(ν)) =n, andω is either equal toα(µ)α(ν) orα(µ)β(ν).

Therefore, the existence of factorisation is guaranteed.

Now we show that the factorisation is unique. First suppose ω=α(µ)α(ν) =α µ⁰

α ν⁰

where d(α(µ)) = d(α(µ⁰)) and d(α(ν)) = d(α(ν⁰)). We consider paths λ=µν and λ⁰ =µ⁰ν⁰. Since α(λ) = ω =α(λ⁰), then λ=λ⁰. This implies µ = µ⁰ and ν = ν⁰ based on the uniquness of factorisation in Λ. Then α(µ) = α(µ⁰) and α(ν) =α(ν⁰). For the case ω = α(µ)β(ν), we get the same result by using the same argument. The conclusion follows.

Remark 3.2. For a directed graph E (that is, for k = 1), the graph T E was constructed by Muhly and Tomforde [7, Definition 3.6] (denoted E_V), and by Sims [15, Section 3] (denotedE). Our notation follows that of Simse because we want to distinguish between paths in TΛ (denoted α(λ) and β(λ)) and those in Λ (denoted λ).

Remark 3.3. Every vertex β(v) satisfies β(v)TΛ¹ = ∅. Then if Λ has a vertex v which receives edges e, f with d(e) 6=d(f), then there is no edge g ∈β(s(e))TΛ^d(f) (or g ∈α(s(e))TΛ^d(f) ifs(e) Λ = ∅), and hence TΛ is not locally convex.

To give an illustration how we construct the k-graph TΛ from a k-graph Λ, we first recall coloured graphs of [4]. By choosing k-different colours c1, . . . , ck, we can view paths in Λ^ei as edges of colour ci. For a k-graph Λ, we call its corresponding coloured graph the skeleton of Λ. For further discussion about k-graphs and their skeletons, see [4].

Example 3.4. Consider the 2-graph Λ which has skeleton

v • e1

e₂ f1

f₂

where e_if_j =f_ie_j for alli, j ∈ {1,2}, the solid edges have degree (1,0) and the dashed edges have degree (0,1). Then the 2-graphTΛ has skeleton

α(v) • •β(v)

α(e₁) α(e2) α(f₁) α(f2)

β(e₂) β(e₁) β(f₂) β(f1)

whereα(ei)α(fj) =α(fi)α(ej) andα(ei)β(fj) =α(fi)β(ej) for alli, j∈ {1,2}, the solid edges have degree (1,0) and the dashed edges have degree (0,1).

The following lemma tells about properties of the k-graph TΛ.

Proposition 3.5. Let Λ be a row-finite k-graph and TΛ be the k-graph as in Proposition 3.1. Then,

(a) TΛ is row-finite.

(b) TΛ is aperiodic.

Proof. To show part (a), take x∈TΛ⁰. If x=β(v) for some v∈Λ⁰, then xTΛ¹ =∅ by Remark 3.3. Suppose x =α(v) for somev ∈Λ⁰. If vΛ¹ =∅, thenxTΛ¹=∅. Otherwise, for 1≤i≤k such thatvΛ^ei 6=∅, we have

|xTΛ^ei| ≤2|vΛ^ei|, which is finite.

For part (b), takeτ, ω∈TΛ such thatτ 6=ωands(τ) =s(ω). We have to show there exists η∈s(τ)TΛ such that MCE (τ η, ωη) =∅. If s(τ) =β(v) for some v∈Λ⁰, then choose η =β(v) and MCE (τ η, ωη) =∅. So suppose s(τ) = α(v) for some v ∈ Λ⁰. If vΛ¹ = ∅, then choose η = α(v) and MCE (τ η, ωη) =∅. Suppose vΛ¹ 6=∅. Take e∈ vΛ¹. If s(e) Λ¹ =∅, then choose η = α(e) and MCE (τ η, ωη) =∅. Otherwise, we have s(e) Λ¹ 6=∅.

Then choose η=β(e) and MCE (τ η, ωη) =∅. Hence,TΛ is aperiodic.

4. Realising T C^∗(Λ) as a Cuntz–Krieger algebra

Let Λ be a row-finitek-graph andTΛ be thek-graph as in Proposition 3.1.

In this section, we show that T C^∗(Λ) is isomorphic to C^∗(TΛ).

Theorem 4.1. Let Λ be a row-finite k-graph and TΛ be the k-graph as in Proposition3.1. Let{t_λ :λ∈Λ}be the universal Toeplitz–Cuntz–Krieger

Λ-family and{s_ω:ω ∈TΛ} be the universal Cuntz–Krieger TΛ-family. For λ∈Λ, let

T_λ:=

(s_α(λ)+s_β(λ) if s(λ) Λ¹ 6=∅ s_α(λ) if s(λ) Λ¹ =∅.

Then there is an isomorphism φT :T C^∗(Λ)→C^∗(TΛ) satisfying φ_T (t_λ) =T_λ

for everyλ∈Λ.

Furthermore, s_α(λ) =φ_T(t_λ) if s(λ) Λ¹ =∅. Meanwhile, if s(λ) Λ¹ 6=∅, we have

s_α(λ)=φ_T



t_λ−t_λ Y

e∈s(λ)Λ¹

(t_v−t_et^∗_e)



,

s_β(λ)=φT



t_λ Y

e∈s(λ)Λ¹

(tv−tet^∗_e)



.

Proof that {T_λ :λ∈Λ} is a Toeplitz–Cuntz–Krieger Λ-family. To avoid an argument by cases, forλ∈Λ withs(λ) Λ¹ =∅, we write

s_β(λ) := 0, so that

T_λ =s_α(λ)+s_β(λ).

First, we want to show{T_λ :λ∈Λ}is a Toeplitz–Cuntz–Krieger Λ-family inC^∗(TΛ). For (TCK1), take v∈Λ⁰. Since

s_α(v) ∪

s_β(v) are mutually orthogonal projections, then Tv is a projection. Meanwhile, for v, w ∈ Λ⁰ withv6=w,

T_vT_w =s_α(v)s_α(w)+s_α(v)s_β(w)+s_β(v)s_α(w)+s_β(v)s_β(w) = 0.

Next we show (TCK2). Takeµ, ν ∈Λ wheres(µ) =r(ν). Then TµTν =s_α(µ)s_α(ν)+s_α(µ)s_β(ν)+s_β(µ)s_α(ν)+s_β(µ)s_β(ν). Ifν is a vertex, the middle terms vanish and we get

TµTν =s_α(µ)+s_β(µ)=Tµ,

as required. Otherwise, the last two terms vanish and we get T_µT_ν =s_α(µ)s_α(ν)+s_α(µ)s_β(ν)=s_α(µν)+s_β(µν) =T_µν, which is (TCK2).

To show (TCK3), take λ, µ∈Λ. Then

(4.1) T_λ^∗T_µ=s^∗_α(λ)s_α(µ)+s^∗_α(λ)s_β(µ)+s^∗_β(λ)s_α(µ)+s^∗_β(λ)s_β(µ).

We give separate arguments for Λ^min(λ, µ) = ∅ and Λ^min(λ, µ) 6= ∅. For case Λ^min(λ, µ) =∅, we have

∅=TΛ^min(α(λ), α(µ)) =TΛ^min(α(λ), β(µ))

=TΛ^min(β(λ), α(µ)) =TΛ^min(β(λ), β(µ)) .

Hence,s^∗_α(λ)s_α(µ)=s^∗_α(λ)s_β(µ) =s^∗_β(λ)s_α(µ) =s^∗_β(λ)s_β(µ) = 0 and then Equa- tion (4.1) becomes

T_λ^∗Tµ= 0 = X

(λ⁰,µ⁰)∈Λ^min(λ,µ)

Tλ⁰T_µ^∗0.

Now suppose Λ^min(λ, µ) 6= ∅. Take (a, b) ∈ Λ^min(λ, µ). We consider several cases: whether a equals s(λ) and/or b equals s(µ). First sup- pose a = s(λ) and b = s(µ). So λ = λs(λ) = µs(µ) = µ. Because α(λ) and β(λ) are paths with the same degree and different sources, then TΛ^min(α(λ), β(λ)) =∅. Thus,

s^∗_β(λ)s_α(λ) = 0 =s^∗_α(λ)s_β(λ) and Equation (4.1) becomes

T_λ^∗T_λ =s^∗_α(λ)s_α(λ)+s^∗_β(λ)s_β(λ)

=s_s(α(λ))+s_s(β(λ))=s_α(s(λ))+s_β(s(λ))

=T_s(λ) =T_s(λ)T^∗

s(λ)

= X

(λ⁰,µ⁰)∈Λ^min(λ,λ)

T_λ⁰T_µ^∗0 (since Λ^min(λ, λ) ={s(λ), s(λ)}).

Next supposea=s(λ) and b6=s(µ). Then λ=µband TΛ^min(α(λ), β(µ)) =∅=TΛ^min(β(λ), β(µ)) sinces(β(µ))TΛ¹=∅. Hence

s^∗_α(λ)s_β(µ)= 0 =s^∗_β(λ)s_β(µ) and Equation (4.1) becomes

T_λ^∗T_µ=s^∗_α(λ)s_α(µ)+s^∗_β(λ)s_α(µ).

Every (α(s(λ)), η)∈TΛ^min(α(λ), α(µ)) has η=α(µ⁰) with (s(λ), µ⁰)∈Λ^min(λ, µ).

Similarly, every (β(s(λ)), η) ∈ TΛ^min(β(λ), α(µ)) has η = β(µ⁰) with (s(λ), µ⁰)∈Λ^min(λ, µ). Thus, by using (TCK3) inC^∗(TΛ),

T_λ^∗Tµ

=s^∗_α(λ)s_α(µ)+s^∗_β(λ)s_α(µ)

= X

(α(s(λ)),η)∈TΛ^min(α(λ),α(µ))

s_α(s(λ))s^∗_η+ X

(β(s(λ)),η)∈TΛ^min(β(λ),α(µ))

s_β(s(λ))s^∗_η

= X

(s(λ),µ⁰)∈Λ^min(λ,µ)

s_α(s(λ))s^∗_α(µ0)+ X

(s(λ),µ⁰)∈Λ^min(λ,µ)

s_β(s(λ))s^∗_β(µ0)

= X

(s(λ),µ⁰)∈Λ^min(λ,µ)

(s_α(s(λ))s^∗_α(µ0)+s_β(s(λ))s^∗_β(µ0))

= X

(s(λ),µ⁰)∈Λ^min(λ,µ)

(s_α(s(λ))+s_β(s(λ)))(s^∗_α(µ0)+s^∗_β(µ0))

= X

(s(λ),µ⁰)∈Λ^min(λ,µ)

T_s(λ)T_µ^∗0 = X

(λ⁰,µ⁰)∈Λ^min(λ,µ)

T_λ⁰T_µ^∗0.

By taking adjoints, we deduce (TCK3) when a6=s(λ) and b=s(µ).

Now we consider the last case, which is a 6= s(λ) and b 6= s(µ). This means we have neitherλ=µbnorµ=λa. Hence,

TΛ^min(α(λ), β(µ)) =TΛ^min(β(λ), α(µ)) =TΛ^min(β(λ), β(µ)) =∅ sinces(β(λ))TΛ¹ =∅=s(β(µ))TΛ¹=∅. Hence,

s^∗_α(λ)s_β(µ)=s^∗_β(λ)s_α(µ)=s^∗_β(λ)s_β(µ)= 0.

On the other hand, we have TΛ^min(α(λ), α(µ))

= α λ⁰

, α µ⁰

, β λ⁰

, β µ⁰

: (λ⁰, µ⁰)∈Λ^min(λ, µ) . Therefore, Equation (4.1) becomes

T_λ^∗Tµ=s^∗_α(λ)s_α(µ)= X

(ω,η)∈TΛ^min(α(λ),α(µ))

sωs^∗_η

= X

(λ⁰,µ⁰)∈Λ^min(λ,µ)

(s_α(λ⁰₎s^∗_α(µ0)+s_β(λ⁰₎s^∗_β(µ0))

= X

(λ⁰,µ⁰)∈Λ^min(λ,µ)

(s_α(λ⁰₎+s_β(λ⁰₎)(s^∗_α(µ0)+s^∗_β(µ0))

= X

(λ⁰,µ⁰)∈Λ^min(λ,µ)

T_λ⁰T_µ^∗0. So for all cases, we have

T_λ^∗T_µ= X

(λ⁰,µ⁰)∈Λ^min(λ,µ)

T_λ⁰T_µ^∗0

and {T_λ:λ∈Λ} satisfies (TCK3).

Proof that φT is injective. Now the universal property ofT C^∗(Λ) gives a homomorphismφ_T :T C^∗(Λ)→C^∗(TΛ) satisfyingφ_T(t_λ) =T_λ for every λ∈Λ.

We show the injectivity of φ_T by using Theorem 2.2. Take v ∈ Λ⁰. We show

Y

e∈vΛ¹

(T_v−T_eT_e^∗)6= 0.

First supposevΛ¹ 6=∅. Take 1≤i≤k such thatvΛ^ei 6=∅. We claim Y

e∈vΛei

(Tv−TeT_e^∗)≥s_β(v).

Since vΛ^ei 6=∅, thenα(v)TΛ^ei 6=∅ and by [11, Lemma 2.7 (iii)], s_α(v)≥ X

g∈α(v)TΛ^ei

sgs^∗_g

= X

e∈vΛ^ei

s_α(e)s^∗_α(e)+ X

e∈vΛ^ei s(e)Λ¹6=∅

s_β(e)s^∗_β(e)

= X

e∈vΛ^ei s(e)Λ¹6=∅

s_α(e)s^∗_α(e)+s_β(e)s^∗_β(e)

+ X

e∈vΛ^ei s(e)Λ¹=∅

s_α(e)s^∗_α(e)

= X

e∈vΛei

s(e)Λ¹6=∅

TeT_e^∗+ X

e∈vΛei

s(e)Λ¹=∅

TeT_e^∗

= X

e∈vΛ^ei

T_eT_e^∗.

Meanwhile, since every e∈vΛ^ei has the same degree, Y

e∈vΛ^ei

(T_v−T_eT_e^∗) =T_v− X

e∈vΛ^ei

T_eT_e^∗

= s_α(v)+s_β(v)

− X

e∈vΛei

T_eT_e^∗

=s_β(v)+

s_α(v)− X

e∈vΛ^ei

TeT_e^∗

≥s_β(v), as claimed. This claim implies

Y

e∈vΛ¹

(Tv−TeT_e^∗)≥ Y

{i:vΛei6=∅}

s_β(v)=s_β(v) 6= 0 sincevΛ¹ 6=∅, as required.

Finally, for v∈Λ⁰ with vΛ¹ =∅, we have T_v =s_α(v)6= 0.

Hence, by Theorem 2.2, φ_T is injective.

Proof that φ_T is surjective. Now we show the surjectivity of φ_T. Since C^∗(TΛ) is generated by{s_τ :τ ∈TΛ}, then it suffices to show that for every τ ∈ TΛ, s_τ ∈ im (φ_T). Recall that for every τ ∈ TΛ, s_τ is either s_α(λ) or s_β(λ) for someλ∈Λ.

Takev∈Λ⁰. First we shows_α(v) ands_β(v)(if it exists) belong to im (φ_T).

IfvΛ¹ =∅, then

s_α(v)=Tv ∈im (φT) .

Next suppose vΛ¹ 6= ∅. First we show that s_β(v) = Q

e∈vΛ¹(T_v−T_eT_e^∗).

Note that for every f ∈ α(v)TΛ¹, the projection s_α(v) −s_fs^∗_f ≤ s_α(v) is othogonal tos_β(v). This implies

Y

f∈α(v)TΛ¹

((s_α(v)+s_β(v))−sfs^∗_f) =s_β(v)+ Y

f∈α(v)TΛ¹

(s_α(v)−sfs^∗_f)

=s_β(v), sincevΛ¹ is an exhaustive set. Hence,

s_β(v) = Y

f∈α(v)TΛ¹

((s_α(v)+s_β(v))−s_fs^∗_f)

= Y

e∈vΛ¹

(Tv−s_α(e)s^∗_α(e)) Y

e∈vΛ¹ s(e)Λ¹6=∅

(Tv−s_β(e)s^∗_β(e))

= Y

e∈vΛ¹ s(e)Λ¹=∅

(T_v−s_α(e)s^∗_α(e)) Y

e∈vΛ¹ s(e)Λ¹6=∅

(T_v−s_α(e)s^∗_α(e))(T_v−s_β(e)s^∗_β(e))

= Y

e∈vΛ¹ s(e)Λ¹=∅

(T_v−s_α(e)s^∗_α(e)) Y

e∈vΛ¹ s(e)Λ¹6=∅

(T_v−(s_α(e)s^∗_α(e)+s_β(e)s^∗_β(e)))

= Y

e∈vΛ¹ s(e)Λ¹=∅

(Tv−TeT_e^∗) Y

e∈vΛ¹ s(e)Λ¹6=∅

(Tv−TeT_e^∗)

= Y

e∈vΛ¹

(Tv−TeT_e^∗) ,

as required, and s_β(v) belongs to im (φT). Furthermore, s_α(v)=T_v−s_β(v) =T_v− Y

e∈vΛ¹

(T_v−T_eT_e^∗)∈im (φ_T) , as required.

Now take λ∈Λ. We have to shows_α(λ) ands_β(λ) (if it exists) belong to im (φT). Ifs(λ) Λ¹ =∅, then

s_α(λ)=s_α(λ)s_α(s(λ)) =TλT_s(λ) =Tλ ∈im (φT) .

Next suppose s(λ) Λ¹ 6= ∅. Then s_β(λ)s_α(s(λ)) = 0 and s_α(λ)s_β(s(λ)) = 0.

Hence,

s_α(λ)=s_α(λ)s_α(s(λ))= s_α(λ)+s_β(λ)

s_α(s(λ))

=T_λ



T_s(λ)− Y

e∈s(λ)Λ¹

(T_s(λ)−T_eT_e^∗)





=Tλ−Tλ

Y

e∈s(λ)Λ¹

(T_s(λ)−TeT_e^∗)∈im (φT) and

s_β(λ)=s_β(λ)s_β(s(λ))= s_α(λ)+s_β(λ)

s_β(s(λ))

=Tλ

Y

e∈s(λ)Λ¹

(T_s(λ)−TeT_e^∗)∈im (φT) .

Therefore, φ_T is surjective and an isomorphism.

Corollary 4.2. Let Λ be a row-finite k-graph and TΛ be the k-graph as in Proposition3.1. Let{t_λ :λ∈Λ}be the universal Toeplitz–Cuntz–Krieger Λ-family and{s_ω:ω ∈TΛ} be the universal Cuntz–Krieger TΛ-family. For τ ∈TΛ, define

Sτ :=







t_λ if τ =α(λ) withs(λ) Λ¹ =∅

t_λ−t_λQ

e∈s(λ)Λ¹(t_v−t_et^∗_e) if τ =α(λ) withs(λ) Λ¹ 6=∅ tλQ

e∈s(λ)Λ¹(tv−tet^∗_e) if τ =β(λ) with s(λ) Λ¹6=∅.

Suppose thatφT :T C^∗(Λ)→C^∗(TΛ)is the isomorphism as in Theorem4.1 and π_S :C^∗(TΛ)→ T C^∗(Λ) is the homomorphism such that π_S(s_τ) =S_τ for τ ∈TΛ. Then φ⁻¹_T =π_S.

Proof. Takeλ∈Λ. By Theorem 4.1, we getφ⁻¹_T s_α(λ)

=t_λifs(λ) Λ¹ =∅.

Meanwhile, ifs(λ) Λ¹ 6=∅, by Theorem 4.1, we have φ⁻¹_T s_α(λ)

=t_λ−t_λ Y

e∈vΛ¹

(tv−tet^∗_e), φ⁻¹_T s_β(λ)

=tλ

Y

e∈vΛ¹

(tv−tet^∗_e).

Hence, φ⁻¹_T (sτ) = Sτ for τ ∈ TΛ. This implies that {S_τ :τ ∈TΛ} is a Cuntz–Krieger TΛ-family, and then φ⁻¹_T =πS.

Remark 4.3. Proposition 3.5 says that TΛ is always aperiodic, and hence the Cuntz–Krieger uniqueness theorem always applies to TΛ. This helps explain why no hypothesis on Λ is required in the uniqueness theorem of [9, Theorem 8.1]. Indeed, we could have deduced that theorem by applying the Cuntz–Krieger uniqueness theorem to TΛ. With our current proof of Theorem 4.1, this argument would be circular, since we used [9, Theorem 8.1] in the proof of Theorem 4.1. However, we could prove Corollary 4.2 directly by showing that {S_τ :τ ∈TΛ} is a Cuntz–Krieger TΛ-family in T C^∗(Λ), hence gives a homomorphismπS :C^∗(TΛ)→T C^∗(Λ), and using the Cuntz–Krieger uniqueness theorem to see that π_S is injective. Then we could deduce [9, Theorem 8.1] from Corollary 4.2, and this would be a legitimate proof. We worked out the details of this approach, but it seemed to require an extensive cases argument, and hence became substantially more complicated.

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(Yosafat E. P. Pangalela)Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand

[email protected]

This paper is available via http://nyjm.albany.edu/j/2016/22-13.html.