Primitive ideals and pure infiniteness of ultragraph $C^*$-algebras

https://doi.org/10.4134/JKMS.j170579 pISSN: 0304-9914 / eISSN: 2234-3008

PRIMITIVE IDEALS AND PURE INFINITENESS OF

ULTRAGRAPH C

^∗

-ALGEBRAS

Hossein Larki

Abstract. Let G be an ultragraph and let C

^∗

(G) be the associated C

^∗

- algebra introduced by Tomforde. For any gauge invariant ideal I

_(H,B)

of C

^∗

(G), we approach the quotient C

^∗

-algebra C

^∗

(G)/I

_(H,B)

by the C

^∗

- algebra of finite graphs and prove versions of gauge invariant and Cuntz- Krieger uniqueness theorems for it. We then describe primitive gauge invariant ideals and determine purely infinite ultragraph C

^∗

-algebras (in the sense of Kirchberg-Rørdam) via Fell bundles.

1. Introduction

In order to bring graph C

^∗

-algebras [7] and Exel-Laca algebras [6] together under one theory, Tomforde introduced in [16] the notion of ultragraphs and as- sociated C

^∗

-algebras. An ultragraph is basically a directed graph in which the range of each edge is allowed to be a nonempty set of vertices rather than a sin- gle vertex. However, the class of ultragraph C

^∗

-algebras are strictly lager than the graph C

^∗

-algebras as well as the Exel-Laca algebras (see [17, Section 5]).

Due to some similarities, some of fundamental results for graph C

^∗

-algebras, such as the Cuntz-Krieger and the gauge invariant uniqueness theorems, sim- plicity, and K-theory computation have been extended to the setting of ultra- graphs [16,17]. In particular, by constructing a specific topological quiver Q(G) from an ultragraph G, Katsura et al. described some properties of the ultra- graph C

^∗

-algebra C

^∗

(G) using those of topological quivers [10]. They showed that every gauge invariant ideal of C

^∗

(G) is of the form I

_(H,B)

corresponding to an admissible pair (H, B) in G.

Recall that for any gauge invariant ideal I

_(H,B)

of a graph C

^∗

-algebra C

^∗

(E), there is a (quotient) graph E/(H, B) such that C

^∗

(E)/I

_(H,B)

∼ = C

^∗

(E/(H, B)) (see [1, 2]). So, the class of graph C

^∗

-algebras contains such quotients, and results and properties of graph C

^∗

-algebras may be applied for their quotients.

For examples, some contexts such as simplicity, K-theory, primitivity, and topo- logical stable rank are directly related to the structure of ideals and quotients.

Received September 5, 2017; Revised July 20, 2018; Accepted September 11, 2018.

2010 Mathematics Subject Classification. 46L55.

Key words and phrases. ultragraph C

^∗

-algebra, primitive ideal, pure infiniteness.

c

2019 Korean Mathematical Society 1

Unlike the C

^∗

-algebras of graphs and topological quivers [13], there are no known ways in the literature for describing quotients of an ultragraph C

^∗

- algebra by structure of the initial ultragraph. So, many graph C

^∗

-algebra’s techniques could not be applied for the ultragraph setting, causing some obsta- cles in studying these C

^∗

-algebras. The initial aim of this article is to analyze the structure of the quotient C

^∗

-algebras C

^∗

(G)/I

_(H,B)

for any gauge invariant ideal I

_(H,B)

of C

^∗

(G). For the sake of convenience, we first introduce the no- tion of quotient ultragraph G/(H, B) and a relative C

^∗

-algebra C

^∗

(G/(H, B)) such that C

^∗

(G)/I

_(H,B)

∼ = C

^∗

(G/(H, B)) and then prove the gauge invariant and the Cuntz-Krieger uniqueness theorems for C

^∗

(G/(H, B)). The uniqueness theorems help us to show when a representation of C

^∗

(G)/I

_(H,B)

is injective.

We see that the structure of C

^∗

(G/(H, B)) is close to that of graph C

^∗

-algebras and we can use them to determine primitive gauge invariant ideals. Moreover, in Section 6, we consider the notion of pure infiniteness for ultragraph C

^∗

- algebras in the sense of Kirchberg-Rørdam [11] which is directly related to the structure of quotients. We should note that the initial idea for definition of quotient ultragraphs has been inspired from [9].

The present article is organized as follows. We begin in Section 2 by giving some definitions and preliminaries about the ultragraphs and their C

^∗

-algebras which will be used in the next sections. In Section 3, for any admissible pair (H, B) in an ultragraph G, we introduce the quotient ultragraph G/(H, B) and an associated C

^∗

-algebra C

^∗

(G/(H, B)). For this, the ultragraph G is modified by an extended ultragraph G and we define an equivalent relation ∼ on G. Then G/(H, B) is the ultragraph G with the equivalent classes {[A] : A ∈ G

⁰

}. In Section 4, by approaching with graph C

^∗

-algebras, the gauge invariant and the Cuntz-Krieger uniqueness theorems will be proved for the quotient ultragraphs C

^∗

-algebras. Moreover, we see that C

^∗

(G/(H, B)) is isometrically isomorphic to the quotient C

^∗

-algebra C

^∗

(G)/I

_(H,B)

.

In Sections 5 and 6, using quotient ultragraphs, some graph C

^∗

-algebra’s techniques will be applied for the ultragraph C

^∗

-algebras. In Section 5, we describe primitive gauge invariant ideals of C

^∗

(G), whereas in Section 6, we characterize purely infinite ultragraph C

^∗

-algebras (in the sense of [11]) via Fell bundles [5, 12].

2. Preliminaries

In this section, we review basic definitions and properties of ultragraph C

^∗

- algebras which will be needed through the paper. For more details, we refer the reader to [10] and [16].

Definition 2.1 ([16]). An ultragraph is a quadruple G = (G

⁰

, G

¹

, r

_G

, s

_G

) con-

sisting of a countable vertex set G

⁰

, a countable edge set G

¹

, the source map

s

_G

: G

¹

→ G

⁰

, and the range map r

_G

: G

¹

→ P(G

⁰

) \ {∅}, where P(G

⁰

) is

the collection of all subsets of G

⁰

. If r

_G

(e) is a singleton vertex for each edge

e ∈ G

¹

, then G is an ordinary (directed) graph.

For our convenience, we use the notation G

⁰

in the sense of [10] rather than [16, 17]. For any set X, a nonempty subcollection of the power set P(X) is said to be an algebra if it is closed under the set operations ∩, ∪, and \. If G is an ultragraph, the smallest algebra in P(G

⁰

) containing {{v} : v ∈ G

⁰

} and {r

_G

(e) : e ∈ G

¹

} is denoted by G

⁰

. We simply denote every singleton set {v}

by v. So, G

⁰

may be considered as a subset of G

⁰

.

Definition 2.2. For each n ≥ 1, a path α of length |α| = n in G is a sequence α = e

1

. . . e

n

of edges such that s(e

i+1

) ∈ r(e

i

) for 1 ≤ i ≤ n − 1. If also s(e

₁

) ∈ r(e

_n

), α is called a loop or a closed path. We write α

⁰

for the set {s

G

(e

_i

) : 1 ≤ i ≤ n}. The elements of G

⁰

are considered as the paths of length zero. The set of all paths in G is denoted by G

^∗

. We may naturally extend the maps s

_G

, r

_G

on G

^∗

by defining s

_G

(A) = r

_G

(A) = A for A ∈ G

⁰

, and r

_G

(α) = r

_G

(e

_n

), s

_G

(α) = s

_G

(e

₁

) for each path α = e

₁

· · · e

_n

.

Definition 2.3 ([16]). Let G be an ultragraph. A Cuntz-Krieger G-family is a set of partial isometries {s

e

: e ∈ G

¹

} with mutually orthogonal ranges and a set of projections {p

A

: A ∈ G

⁰

} satisfying the following relations:

(UA1) p

_∅

= 0, p

A

p

B

= p

A∩B

, and p

A∪B

= p

A

+ p

B

− p

A∩B

for all A, B ∈ G

⁰

, (UA2) s

^∗_e

s

e

= p

r_G(e)

for e ∈ G

¹

,

(UA3) s

e

s

^∗_e

≤ p

s_G(e)

for e ∈ G

¹

, and (UA4) p

_v

= P

sG(e)=v

s

_e

s

^∗_e

whenever 0 < |s

⁻¹_G

(v)| < ∞.

The C

^∗

-algebra C

^∗

(G) of G is the (unique) C

^∗

-algebra generated by a universal Cuntz-Krieger G-family.

By [16, Remark 2.13], we have

C

^∗

(G) = span s

α

p

A

s

^∗_β

: α, β ∈ G

^∗

, A ∈ G

⁰

, and r

_G

(α) ∩ r

_G

(β) ∩ A 6= ∅ , where s

_α

:= s

_e1

· · · s

en

if α = e

₁

· · · e

n

, and s

_α

:= p

_A

if α = A.

Remark 2.4. As noted in [16, Section 3], every graph C

^∗

-algebra is an ultra- graph C

^∗

-algebra. Recall that if E = (E

⁰

, E

¹

, r

_E

, s

_E

) is a directed graph, a collection {s

e

, p

v

: v ∈ E

⁰

, e ∈ E

¹

} containing mutually orthogonal projections p

v

and partial isometries s

e

is called a Cuntz-Krieger E-family if

(GA1) s

^∗_e

s

e

= p

_rE(e)

for all e ∈ E

¹

, (GA2) s

e

s

^∗_e

≤ p

_sE(e)

for all e ∈ E

¹

, and (GA3) p

v

= P

s_E(e)=v

s

e

s

^∗_e

for every vertex v ∈ E

⁰

with 0 < |s

⁻¹_E

(v)| < ∞.

We denote by C

^∗

(E) the universal C

^∗

-algebra generated by a Cuntz-Krieger E-family.

By the universal property, C

^∗

(G) admits the gauge action of the unit circle

T. By an ideal, we mean a closed two-sided ideal. Using the properties of

quiver C

^∗

-algebras [10], the gauge invariant ideals of C

^∗

(G) were characterized

in [10, Theorem 6.12] via a one-to-one correspondence with the admissible pairs

of G as follows.

Definition 2.5. A subset H ⊆ G

⁰

is said to be hereditary if the following properties holds:

(H1) s

_G

(e) ∈ H implies r

_G

(e) ∈ H for all e ∈ G

¹

. (H2) A ∪ B ∈ H for all A, B ∈ H.

(H3) If A ∈ H, B ∈ G

⁰

, and B ⊆ A, then B ∈ H.

Moreover, a subset H ⊆ G

⁰

is called saturated if for any v ∈ G

⁰

with 0 <

|s

⁻¹_G

(v)| < ∞, then {r

_G

(e) : s

_G

(e) = v} ⊆ H implies v ∈ H. The saturated hereditary closure of a subset H ⊆ G

⁰

is the smallest hereditary and saturated subset H of G

⁰

containing H.

Let H be a saturated hereditary subset of G

⁰

. The set of breaking vertices of H is denoted by

B

_H

:= w ∈ G

⁰

: |s

⁻¹_G

(w)| = ∞ but 0 < |r

_G

(s

⁻¹_G

(w)) ∩ (G

⁰

\ H)| < ∞ . An admissible pair (H, B) in G is a saturated hereditary set H ⊆ G

⁰

together with a subset B ⊆ B

H

. For any admissible pair (H, B) in G, we define the ideal I

(H,B)

of C

^∗

(G) generated by

{p

A

: A ∈ G

⁰

} ∪ p

^H_w

: w ∈ B , where p

^H_w

:= p

w

− P

s_G(e)=w, r_G(e) /∈H

s

e

s

^∗_e

. Note that the ideal I

_(H,B)

is gauge invariant and [10, Theoerm 6.12] implies that every gauge invariant ideal I of C

^∗

(G) is of the form I

(H,B)

by setting

H := {A : p

A

∈ I} and B := w ∈ B

H

: p

^H_w

∈ I . 3. Quotient ultragraphs and their C

^∗

-algebras

In this section, for any admissible pair (H, B) in an ultragraph G, we intro- duce the quotient ultragraph G/(H, B) and its relative C

^∗

-algebra C

^∗

(G/(H, B)) . We will show in Proposition 4.6 that C

^∗

(G/(H, B)) is isomorphic to the quo- tient C

^∗

-algebra C

^∗

(G)/I

_(H,B)

.

Let us fix an ultragraph G = (G

⁰

, G

⁰

, r

_G

, s

_G

) and an admissible pair (H, B) in G. For defining our quotient ultragraph G/(H, B), we first modify G by an extended ultragraph G such that their C

^∗

-algebras coincide. For this, add the vertices {w

⁰

: w ∈ B

H

\ B} to G

⁰

and denote A := A ∪ {w

⁰

: w ∈ A ∩ (B

H

\ B)}

for each A ∈ G

⁰

. We now define the new ultragraph G = (G

⁰

, G

¹

, r

_G

, s

_G

) by G

⁰

:= G

⁰

∪ {w

⁰

: w ∈ B

H

\ B},

G

¹

:= G

¹

, the source map

s

_G

(e) :=

 (s

_G

(e))

⁰

if s

_G

(e) ∈ B

H

\ B and r

_G

(e) ∈ H s

_G

(e) otherwise,

and the rang map r

_G

(e) := r

_G

(e) for every e ∈ G

¹

. In Proposition 3.3 below,

we will see that the C

^∗

-algebras of G and G coincide.

Example 3.1. Suppose G is the ultragraph

w u

v

f A (∞) e

e g

H

where (∞) indicates infinitely many edges. If H is the saturated hereditary subset of G

⁰

containing {v} and A, then we have B

_H

= {w}. For B := ∅, consider the admissible pair (H, ∅) in G. Then the ultragraph G associated to (H, ∅) would be

w w

⁰

u

v

A (∞)

f e

e g

e

H

Indeed, since B

_H

\ B = {w}, for constructing G we first add a vertex w

⁰

to G.

We then define

r

G

(f ) := A = A,

r

_G

(e) := {v, w} = {v, w, w

⁰

}, and r

_G

(g) := {u} = {u}.

For the source map s

G

, for example, since s

G

(f ) ∈ B

H

\ B and r

G

(f ) ∈ H, we may define s

G

(f ) := w

⁰

. Note that the range of each edge emitted by w

⁰

belongs to H.

As usual, we write G

⁰

for the algebra generated by the elements of G

⁰

∪ {r

_G

(e) : e ∈ G

¹

}. Note that A = A for every A ∈ H, and hence, H would be a saturated hereditary subset of G

⁰

as well. Moreover, the set of breaking vertices of H in G coincides with B (meaning B

^G_H

= B).

Remark 3.2. Suppose that C

^∗

(G) is generated by a Cuntz-Krieger G-family

{s

e

, p

A

: A ∈ G

⁰

, e ∈ G

¹

}. If a family M = {S

e

, P

v

, P

_A

: v ∈ G

⁰

, A ∈ G

⁰

, e ∈

G

¹

} in a C

^∗

-algebra X satisfies relations (UA1)-(UA4) in Definition 2.3, we

may generate a Cuntz-Krieger G-family N = {S

e

, P

_A

: A ∈ G

⁰

, e ∈ G

¹

} in

X. For this, since G

⁰

is the algebra generated by {v, w

⁰

, r

_G

(e) : v ∈ G

⁰

, w ∈

B

H

\ B, e ∈ G

¹

}, we may use the definitions P

A∩C

:= P

A

P

C

,

P

_A∪C

:= P

_A

+ P

_C

− P

_A

P

_C

, P

A\C

:= P

A

− P

A

P

C

,

to generate each projection P

A

, A ∈ G

⁰

, by finitely many operations. Then N would be a Cuntz-Krieger G-family in X, and the C

^∗

-subalgebras generated by M and N coincide.

Proposition 3.3. Let G be an ultragraph, and let (H, B) be an admissible pair in G. If G is the extended ultragraph as above, then C

^∗

(G) ∼ = C

^∗

(G).

Proof. Suppose that C

^∗

(G) = C

^∗

(t

e

, q

A

) and C

^∗

(G) = C

^∗

(s

e

, p

C

). If we define P

v

:= q

v

for v ∈ G

⁰

\ (B

H

\ B),

P

w

:= P

s_G(e)=w r_G(e) /∈H

t

e

t

^∗_e

for w ∈ B

H

\ B, P

_w⁰

:= q

_w

− P

sG(e)=w rG(e) /∈H

t

_e

t

^∗_e

for w ∈ B

_H

\ B,

P

_A

:= q

A

for A ∈ G

⁰

,

S

e

:= t

e

for e ∈ G

¹

,

then, by Remark 3.2, the family n

P

_v

, P

_w

, P

_w⁰

, P

_A

, S

_e

: v ∈ G

⁰

\ (B

_H

\ B), w ∈ B

_H

\ B, A ∈ G

⁰

, e ∈ G

¹

o induces a Cuntz-Krieger G-family in C

^∗

(G). Since all vertex projections of this family are nonzero (which follows all set projections P

A

are nonzero for

∅ 6= A ∈ G

⁰

), the gauge-invariant uniqueness theorem [16, Theorem 6.8] implies that the ∗-homomorphism φ : C

^∗

(G) → C

^∗

(G) with φ(p

_∗

) = P

_∗

and φ(s

_∗

) = S

_∗

is injective. On the other hand, the family generates C

^∗

(G), and hence, φ is an

isomorphism. 

To define a quotient ultragraph G/(H, B), we use the following equivalent relation on G.

Definition 3.4. Suppose that (H, B) is an admissible pair in G, and that G is the extended ultragraph as above. We define the relation ∼ on G

⁰

by

A ∼ C ⇐⇒ ∃V ∈ H such that A ∪ V = C ∪ V.

Note that A ∼ C if and only if both sets A \ C and C \ A belong to H.

The following lemma may be proved by a tedious, but straightforward com- putations.

Lemma 3.5. The relation ∼ is an equivalent relation on G

⁰

. Furthermore, the operations

[A] ∪ [C] := [A ∪ C], [A] ∩ [C] := [A ∩ C], and [A] \ [C] := [A \ C]

are well-defined on the equivalent classes {[A] : A ∈ G

⁰

}.

Definition 3.6. Let G be an ultragraph, let (H, B) be an admissible pair in G, and consider the equivalent relation of Definition 3.4 on the extended ultragraph G = (G

⁰

, G

¹

, r

_G

, s

_G

). The quotient ultragraph of G by (H, B) is the quintuple G/(H, B) = (Φ(G

⁰

), Φ(G

⁰

), Φ(G

¹

), r, s), where

Φ(G

⁰

) := [v] : v ∈ G

⁰

\ H ∪ {[w

⁰

] : w ∈ B

H

\ B} , Φ(G

⁰

) := n

[A] : A ∈ G

⁰

o , Φ(G

¹

) := n

e ∈ G

¹

: r

G

(e) / ∈ H o ,

and r : Φ(G

¹

) → Φ(G

⁰

), s : Φ(G

¹

) → Φ(G

⁰

) are the range and source maps defined by

r(e) := [r

_G

(e)] and s(e) := [s

_G

(e)].

We refer to Φ(G

⁰

) as the vertices of G/(H, B).

Remark 3.7. Lemma 3.5 implies that Φ(G

⁰

) is the smallest algebra containing

[v], [w

⁰

] : v ∈ G

⁰

\ H, w ∈ B

H

\ B ∪ n

[r

_G

(e)] : e ∈ G

¹

o . Notation.

(1) For every vertex v ∈ G

⁰

\ H, we usually denote [v] instead of [{v}].

(2) For A, C ∈ G

⁰

, we write [A] ⊆ [C] whenever [A] ∩ [C] = [A].

(3) Through the paper, we will denote the range and the source maps of G by r

_G

, s

_G

, those of G by r

_G

, s

_G

, and those of G/(H, B) by r, s.

Now we introduce representations of quotient ultragraphs and their relative C

^∗

-algebras.

Definition 3.8. Let G/(H, B) be a quotient ultragraph. A representation of G/(H, B) is a set of partial isometries {T

e

: e ∈ Φ(G

¹

)} and a set of projections {Q

_[A]

: [A] ∈ Φ(G

⁰

)} which satisfy the following relations:

(QA1) Q

_[∅]

= 0, and for [A], [C] ∈ Φ(G

⁰

), Q

_[A∩C]

= Q

_[A]

Q

_[C]

and Q

_[A∪C]

= Q

_[A]

+ Q

_[C]

− Q

_[A∩C]

.

(QA2) T

_e^∗

T

f

= δ

e,f

Q

_r(e)

for e, f ∈ Φ(G

¹

).

(QA3) T

e

T

_e^∗

≤ Q

_s(e)

for e ∈ Φ(G

¹

).

(QA4) Q

[v]

= P

s(e)=[v]

T

e

T

_e^∗

, whenever 0 < |s

⁻¹

([v])| < ∞.

We denote by C

^∗

(G/(H, B)) the universal C

^∗

-algebra generated by a repre- sentation {t

e

, q

[A]

: [A] ∈ Φ(G

⁰

), e ∈ Φ(G

¹

)} which exists by Theorem 3.10 below.

Note that if α = e

1

· · · e

n

is a path in G such that r

G

(α) / ∈ H, then the hereditary property of H yields r

G

(e

i

) / ∈ H, and so e

i

∈ Φ(G

¹

) for all 1 ≤ i ≤ n.

In this case, we denote t

_α

:= t

_e1

· · · t

en

. Moreover, we define (G/(H, B))

^∗

:= {[A] : [A] 6= [∅]} ∪ n

α ∈ G

^∗

: r(α) 6= [∅] o

as the set of finite paths in G/(H, B) and we can extend the maps s, r on (G/(H, B))

^∗

by setting

s([A]) := r([A]) := [A] and s(α) := s(e

₁

), r(α) := r(e

_n

).

The proof of next lemma is similar to the arguments of [16, Lemmas 2.8 and 2.9].

Lemma 3.9. Let G/(H, B) be a quotient ultragraph and let {T

e

, Q

_[A]

} be a representation of G/(H, B). Then any nonzero word in T

e

, Q

_[A]

, and T

_f^∗

may be written as a finite linear combination of the forms T

α

Q

_[A]

T

_β^∗

for α, β ∈ (G/(H, B))

^∗

and [A] ∈ Φ(G

⁰

) with [A] ∩ r(α) ∩ r(β) 6= [∅].

Theorem 3.10. Let G/(H, B) be a quotient ultragraph. Then there exists a (unique up to isomorphism) C

^∗

-algebra C

^∗

(G/(H, B)) generated by a universal representation {t

e

, q

_[A]

: [A] ∈ Φ(G

⁰

), e ∈ Φ(G

¹

)} for G/(H, B). Furthermore, all the t

e

’s and q

_[A]

’s are nonzero for [∅] 6= [A] ∈ Φ(G

⁰

) and e ∈ Φ(G

¹

).

Proof. By a standard argument similar to the proof of [16, Theorem 2.11], we may construct such universal C

^∗

-algebra C

^∗

(G/(H, B)). Note that the universality implies that C

^∗

(G/(H, B)) is unique up to isomorphism. To show the last statement, we generate an appropriate representation for G/(H, B) as follows. Suppose C

^∗

(G) = C

^∗

(s

e

, p

A

) and consider I

_(H,B)

as an ideal of C

^∗

(G) by the isomorphism in Proposition 3.3. If we define

 Q

_[A]

:= p

A

+ I

_(H,B)

for [A] ∈ Φ(G

⁰

), T

e

:= s

e

+ I

_(H,B)

for e ∈ Φ(G

¹

),

then the family {T

_e

, Q

_[A]

: [A] ∈ Φ(G

⁰

), e ∈ Φ(G

¹

)} is a representation for G/(H, B) in the quotient C

^∗

-algebra C

^∗

(G)/I

_(H,B)

. Note that the definition of Q

_[A]

’s is well-defined. Indeed, if A

1

∪ V = A

2

∪ V for some V ∈ H, then p

A₁

+ p

_{V \A1}

= p

A₂

+ p

_{V \A2}

and hence p

A₁

+ I

_(H,B)

= p

A₂

+ I

_(H,B)

by the facts V \ A

1

, V \ A

2

∈ H.

Moreover, all elements Q

_[A]

and T

e

are nonzero for [∅] 6= [A] ∈ Φ(G

⁰

), e ∈ Φ(G

¹

). In fact, if Q

_[A]

= 0, then p

A

∈ I

_(H,B)

and we get A ∈ H by [10, Theorem 6.12]. Also, since T

_e^∗

T

e

= Q

_r(e)

6= 0, all partial isometries T

e

are nonzero.

Now suppose that C

^∗

(G/(H, B)) is generated by the family {t

e

, q

_[A]

: [A] ∈ Φ(G

⁰

), e ∈ Φ(G

¹

)}. By the universality of C

^∗

(G/(H, B)), there is a ∗-homo- morphism φ : C

^∗

(G/(H, B)) → C

^∗

(G)/I

_(H,B)

such that φ(t

_e

) = T

_e

and φ(q

_[A]

)

= Q

_[A]

, and thus, all elements of {t

_e

, q

_[A]

: [∅] 6= [A] ∈ Φ(G

⁰

), e ∈ Φ(G

¹

)} are

nonzero. 

Note that, by a routine argument, one may obtain

C

^∗

(G/(H, B)) = spant

α

q

_[A]

t

^∗_β

: α, β ∈ (G/(H, B))

^∗

, r(α) ∩ [A] ∩ r(β) 6= [∅] .

4. Uniqueness theorems

After defining the C

^∗

-algebras of quotient ultragraphs, in this section, we prove the gauge invariant and the Cuntz-Krieger uniqueness theorems for them.

To do this, we approach to a quotient ultragraph C

^∗

-algebra by graph C

^∗

- algebras and then apply the corresponding uniqueness theorems for graph C

^∗

- algebras. This approach is a developed version of the dual graph method of [14, Section 2] and [16, Section 5] with more complications. In particular, we show that the C

^∗

-algebra C

^∗

(G/(H, B)) is isomorphic to the quotient C

^∗

(G)/I

_(H,B)

, and the uniqueness theorems may applied for such quotients.

We fix again an ultragraph G, an admissible pair (H, B) in G, and the quotient ultragraph G/(H, B) = (Φ(G

⁰

), Φ(G

⁰

), Φ(G

¹

), r, s).

Definition 4.1. We say that a vertex [v] ∈ Φ(G

⁰

) is a sink if s

⁻¹

([v]) = ∅. If [v] only emits finitely many edges of Φ(G

¹

), [v] is called a regular vertex. Any non-regular vertex is called a singular vertex. The set of singular vertices in Φ(G

⁰

) is denoted by

Φ

sg

(G

⁰

) := [v] ∈ Φ(G

⁰

) : |s

⁻¹

([v])| = 0 or ∞ .

Let F be a finite subset of Φ

sg

(G

⁰

) ∪ Φ(G

¹

). Write F

⁰

:= F ∩ Φ

sg

(G

⁰

) and F

¹

:= F ∩ Φ(G

¹

) = {e

1

, . . . , e

n

}. We want to construct a special graph G

F

such that C

^∗

(G

F

) is isomorphic to C

^∗

(t

e

, q

_[v]

: [v] ∈ F

⁰

, e ∈ F

¹

). For each ω = (ω

1

, . . . , ω

n

) ∈ {0, 1}

ⁿ

\ {0

ⁿ

}, we write

r(ω) := \

ωi=1

r(e

i

) \ [

ωj=0

r(e

j

) and R(ω) := r(ω) \ [

[v]∈F⁰

[v].

Note that r(ω) ∩ r(ν) = [∅] for distinct ω, ν ∈ {0, 1} \ {0

ⁿ

}. If Γ

0

:= ω ∈ {0, 1}

ⁿ

\ {0

ⁿ

} : ∃[v

1

], . . . , [v

m

] ∈ Φ(G

⁰

) such that

R(ω) =

m

[

i=1

[v

_i

] and ∅ 6= s

⁻¹

([v

_i

]) ⊆ F

¹

for 1 ≤ i ≤ m , we consider the finite set

Γ := {ω ∈ {0, 1}

ⁿ

\ {0

ⁿ

} : R(ω) 6= [∅] and ω / ∈ Γ

0

} .

Now we define the finite graph G

F

= (G

⁰_F

, G

¹_F

, r

F

, s

F

) containing the ver- tices G

⁰_F

:= F

⁰

∪ F

¹

∪ Γ and the edges

G

¹_F

:= (e, f ) ∈ F

¹

× F

¹

: s(f ) ⊆ r(e)

∪ (e, [v]) ∈ F

¹

× F

⁰

: [v] ⊆ r(e)

∪ (e, ω) ∈ F

¹

× Γ : ω

i

= 1 when e = e

i

with the source map s

F

(e, f ) = s

F

(e, [v]) = s

F

(e, ω) = e, and the range map

r

F

(e, f ) = f , r

F

(e, [v]) = [v], r

F

(e, ω) = ω.

Proposition 4.2. Let G/(H, B) be a quotient ultragraph and let F be a finite subset of Φ

sg

(G

⁰

) ∪ Φ(G

¹

). If C

^∗

(G/(H, B)) = C

^∗

(t

e

, q

_[A]

), then the elements

Q

e

:= t

e

t

^∗_e

, Q

[v]

:= q

[v]

(1 − P

e∈F¹

t

e

t

^∗_e

), Q

ω

:= q

R(ω)

(1 − P

e∈F¹

t

e

t

^∗_e

) T

_{(e,f )}

:= t

_e

Q

_f

, T

_(e,[v])

:= t

_e

Q

_[v]

, T

_(e,ω)

:= t

_e

Q

_ω

form a Cuntz-Krieger G

F

-family generating the C

^∗

-subalgebra C

^∗

(t

e

, q

[v]

: [v] ∈ F

⁰

, e ∈ F

¹

) of C

^∗

(G/(H, B)). Moreover, all projections Q

∗

are nonzero.

Proof. We first note that all the projections Q

e

, Q

_[v]

, and Q

ω

are nonzero.

Indeed, each [v] ∈ F

⁰

is a singular vertex in G/(H, B), so Q

_[v]

is nonzero.

Also, by definition, for every ω ∈ Γ we have ω / ∈ Γ

0

and R(ω) 6= [∅]. Hence, for any ω ∈ Γ, if there is an edge f ∈ Φ(G

¹

) \ F

¹

with s(f ) ⊆ R(ω), then 0 6= t

f

t

^∗_f

≤ Q

ω

. If there is a sink [w] such that [w] ⊆ R(ω) = r(ω) \ S F

⁰

, then 0 6= q

_[w]

≤ q

_R(ω)

(1 − P

e∈F¹

t

e

t

^∗_e

) = Q

ω

. Thus Q

ω

is nonzero in either case. In addition, the projections Q

e

, Q

_[v]

, and Q

ω

are mutually orthogonal because of the factor 1 − P

e∈F¹

t

e

t

^∗_e

and the definition of R(ω).

Now we show the collection {T

x

, Q

a

: a ∈ G

⁰_F

, x ∈ G

¹_F

} is a Cuntz-Krieger G

F

-family by checking the relations (GA1)-(GA3) in Remark 2.4.

(GA1): Since Q

[v]

, Q

ω

≤ q

r(e)

for (e, [v]), (e, ω) ∈ G

¹_F

, we have T

_{(e,f )}^∗

T

_{(e,f )}

= Q

f

t

^∗_e

t

e

Q

f

= t

f

t

^∗_f

q

_r(e)

t

f

t

^∗_f

= t

f

q

_{r(f )}

t

^∗_f

= Q

f

,

T

_(e,[v])^∗

T

_(e,[v])

= Q

_[v]

t

^∗_e

t

e

Q

_[v]

= Q

_[v]

q

_r(e)

Q

_[v]

= Q

_[v]

, and

T

_(e,ω)^∗

T

_(e,ω)

= Q

ω

t

^∗_e

t

e

Q

ω

= Q

ω

q

_r(e)

Q

ω

= Q

ω

. (GA2): This relation may be checked similarly.

(GA3): Note that any element of F

⁰

∪Γ is a sink in G

F

. So, fix some e

i

∈ F

¹

as a vertex of G

⁰_F

. Write q

_F0

:= P

[v]∈F⁰

q

_[v]

. We compute (i) q

_r(ei)

X

f ∈F¹ s(f )⊆r(ei)

Q

f

= q

_r(ei)

X

f ∈F¹ s(f )⊆r(ei)

t

f

t

^∗_f

= q

_r(ei)

X

f ∈F¹

t

f

t

^∗_f

;

(ii) q

_r(ei)

X

[v]∈F⁰, [v]⊆r(ei)

Q

_[v]

= q

_r(ei)

X

[v]∈F⁰

q

_[v]

(1 − X

e∈F¹

t

e

t

^∗_e

)

= q

r(e_i)

q

F⁰

(1 − X

e∈F¹

t

e

t

^∗_e

);

(iii) X

ω∈Γ,ω_i=1

Q

ω

= X

ω∈Γ,ω_i=1

q

R(ω)

(1 − X

e∈F¹

t

e

t

^∗_e

) = X

ω_i=1

q

R(ω)

(1 − X

e∈F¹

t

e

t

^∗_e

), because P

ω_i=1

q

R(ω)

= q

r(e_i)

(1 − q

F⁰

).

We can use these relations to get X

s(f )⊆r(ei)

T

_{(ei,f )}

+ X

[v]∈F⁰, [v]⊆r(ei)

T

_(ei,[v])

+ X

ω∈Γ, ω_i=1

T

_(ei,ω)

(4.1)

= t

_ei

q

_r(ei)

X

e∈F¹

t

_e

t

^∗_e

+ q

_r(ei)

q

_F0

( X

e∈F¹

t

_e

t

^∗_e

) + q

_r(ei)

(1 − q

_F0

)( X

e∈F¹

t

_e

t

^∗_e

)

!

= t

_ei

q

_r(ei)

X

e∈F¹

t

_e

t

^∗_e

+ (q

_F0

+ 1 − q

_F0

)(1 − X

e∈F¹

t

_e

t

^∗_e

)

!

= t

_ei

.

Now if e

_i

is not a sink as a vertex in G

_F

(i.e., |{x ∈ G

¹_F

: s

_F

(x) = e

_i

}| > 0), we conclude that

X

f ∈F¹, s(f )⊆r(ei)

T

_{(ei,f )}

T

_(e^∗

i,f )

+ X

[v]∈F⁰, [v]⊆r(ei)

T

_(ei,[v])

T

_(e^∗

i,[v])

+ X

ω∈Γ, ω_i=1

T

_(ei,ω)

T

_(e^∗

i,ω)

= X

t

_ei

Q

_f

t

^∗_e

i

+ X

t

_ei

Q

_[v]

t

^∗_e

i

+ X

t

_ei

Q

_ω

t

^∗_e

i

= t

_ei

q

_r(ei)

( X

Q

_f

+ X

Q

_[v]

+ X Q

_ω

)t

^∗_e

i

= t

_ei

t

^∗_e

i

= Q

_ei

,

which establishes the relation (GA3).

Furthermore, equation (4.1) in above says that t

_ei

∈ C

^∗

(T

_∗

, Q

_∗

) for every e

i

∈ F

¹

. Also, for each [v] ∈ F

⁰

, we have

Q

_[v]

+ X

e∈F¹,s(e)=[v]

Q

e

= t

_[v]

(1 − X

e∈F¹

t

e

t

^∗_e

) + X

e∈F¹,s(e)=[v]

t

e

t

^∗_e

= t

_[v]

− t

[v]

X

e∈F¹

t

_e

t

^∗_e

+ t

_[v]

X

e∈F¹

t

_e

t

^∗_e

= t

_[v]

.

Therefore, the family {T

_x

, Q

_a

: a ∈ G

⁰_F

, x ∈ G

¹_F

} generates the C

^∗

-subalgebra C

^∗

({t

e

, q

_[v]

: e ∈ F

¹

, [v] ∈ F

⁰

}) of C

^∗

(G/(H, B)) and the proof is complete.  Corollary 4.3. If F is a finite subset of Φ

sg

(G

⁰

) ∪ Φ(G

¹

), then C

^∗

(G

F

) is isometrically isomorphic to the C

^∗

-subalgebra of C

^∗

(G/(H, B)) generated by {t

e

, q

_[v]

: [v] ∈ F

⁰

, e ∈ F

¹

}.

Proof. Suppose that X is the C

^∗

-subalgebra generated by {t

_e

, q

_[v]

: [v] ∈

F

⁰

, e ∈ F

¹

} and let {T

_x

, Q

_a

: a ∈ G

⁰_F

, x ∈ G

¹_F

} be the Cuntz-Krieger G

_F

-

family in Proposition 4.2. If C

^∗

(G

F

) = C

^∗

(s

x

, p

a

), then there exists a ∗-

homomorphism φ : C

^∗

(G

F

) → X with φ(p

a

) = Q

a

and φ(s

x

) = T

x

for every

a ∈ G

⁰_F

, x ∈ G

¹_F

. Since each Q

a

is nonzero by Proposition 4.2, the gauge

invariant uniqueness theorem implies that φ is injective. Moreover, the family

{T

x

, Q

a

} generates X, so φ is an isomorphism. 

Note that if F

₁

⊆ F

₂

are two finite subsets of Φ

_sg

(G

⁰

)∪Φ(G

¹

) and X

₁

, X

₂

are the C

^∗

-subalgebras of C

^∗

(G/(H, B)) associated to G

F₁

and G

F₂

, respectively, we then have X

1

⊆ X

2

by Proposition 4.2.

Remark 4.4. Using relations (QA1)-(QA4) in Definition 3.8, each q

_[A]

for [A] ∈ Φ(G

⁰

), can be produced by the elements of

{q

_[v]

: [v] ∈ Φ

sg

(G

⁰

)} ∪ {t

e

: e ∈ Φ(G

¹

)}

with finitely many operations. So, the ∗-subalgebra of C

^∗

(G/(H, B)) generated by

{q

_[v]

: [v] ∈ Φ

sg

(G

⁰

)} ∪ {t

e

: e ∈ Φ(G

¹

)}

is dense in C

^∗

(G/(H, B)).

As for graph C

^∗

-algebras, we can apply the universal property to have a strongly continuous gauge action γ : T → Aut(C

^∗

(G/(H, B))) such that

γ

z

(t

e

) = zt

e

and γ

z

(q

_[A]

) = q

_[A]

for every [A] ∈ Φ(G

⁰

), e ∈ Φ(G

¹

), and z ∈ T. Now we are ready to prove the uniqueness theorems.

Theorem 4.5 (The Gauge Invariant Uniqueness Theorem). Let G/(H, B) be a quotient ultragraph and let {T

e

, Q

_[A]

} be a representation for G/(H, B) such that Q

_[A]

6= 0 for [A] 6= [∅]. If π

T ,Q

: C

^∗

(G/(H, B)) → C

^∗

(T

e

, Q

_[A]

) is the

∗-homomorphism satisfying π

T ,Q

(t

e

) = T

e

, π

T ,Q

(q

_[A]

) = Q

_[A]

, and there is a strongly continuous action β of T on C

^∗

(T

e

, Q

_[A]

) such that β

z

◦π

T ,Q

= π

T ,Q

◦γ

z

for every z ∈ T, then π

^{T ,Q}

is faithful.

Proof. Select an increasing sequence {F

_n

} of finite subsets of Φ

_sg

(G

⁰

) ∪ Φ(G

¹

) such that ∪

^∞_n=1

F

_n

= Φ

_sg

(G

⁰

) ∪ Φ(G

¹

). For each n, Corollary 4.3 gives an isomorphism

π

n

: C

^∗

(G

F_n

) → C

^∗

({t

e

, q

[v]

: [v] ∈ F

⁰

, e ∈ F

¹

})

that respects the generators. We can apply the gauge invariant uniqueness theorem for graph C

^∗

-algebras to see that the homomorphism

π

T ,Q

◦ π

n

: C

^∗

(G

Fn

) → C

^∗

(T

e

, Q

[A]

)

is faithful. Hence, for every F

n

, the restriction of π

T ,Q

on the ∗-subalgebra of C

^∗

(G/(H, B)) generated by {t

e

, q

_[v]

: [v] ∈ F

_n⁰

, e ∈ F

_n¹

} is faithful. This turns out that π

T ,Q

is injective on the ∗-subalgebra C

^∗

(t

e

, q

_[v]

: [v] ∈ Φ

sg

(G

⁰

), e ∈ Φ(G

¹

)). Since, this subalgebra is dense in C

^∗

(G/(H, B)), we conclude that π

T ,Q

is faithful. 

Proposition 4.6. Let G be an ultragraph. If (H, B) is an admissible pair in

G, then C

^∗

(G/(H, B)) ∼ = C

^∗

(G)/I

_(H,B)

.

Proof. Using Proposition 3.3, we can consider I

_(H,B)

as an ideal of C

^∗

(G).

Suppose that C

^∗

(G) = C

^∗

(s

_e

, p

_A

) and C

^∗

(G/(H, B)) = C

^∗

(t

_e

, q

_[A]

). If we define

T

e

:= s

e

+ I

_(H,B)

and Q

_[A]

:= p

A

+ I

_(H,B)

for every [A] ∈ Φ(G

⁰

) and e ∈ Φ(G

¹

), then the family {T

e

, Q

_[A]

} is a rep- resentation for G/(H, B) in C

^∗

(G)/I

_(H,B)

. So, there is a ∗-homomorphism φ : C

^∗

(G/(H, B)) → C

^∗

(G)/I

_(H,B)

such that φ(t

e

) = T

e

and φ(q

_[A]

) = Q

_[A]

. Moreover, all Q

_[A]

with [A] 6= [∅] are nonzero because p

A

+ I

_(H,B)

= I

_(H,B)

implies A ∈ H. Then, an application of Theorem 4.5 yields that φ is faithful.

On the other hand, the family {T

e

, Q

[A]

: [A] ∈ Φ(G

⁰

), e ∈ Φ(G

¹

)} generates the quotient C

^∗

(G)/I

(H,B)

, and hence, φ is surjective as well. Therefore, φ is

an isomorphism and the result follows. 

To prove a version of Cuntz-Krieger uniqueness theorem, we extend Condi- tion (L) for quotient ultragraphs.

Definition 4.7. We say that G/(H, B) satisfies Condition (L) if for every loop α = e

1

· · · e

n

in G/(H, B), at least one of the following conditions holds:

(i) r(e

_i

) 6= s(e

_i+1

) for some 1 ≤ i ≤ n, where e

_i+1

:= e

₁

(or equivalently, r(e

i

) \ s(e

i+1

) 6= [∅]).

(ii) α has an exit; that means, there exists f ∈ Φ(G

¹

) such that s(f ) ⊆ r(e

i

) and f 6= e

i+1

for some 1 ≤ i ≤ n.

Lemma 4.8. Let F be a finite subset of Φ

_sg

(G

⁰

) ∪ Φ(G

¹

). If G/(H, B) satisfies Condition (L), then so does the graph G

_F

.

Proof. Suppose that G/(H, B) satisfies Condition (L). As the elements of F

⁰

∪ Γ are sinks in G

F

, every loop in G

F

is of the form α = (e e

1

, e

2

) · · · (e

n

, e

1

) corresponding with a loop α = e

1

· · · e

n

in G/(H, B). So, fix a loop α = e (e

1

, e

2

) · · · (e

n

, e

1

) in G

F

. Then α = e

1

· · · e

n

is a loop in G/(H, B) and by Condition (L), one of the following holds:

(i) r(e

i

) 6= s(e

i+1

) for some 1 ≤ i ≤ n, where e

i+1

:= e

1

, or

(ii) there exists f ∈ Φ(G

¹

) such that s(f ) ⊆ r(e

i

) and f 6= e

i+1

for some 1 ≤ i ≤ n.

We can suppose in the case (i) that s(e

_i+1

) ( r(e

i

) and r(e

_i

) emits only the edge e

_i+1

in G/(H, B). Then, by the definition of Γ, there exists either [v] ∈ F

⁰

with [v] ⊆ r(e

_i

) \ s(e

_i+1

), or ω ∈ Γ with ω

_i

= 1. Thus, either (e

_i

, [v]) or (e

_i

, ω) is an exit for the loop α in G e

F

, respectively.

Now assume case (ii) holds. If f ∈ F

¹

, then (e

i

, f ) is an exit for α. If f / e ∈ F

¹

, for [v] := s(f ) we have either [v] / ∈ F

⁰

or

∃ω ∈ Γ with ω

i

= 1 such that [v] ⊆ R(ω).

Hence, (e

i

, [v]) or (e

i

, ω) is an exit for α, respectively. Consequently, in any e

case, α has an exit. e 

Theorem 4.9 (The Cuntz-Krieger Uniqueness Theorem). Suppose that G/(H, B) is a quotient ultragraph satisfying Condition (L). If {T

e

, Q

A

} is a Cuntz-Krieger representation for G/(H, B) in which all the projection Q

_[A]

are nonzero for [A] 6= [∅], then the ∗-homomorphism π

T ,Q

: C

^∗

(G/(H, B)) → C

^∗

(T

e

, Q

_[A]

) with π

T ,Q

(t

e

) = T

e

and π

T ,Q

(q

_[A]

) = Q

_[A]

is an isometrically iso- morphism.

Proof. It suffices to show that π

_{T ,Q}

is faithful. Similar to Theorem 4.5, choose an increasing sequence {F

_n

} of finite sets such that ∪

^∞_n=1

F

_n

= Φ

_sg

(G

⁰

)∪Φ(G

¹

).

By Corollary 4.3, there are isomorphisms π

n

: C

^∗

(G

F_n

) → C

^∗

({t

e

, q

_[v]

: [v] ∈ F

_n⁰

, e ∈ F

_n¹

}) that respect the generators. Since all the graphs G

F_n

satisfy Condition (L) by Lemma 4.8, the Cuntz-Krieger uniqueness theorem for graph C

^∗

-algebras implies that the ∗-homomorphisms

π

T ,Q

◦ π

n

: C

^∗

(G

F_n

) → C

^∗

(T

e

, Q

_[A]

)

are faithful. Therefore, π

T ,Q

is faithful on the subalgebra C

^∗

(t

e

, q

_[v]

: [v] ∈ Φ

sg

(G

⁰

), e ∈ Φ(G

¹

)) of C

^∗

(G/(H, B)). Since this subalgebra is dense in C

^∗

(G/(H, B)), we conclude that π

T ,Q

is a faithful homomorphism. 

5. Primitive ideals in C

^∗

(G)

In this section, we apply quotient ultragraphs to describe primitive gauge invariant ideals of an ultragraph C

^∗

-algebra. Recall that since every ultragraph C

^∗

-algebra C

^∗

(G) is separable (as assumed G

⁰

to be countable), a prime ideal of C

^∗

(G) is primitive and vice versa [3, Corollaire 1].

To prove Proposition 5.4 below, we need the following simple lemmas.

Lemma 5.1. Let G/(H, B) = (Φ(G

⁰

), Φ(G

⁰

), Φ(G

¹

), r, s) be a quotient ultra- graph of G. If G/(H, B) does not satisfy Condition (L), then C

^∗

(G/(H, B)) contains an ideal Morita-equivalent to C(T).

Proof. Suppose that γ = e

₁

· · · e

n

is a loop in G/(H, B) without exits and r(e

_i

) = s(e

_i+1

) for 1 ≤ i ≤ n. If C

^∗

(G/(H, B)) = C

^∗

(t

_e

, q

_[A]

), for each i we have

t

^∗_ei

t

e_i

= q

_r(ei)

= q

_s(ei+1)

= t

e_i+1

t

^∗_ei+1

.

Write [v] := s(γ) and let I

γ

be the ideal of C

^∗

(G/(H, B)) generated by q

_[v]

. Since γ has no exits in G/(H, B) and we have

q

_s(ei)

= (t

e_i

· · · t

e_n

)q

_[v]

(t

^∗_en

· · · t

^∗_ei

) (1 ≤ i ≤ n), an easy argument shows that

I

γ

= span t

α

q

[v]

t

^∗_β

: α, β ∈ (G/(H, B))

^∗

, [v] ⊆ r(α) ∩ r(β) . So, we get

q

_[v]

I

_γ

q

_[v]

= span (t

γ

)

ⁿ

q

_[v]

(t

^∗_γ

)

^m

: m, n ≥ 0 ,

where (t

_γ

)

⁰

= (t

^∗_γ

)

⁰

:= q

_[v]

. We show that q

_[v]

I

_γ

q

_[v]

is a full corner in I

_γ

which is

isometrically isomorphic to C(T). For this, let E be the graph with one vertex

w and one loop f . If we set Q

_w

:= q

_[v]

and T

_f

:= t

_γ

(= t

_γ

q

_[v]

), then {T

_f

, Q

_w

} is a Cuntz-Krieger E-family in q

_[v]

I

_γ

q

_[v]

. Assume C

^∗

(E) = C

^∗

(s

_f

, p

_w

). Since Q

_w

6= 0, the gauge-invariant uniqueness theorem for graph C

^∗

-algebras implies that the ∗-homomorphism φ : C

^∗

(E) → q

_[v]

I

γ

q

_[v]

with p

w

7→ Q

w

and s

f

7→ T

f

is faithful. Moreover, the C

^∗

-algebra q

_[v]

I

γ

q

_[v]

is generated by {T

f

, Q

w

}, and hence φ is an isomorphism. As we know C

^∗

(E) ∼ = C(T), q

[v]

I

γ

q

_[v]

is isomorphic to C(T). Moreover, since q

[v]

generates I

γ

, the corner q

_[v]

I

γ

q

_[v]

is full in I

γ

. Thus, I

γ

is Morita-equivalent to q

_[v]

I

γ

q

_[v]

∼ = C(T) and the proof is complete.  Lemma 5.2. If G/(H, B) satisfies Condition (L), then any nonzero ideal in C

^∗

(G/(H, B)) contains projection q

_[A]

for some [A] 6= [∅].

Proof. Take an arbitrary ideal J in C

^∗

(G/(H, B)). If there are no q

_[A]

∈ J with [A] 6= [∅], then Theorem 4.9 implies that the quotient homomorphism φ : C

^∗

(G/(H, B)) → C

^∗

(G/(H, B))/J is injective. Hence, we have J = ker φ =

(0). 

Definition 5.3. Let G be an ultragraph. For two sets A, C ∈ G

⁰

, we write A ≥ C if either A ⊇ C, or there exists α ∈ G

^∗

with |α| ≥ 1 such that s(α) ∈ A and C ⊆ r(α). We simply write A ≥ v, v ≥ C, and v ≥ w if A ≥ {v}, {v} ≥ C, and {v} ≥ {w}, respectively. A subset M ⊆ G

⁰

is said to be downward directed whenever for every A

1

, A

2

∈ M , there exists ∅ 6= C ∈ M such that A

1

, A

2

≥ C.

Proposition 5.4. Let H be a saturated hereditary subset of G

⁰

. Then the ideal I

_(H,BH)

in C

^∗

(G) is primitive if and only if the quotient ultragraph G/(H, B

H

) satisfies Condition (L) and the collection G

⁰

\ H is downward directed.

Proof. Let I

_(H,BH)

be a primitive ideal of C

^∗

(G). Since C

^∗

(G)/I

_(H,BH)

∼ = C

^∗

(G/(H, B

H

)), the zero ideal in C

^∗

(G/(H, B

H

)) is primitive. If G/(H, B

H

) does not satisfy Condition (L), then C

^∗

(G/(H, B

H

)) contains an ideal J Morita- equivalent to C(T) by Lemma 5.1. Select two ideals I

¹

, I

2

in C(T) with I

¹

∩I

2

= (0), and let J

1

, J

2

be their corresponding ideals in J . Then J

1

and J

2

are two nonzero ideals of C

^∗

(G/(H, B

H

)) with J

1

∩J

2

= (0), contradicting the primness of C

^∗

(G/(H, B

H

)). Therefore, G/(H, B) satisfies Condition (L).

Now we show that M := G

⁰

\ H is downward directed. For this, we take two arbitrary sets A

1

, A

2

∈ M and consider the ideals

J

1

:= C

^∗

(G/(H, B

H

))q

_[A1]

C

^∗

(G/(H, B

H

)) and

J

₂

:= C

^∗

(G/(H, B

_H

))q

_[A2]

C

^∗

(G/(H, B

_H

))

in C

^∗

(G/(H, B

H

)) generated by q

[A₁]

and q

[A₂]

, respectively. Since A

1

, A

2

∈ H, / the projections q

[A₁]

, q

[A₂]

are nonzero by Theorem 3.10, and so are the ideals J

1

, J

2

. The primness of C

^∗

(G/(H, B

H

)) implies that the ideal

J

1

J

2

= C

^∗

(G/(H, B

H

)) q

_[A1]

C

^∗

(G/(H, B

H

)) q

_[A2]

C

^∗

(G/(H, B

H

))

is nonzero, and hence q

_[A1]

C

^∗

(G/(H, B

_H

))q

_[A2]

6= {0}. As the set span t

α

q

_[D]

t

^∗_β

: α, β ∈ (G/(H, B))

^∗

, r(α) ∩ [D] ∩ r(β) 6= [∅]

is dense in C

^∗

(G/(H, B

H

)), there exist α, β ∈ (G/(H, B

H

))

^∗

and [D] ∈ Φ(G

⁰

) such that q

[A₁]

(t

α

q

[D]

t

^∗_β

)q

[A₂]

6= 0. In this case, we must have s(α) ⊆ [A

1

] and s(β) ⊆ [A

2

] and thus, A

1

, A

2

≥ C for C := r

G

(α) ∩ D ∩ r

G

(β). Therefore, G

⁰

\ H is downward directed.

For the converse, we assume that G/(H, B

_H

) satisfies Condition (L) and the collection M = G

⁰

\ H is downward directed. Fix two nonzero ideals J

1

, J

₂

of C

^∗

(G/(H, B

_H

)). By Lemma 5.2, there are nonzero projections q

_[A1]

∈ J

₁

and q

_[A2]

∈ J

2

. Then A

₁

, A

₂

∈ H and, since M is downward directed, there / exists C ∈ M such that A

₁

, A

₂

≥ C. Hence, the ideal J

₁

∩ J

₂

contains the nonzero projection q

_[C]

. Since J

₁

and J

₂

were arbitrary, this concludes that the C

^∗

-algebra C

^∗

(G/(H, B

_H

)) is primitive and I

_(H,BH)

is a primitive ideal in

C

^∗

(G) by Proposition 4.6. 

The following proposition describes another kind of primitive ideals in C

^∗

(G).

Proposition 5.5. Let (H, B) be an admissible pair in G and let B = B

H

\{w}.

Then the ideal I

_(H,B)

in C

^∗

(G) is primitive if and only if A ≥ w for all A ∈ G

⁰

\ H.

Proof. Suppose that I

(H,B)

is a primitive ideal and take an arbitrary A ∈ G

⁰

\H.

If A := A ∪ {v

⁰

: v ∈ A ∩ (B

H

\ B)}, then q

_[A]

and q

[w⁰]

are two nonzero projections in C

^∗

(G/(H, B)). If we consider ideals J

_[A]

:= hq

_[A]

i and J

[w⁰]

:=

hq

[w⁰]

i in C

^∗

(G/(H, B)), then the primness of C

^∗

(G/(H, B)) ∼ = C

^∗

(G)/I

_H,B

implies that the ideal

J

_[A]

J

[w⁰]

= C

^∗

(G/(H, B))q

_[A]

C

^∗

(G/(H, B))q

[w⁰]

C

^∗

(G/(H, B))

is nonzero, and hence q

_[A]

C

^∗

(G/(H, B))q

[w⁰]

6= {0}. So, there exist α, β ∈ (G/(H, B))

^∗

such that q

_[A]

t

α

t

^∗_β

q

[w⁰]

6= 0. Since [w

⁰

] is a sink in G/(H, B), we must have q

_[A]

t

_α

q

_[w0]

6= 0. If |α| = 0, then [w

⁰

] ⊆ [A], w

⁰

∈ A and w ∈ A.

If |α| ≥ 1, then s(α) ⊆ [A] and [w

⁰

] ⊆ r(α), which follow s

_G

(α) ∈ A and w ∈ r

_G

(α). Therefore, we obtain A ≥ w in either case.

Conversely, assume A ≥ w for every A ∈ G

⁰

\ H. Then the collection G

⁰

\ H is downward directed. Moreover, for every [∅] 6= [A] ∈ Φ(G

⁰

), there exists α ∈ (G/(H, B))

^∗

such that s(α) ⊆ [A] and [w

⁰

] ⊆ r(α). As [w

⁰

] is a sink in G/(H, B), we see that the quotient ultragraph G/(H, B) satisfies Condition (L). Now similar to the proof of Proposition 5.4, we can show that I

_(H,B)

is a

primitive ideal. 

Recall that each loop in G/(H, B) comes from a loop in the initial ultragraph

G. So, to check Condition (L) for a quotient ultragraph G/(H, B), we can use

the following.

Definition 5.6. Let H be a saturated hereditary subset of G

⁰

. For simplicity, we say that a path α = e

1

· · · e

n

lies in G \ H whenever r

_G

(α) ∈ G

⁰

\ H. We also say that α has an exit in G \ H if either r

_G

(e

i

) \ s

_G

(e

i+1

) ∈ G

⁰

\ H for some i, or there is an edge f with r

_G

(f ) ∈ G

⁰

\ H such that s

_G

(f ) = s

_G

(e

i

) and f 6= e

i

, for some 1 ≤ i ≤ n.

It is easy to verify that a quotient ultragraph G/(H, B) satisfies Condition (L) if and only if every loop in G \ H has an exit in G \ H. Hence we have:

Theorem 5.7 (See [1, Theorem 4.7]). Let G be an ultragraph. A gauge in- variant ideal I

_(H,B)

of C

^∗

(G) is primitive if and only if one of the following holds:

(1) B = B

H

, G

⁰

\ H is downward directed, and every loop in G \ H has an exit in G \ H.

(2) B = B

H

\ {w} for some w ∈ B

H

, and A ≥ w for all A ∈ G

⁰

\ H.

Proof. Let I

(H,B)

be a primitive ideal in C

^∗

(G). Then C

^∗

(G/(H, B)) ∼ = C

^∗

(G)/I

_(H,B)

is a primitive C

^∗

-algebra. We claim that |B

H

\ B| ≤ 1. Indeed, if w

1

, w

2

are two distinct vertices in B

H

\ B, similar to the proof of Propositions 5.4 and 5.5, the primitivity of C

^∗

(G/(H, B)) implies that the corner q

_[w⁰

1]

C

^∗

(G/(H, B))q

_[w⁰

2]

is nonzero. So, there exist α, β ∈ (G/(H, B))

^∗

such that q

[w⁰₁]

t

α

t

^∗_β

q

[w⁰₂]

6= 0. But we must have |α| = |β| = 0 because [w

₁⁰

], [w

₂⁰

] are two sinks in G/(H, B). Hence, q

[w⁰₁]

q

[w⁰₂]

6= 0 which is impossible because q

[w₁⁰]

q

[w⁰₂]

= q

[{w₁⁰}∩{w₂⁰}]

= q

[∅]

= 0.

Thus, the claim holds. Now we may apply Propositions 5.4 and 5.5 to obtain

the result. 

Following [10, Definition 7.1], we say that an ultragraph G satisfies Condition (K) if every vertex v ∈ G

⁰

either is the base of no loops, or there are at least two loops α, β in G based at v such that neither α nor β is a subpath of the other. In view of [10, Proposition 7.3], if G satisfies Condition (K), then all ideals of C

^∗

(G) are of the form I

_(H,B)

. So, in this case, Theorem 5.7 describes all primitive ideals of C

^∗

(G).

6. Purely infinite ultragraph C

^∗

-algebras via Fell bundles

Mark Tomforde in [17] determined ultragraph C

^∗

-algebras in which every

hereditary subalgebra contains infinite projections. Here, we consider the no-

tion of “pure infiniteness” in the sense of Kirchberg-Rørdam [11], and generalize

[8, Theorem 2.3] to ultragraph setting. In view of Proposition 3.14 and The-

orem 4.16 of [11], a (not necessarily simple) C

^∗

-algebra A is purely infinite if

and only if for every a ∈ A

⁺

\ {0} and closed two-sided ideal I E A, a + I

in the quotient A/I is either zero or infinite (in this case, a is called properly

infinite). Recall from [11, Definition 3.2] that an element a ∈ A

⁺

\ {0} is called

infinite if there is b ∈ A

⁺

\ {0} such that a ⊕ b . a ⊕ 0 in the matrix algebra

M

2

(A).