On the stability of a fixed point algebra $C^*(E)^gamma$ of a gauge action on a graph $C^*$-algebra

DOI 10.4134/JKMS.2009.46.3.657

ON THE STABILITY OF A FIXED POINT ALGEBRA C ^∗ (E) ^γ

OF A GAUGE ACTION ON A GRAPH C ^∗ -ALGEBRA

Ja A Jeong

Abstract. The fixed point algebra C

^∗

(E)

^γ

of a gauge action γ on a graph C

^∗

-algebra C

^∗

(E) and its AF subalgebras C

^∗

(E)

^γv

associated to each vertex v do play an important role for the study of dynamical prop- erties of C

^∗

(E). In this paper, we consider the stability of C

^∗

(E)

^γ

(an AF algebra is either stable or equipped with a (nonzero bounded) trace). It is known that C

^∗

(E)

^γ

is stably isomorphic to a graph C

^∗

- algebra C

^∗

(E

Z

× E) which we observe being stable. We first give an explicit isomorphism from C

^∗

(E)

^γ

to a full hereditary C

^∗

-subalgebra of C

^∗

(E

N

× E) (⊂ C

^∗

(E

Z

× E)) and then show that C

^∗

(E

N

× E) is stable whenever C

^∗

(E)

^γ

is so. Thus C

^∗

(E)

^γ

cannot be stable if C

^∗

(E

N

× E) admits a trace. It is shown that this is the case if the vertex matrix of E has an eigenvector with an eigenvalue λ > 1. The AF algebras C

^∗

(E)

^γv

are shown to be nonstable whenever E is irreducible. Several examples are discussed.

1. Introduction

Let E be a row finite directed graph and C ^∗ (E) be the graph C ^∗ -algebra of E generated by a universal Cuntz-Krieger E family {p v , s e } (for example, see [1, 3, 18, 19, 22]). Then by the universal property, the gauge action γ of T, γ z (p v ) = p v , γ z (s e ) = zs e , is well defined and the fixed point algebra C ^∗ (E) ^γ turns out to be an AF algebra. In fact, it is known in [17] using results of [25] and [19] on groupoid C ^∗ -algebras that C ^∗ (E) ^γ is strong Morita equivalent (hence stably isomorphic by [7]) to the graph C ^∗ -algebra C ^∗ (E Z × E) of the Cartesian product graph E Z × E (E Z × E is the graph Z × E in [17]). Since E Z × E has no loops, its graph C ^∗ -algebra C ^∗ (E Z × E) is an AF algebra ([18]).

In this paper we are concerned with the question whether C ^∗ (E) ^γ is in fact isomorphic to C ^∗ (E Z × E). For this, we observe that C ^∗ (E Z × E) is always stable, that is, C ^∗ (E Z × E) ∼ = C ^∗ (E Z × E) ⊗ K, where K is the C ^∗ -algebra of compact operators on a separable infinite dimensional Hilbert space. Thus

Received February 21, 2008.

2000 Mathematics Subject Classification. 46L05, 46L55.

Key words and phrases. graph C

^∗

-algebra, stable C

^∗

-algebra, fixed point algebra, full hereditary C

^∗

-subalgebra.

Research supported by KRF-2005-041-C00030.

°2009 The Korean Mathematical Societyc

657

the question is equivalent to asking if C ^∗ (E) ^γ is stable. But we will see that C ^∗ (E) ^γ may not be stable (then it should admit a nonzero bounded trace since every AF algebra is either stable or equipped with such a nonzero bounded trace by [4, 24]).

The fixed point algebra C ^∗ (E) ^γ and its AF subalgebras C ^∗ (E) ^γ _v associated to each vertex v of E do play an important role for the study of the dynamical properties of C ^∗ (E). For example, if E is locally finite, C ^∗ (E) ^γ contains a C ^∗ - subalgebra isomorphic to the commutative C ^∗ -algebra C 0 (X E ) of continuous functions (vanishing at infinity) on the locally compact shift space X E of one- sided infinite paths, and it is shown in [13] that if X _E and X _F are topologically conjugate, the graph C ^∗ -algebras C ^∗ (E) and C ^∗ (F ) are isomorphic. Moreover, for E irreducible, the topological entropy ht(Φ E ) (in the sense of [8, 28]) of the canonical completely positive map Φ E on C ^∗ (E) is equal to that of the restriction Φ E | _C

^∗

_(E)

^γ

([15]). C ^∗ (E) ^γ _v is a Φ E -invariant subalgebra of C ^∗ (E) ^γ such that the topological entropy ht(Φ E | _C

∗

(E)

^γv

) is equal to the loop entropy of the graph E if E is a locally finite irreducible infinite graph [13]. The restriction of Φ E onto the commutative subalgebra isomorphic to C 0 (X E ) corresponds to the ∗-homomorphism on C 0 (X E ) induced by the continuous shift map on X E . It is known in [7] that every full hereditary C ^∗ -subalgebra B of a C ^∗ -algebra A is stably isomorphic to A. We will define an isomorphism from C ^∗ (E) ^γ onto a full hereditary C ^∗ -subalgebra A γ of a graph C ^∗ -algebra C ^∗ (E N × E) which itself can be viewed as a full hereditary C ^∗ -subalgebra of C ^∗ (E Z × E) (thus C ^∗ (E) ^γ is stably isomorphic to C ^∗ (E Z × E) as proved in [17]). The isomorphism is obtained by using the fact that C ^∗ (E) ^γ can be identified with a full hereditary C ^∗ -subalgebra of the crossed product C ^∗ (E) × γ T because T is compact ([16, 26]) and the concrete isomorphism between C ^∗ (E) × γ T and C ^∗ (E Z × E) constructed in [15]. (It was already known in [17] that these two algebras C ^∗ (E) × γ T and C ^∗ (E Z × E) are isomorphic, but with no explicit isomorphism.) The ideal structure of C ^∗ (E) ^γ has been studied in [20].

We show in Theorem 4.2 that if C ^∗ (E) ^γ is stable, so is C ^∗ (E N × E), which implies that the C ^∗ -algebras A γ (∼ = C ^∗ (E) ^γ ) ⊂ C ^∗ (E N × E) ⊂ C ^∗ (E Z × E) are all isomorphic if and only if C ^∗ (E) ^γ is stable. (In particular, C ^∗ (E) ^γ can be realized as a graph C ^∗ -algebra.) By an example we also show that the converse of the theorem may not be true. Theorem 4.2 is useful especially when we want to prove nonstability of C ^∗ (E) ^γ . Of course, C ^∗ (E) ^γ is possibly stable.

Actually a locally finite irreducible (infinite) graph E is given for which C ^∗ (E) ^γ is stable (we prove that C ^∗ (E N × E) cannot admit a nonzero bounded trace).

In Theorem 5.1, we give a condition in terms of the vertex matrix of E under

which C ^∗ (E N × E) admits a bounded trace, hence C ^∗ (E) ^γ is not stable by

Theorem 4.2. Examples of E with nonstable C ^∗ (E) ^γ are discussed. Finally

we prove that the AF subalgebras C ^∗ (E) ^γ _v of C ^∗ (E) ^γ are not stable if E is

irreducible.

2. Preliminaries

Crossed products by compact groups and fixed point algebras. Let A be a C ^∗ -algebra and α be an action of a compact group G on A. Then the ∗-algebra C(G, A) of continuous functions from G to A with the following convolution (as multiplication) and involution

f ∗ g(t) = Z

G

f (s)α s (g(s ⁻¹ t))ds, f ^∗ (t) = α t (f (t ⁻¹ ) ^∗ )

is dense in the crossed product A× α G, where ds is the normalized Haar measure on G (see [21, 7.7] or [10, 8.3.1]). If ˜ A denotes the smallest unitization of A (so ˜ A = A if A is unital), every continuous function h : G → ˜ A belongs to the multiplier algebra of A × α G. In particular, the constant function 1 G : G → ˜ A given by 1 G (s) = 1, s ∈ G, is a projection of the multiplier algebra of A × α G ([26]). Thus 1 G (A × α G)1 G is a hereditary C ^∗ -subalgebra of A × α G.

Remark 2.1. Let α be an action of a compact group G on a C ^∗ -algebra A.

(i) For a function f ∈ C(G)(⊂ C(G, ˜ A)) and an element x ∈ A, define f · x ∈ C(G, A) by

(f · x)(s) = f (s)x, s ∈ G.

Then span{f · x | f ∈ C(G), x ∈ A} is dense in A × α G.

(ii) If A ^α := {a ∈ A | α g (a) = a for all g ∈ G} is the fixed point algebra of α, identifying x ∈ A ^α and the constant function 1 G · x in C(G, A) with the value x everywhere we see that

(1) x 7→ 1 G · x : A ^α → 1 G (A × α G)1 G

is an isomorphism of A ^α onto the hereditary subalgebra 1 G (A × α G)1 G

of A × α G ([26]).

Graph C ^∗ -algebras. A directed graph E = (E ⁰ , E ¹ , r, s) consists of the vertex set E ⁰ , the edge set E ¹ , and the range, source maps r, s : E ¹ → E ⁰ . E is called row finite if each vertex of E emits only finitely many edges and locally finite if it is row finite and each vertex receives only finitely many edges. By E ⁿ we denote the set of all finite paths α = e 1 · · · e n (r(e i ) = s(e i+1 ), 1 ≤ i ≤ n − 1) of length n (|α| = n) (Vertices are finite paths of length 0). Then E ^∗ = ∪ n≥0 E ⁿ denotes the set of all finite paths. Infinite paths e 1 e 2 e 3 · · · or · · · e 3 e 2 e 1 can be considered and the maps r or s naturally extend to E ^∗ and the infinite paths.

A vertex v is called a sink if s ⁻¹ (v) = ∅ and a source if r ⁻¹ (v) = ∅. In this paper, we consider only row finite graphs. For v, w ∈ E ⁰ , we write v À w if there is a path α ∈ E ^∗ with s(α) = v and r(α) = w.

Now we collect some definitions from [18] and [19] that we will be using below:

(i) E is irreducible if v À w for any v, w ∈ E ⁰ .

(ii) A finite path β is a loop if s(β) = r(β) and |β| > 0.

(iii) An exit of a subgraph F of E is an edge e ∈ E ¹ with s(e) ∈ F ⁰ and r(e) / ∈ F ⁰ . E has property (L) if every loop has an exit. A graph with no loops has the property vacuously.

(iv) E has property (K) if for any vertex v and a loop β = β 1 β 2 · · · β |β| with s(β) = v there is another loop α = α 1 α 2 · · · α |α| with s(α) = v such that α i 6= β i for some i ≤ min{|α|, |β|}. (K) implies (L).

It is now well known ([3, 18, 19, 22]) that there exists a universal C ^∗ -algebra C ^∗ (E), called the graph C ^∗ -algebra, associated with a row finite graph E gen- erated by a Cuntz-Krieger E-family which consists of operators {s e , p v | e ∈ E ¹ , v ∈ E ⁰ } such that {s e } e∈E

¹

are partial isometries and {p v } v∈E

⁰

are mutu- ally orthogonal projections satisfying the relations

s ^∗ _e s e = p r(e) and p v = X

s(e)=v

s e s ^∗ _e if s ⁻¹ (v) 6= ∅.

(We simply write C ^∗ (E) = C ^∗ (s e , p v ) if C ^∗ (E) is generated by {s e , p v | e ∈ E ¹ , v ∈ E ⁰ }.) For each α = α 1 α 2 · · · α |α| ∈ E ^∗ , α i ∈ E ¹ , s α denotes the partial isometry s α

1

s α

2

· · · s α

|α|

(s v = s ^∗ _v = p v for v ∈ E ⁰ ). Note that for every α ∈ E ^∗ ,

s α s ^∗ _α ≤ p _s(α) and s ^∗ _α s α = p _r(α) .

Remark 2.2. Let C ^∗ (E) = C ^∗ (s e , p v ) be the graph C ^∗ -algebra associated with a row finite graph E. We will need the following basic facts which can be easily found in [1], [3], [18], [19], [22], etc.

(i) C ^∗ (E) = span{s α s ^∗ _β | α, β ∈ E ^∗ } since

s ^∗ _α s β =

 



s ^∗ _µ , if α = βµ, s ν , if β = αν, 0, otherwise.

Also s α s ^∗ _β = 0 if r(α) 6= r(β).

(ii) For each p v ∈ C ^∗ (E) and n ∈ N, if E is row finite, p v = X

s(α)=v

|α|=n

s α s ^∗ _α .

(iii) Let E ⁰ := {v 1 , v 2 , v 3 , . . . }. Then the set of projections { P _n

i=1 p v

i

| n ≥ 1} forms an approximate identity for C ^∗ (E). C ^∗ (E) is unital if and only if E ⁰ is finite.

(iv) If E has property (L), in particular if E has no loops, every Cuntz- Krieger E-family of nonzero operators generates a C ^∗ -algebra isomor- phic to C ^∗ (E).

(v) If V ⊂ E ⁰ is a hereditary subset (v ∈ V , v À w implies w ∈ V ),

I(V ) := span{s α s ^∗ _β | r(α) = r(β) ∈ V } is an ideal of C ^∗ (E). Fur-

thermore for E with property (K), V → I(V ) constitutes a bijection

between the set of saturated hereditary vertex subsets of E ⁰ and the ideals of C ^∗ (E) (V ⊂ E ⁰ is saturated if r(s ⁻¹ (v)) ⊂ V implies v ∈ V ).

Let G be a countable group. Recall ([17]) that for a graph E and a function c : E ¹ → G, the skew product graph E(c) is defined to be (G × E ⁰ , G × E ¹ , r, s), where

s(g, e) = (g, s(e)) and r(g, e) = (gc(e), r(e)).

For two graphs E and F , the Cartesian product is the graph E × F = (E ⁰ × F ⁰ , E ¹ × F ¹ , r, s),

where r(e, f ) = (r(e), r(f )) and s(e, f ) = (s(e), s(f )). For example, if

• _ _ _ _ _ // • _ _ _ _ _ // • _ _ _ _ _ // • _ _ _ _ _ // • · · · ,

· · · E Z :

−1 0 1 2 3

• _ _ _ _ _ // • _ _ _ _ _ // • · · · , E N :

1 2 3

then E N × E Z is as follows;

• E N × E Z :

•

(1, −1) (1, 0) (1, 1)

• • •

• • • • •

• • • • •

K K K K K K

%% K K K K K K

%% K K K K K K

%% K K K K K K

%% · · ·

K K K K K K

%% K K K K K K

%% K K K K K K

%% K K K K K K

%%

· · ·

.. .

Note that E Z × E or E N × E have no loops for every E. Moreover, E Z × E = E(c) if c : E ¹ → Z is given c(e) = 1. For ease of notation, we denote an edge x of E Z × E by (n, e) (n ∈ Z, e ∈ E ¹ ) if s(x) = (n, s(e)) and r(x) = (n + 1, r(e)).

For paths of E Z × E (or E N × E), we use similar notations, namely we write (n, α) for a path (n, α 1 )(n + 1, α 2 ) · · · (n + |α| − 1, α _|α| ).

3. C ^∗ (E) ^γ , C ^∗ (E N × E), and C ^∗ (E Z × E)

By the universal property of C ^∗ (E) = C ^∗ (s e , p v ), there exists an action γ (called the gauge action) of T on C ^∗ (E) given by

γ z (s e ) = zs e , γ z (p v ) = p v , z ∈ T.

The fixed point algebra of γ is

C ^∗ (E) ^γ = span{s _α s ^∗ _β | α, β ∈ E ^∗ , |α| = |β|}.

Applying some results of [25] on groupoid C ^∗ -algebras it is proved in [17] that

C ^∗ (E Z × E) ∼ = C ^∗ (E) × γ T. But one can also give an explicit isomorphism:

Proposition 3.1 ([15]). Let E be a row-finite graph with no sinks. If C ^∗ (E) = C ^∗ (p v , s e ) and C ^∗ (E Z × E) = C ^∗ (p _(n,v) , s _(n,e) ), then there is an isomorphism η : C ^∗ (E Z × E) → C ^∗ (E) × γ T such that

η(p _(m,v) ) = z ^m · p v , η(s _(m,e) ) = z ^m · s e , where m ∈ Z, v ∈ E ⁰ , and e ∈ E ¹ .

Since the graph E N × E has property (L) for every E, C ^∗ (E N × E) can be identified with the C ^∗ -subalgebra

C ^∗ (E N × E) = C ^∗ {p _(n,v) , s _(n,e) | n ∈ N, v ∈ E ⁰ , e ∈ E ¹ } of C ^∗ (E Z × E) (Remark 2.2.(iv)).

Proposition 3.2. Let E be a row-finite graph with no sinks.

(i) If F is a subgraph of E with no exits, then B F := span{s α s ^∗ _β | α, β ∈ F ^∗ } is a hereditary C ^∗ -subalgebra of C ^∗ (E) that generates the ideal

I(F ⁰ ) = span{s α s ^∗ _β ∈ C ^∗ (E) | r(α) = r(β) ∈ F ⁰ }.

(ii) C ^∗ (E N × E) is a full hereditary C ^∗ -subalgebra of C ^∗ (E Z × E).

Proof. (i) From B F ·C ^∗ (E) ⊂ B F we see that B F is a hereditary C ^∗ -subalgebra of C ^∗ (E). Since F ⁰ is a hereditary vertex subset, by Remark 2.2.(v), I(F ⁰ ) is an ideal of C ^∗ (E). B F ⊂ I(F ⁰ ) is obvious and I(F ⁰ ) is generated by B F

because s α s ^∗ _β = s α p _r(α) s ^∗ _β and p _r(α) ∈ B F if s α s ^∗ _β ∈ I(F ⁰ ).

(ii) Let C ^∗ (E Z × E) = C ^∗ (p _(n,v) , s _(n,e) ), n ∈ Z, v ∈ E ⁰ , and e ∈ E ¹ . Since E N × E has no exits, by (i), C ^∗ (E N × E) is a hereditary subalgebra generating the ideal

I = span{s _(n,α) s ^∗ _(m,β) | r(n, α) = r(m, β) ∈ (E N × E) ⁰ }.

Since E has no sinks, every element of the form s µ s ^∗ _ν ∈ C ^∗ (E Z × E) can be written as the finite sum of elements s α s ^∗ _β with r(α) = r(β) ∈ (E N × E) ⁰ by

Remark 2.2.(ii), so that s µ s ^∗ _ν ∈ I. ¤

Proposition 3.3. Let E be a locally finite graph with no sinks and sources.

Then C ^∗ (E) ^γ is isomorphic to the full hereditary C ^∗ -subalgebra A γ := span{s _(1,α) s ^∗ _(1,β) | α, β ∈ E ^∗ and |α| = |β|}

of C ^∗ (E N × E).

Proof. Let η : C ^∗ (E _Z ×E) → C ^∗ (E)× _γ T be the isomorphism of Proposition 3.1.

We show that η(A γ ) = B γ , where

B γ := span{1 G · s α s ^∗ _β | α, β ∈ E ^∗ and |α| = |β|}

is isomorphic to C ^∗ (E) ^γ by (1) of Remarks 2.1. Then, since the hereditary C ^∗ -

subalgebra B γ is full in C ^∗ (E) × γ T by [16, Proposition 5.4 and Theorem 6.3],

so is A γ = η ⁻¹ (B γ ) in C ^∗ (E Z × E).

First note that if x = s α s ^∗ _β , y = s µ s ^∗ _ν , f (z) = z ⁿ , and g(z) = z ^k , then (f · x) ∗ (g · y)(z) = z ^k ¡

xy Z

T

w n−k+|µ|−|ν| dw ¢ , (f · x) ^∗ (z) = f (z)γ z (x) ^∗

from which we have for α = α 1 α 2 · · · α n ∈ E ⁿ ,

η(s _(1,α) ) = η(s _(1,α

₁

₎ ) ∗ η(s _(2,α

₂

₎ ) ∗ · · · ∗ η(s _(n,α

_n

₎ )

= (z ¹ · s _α

₁

) ∗ (z ² · s _α

₂

) ∗ · · · ∗ (z ⁿ · s _α

_n

)

= z ⁿ · s α . Thus if α, β ∈ E ⁿ , then

η(s _(1,α) s ^∗ _(1,β) ) = η(s _(1,α) ) ∗ η(s _(1,β) ) ^∗

= (z ⁿ · s α ) ∗ (z ⁿ · s β ) ^∗ = (z ⁿ · s α ) ∗ (1 G · s ^∗ _β ) = 1 G · s α s ^∗ _β . ¤ Remark 3.4. It is known in [17] that C ^∗ (E) ^γ and C ^∗ (E Z ×E) are stably isomor- phic (or strong Morita equivalent) if E is a row finite graph with no sinks, which also immediately follows from Proposition 3.2 and Proposition 3.3 above since every C ^∗ -algebra is stably isomorphic to its full hereditary C ^∗ -subalgebras.

4. Stable case

Note that the algebras C ^∗ (E) ^γ , C ^∗ (E Z × E), and C ^∗ (E N × E) are all AF.

An AF algebra is known to be stable (A ∼ = A ⊗ K) unless it admits a nonzero bounded trace [4, 24].

The following lemma is immediate from [11, Lemma 2.1] and [12, Theo- rem 3.3]. For two projections p, q, we write p . q if p is equivalent to a subprojection of q.

Lemma 4.1. Let A be a C ^∗ -algebra with an approximate identity (p n ) n≥1

consisting of projections with p 1 ≤ p 2 ≤ · · · . Then we have the following:

(i) A is stable if and only if for every n, there is an m > n such that p n . p m − p n .

(ii) For a row-finite graph E, C ^∗ (E) = C ^∗ (p v , s e ) is stable if and only if for each finite subset V ⊂ E ⁰ , there is a finite set W ⊂ E ⁰ with V ∩ W = ∅ such that P

v∈V p v . P

w∈W p w . For a finite subset V ⊂ E ⁰ , let p V := P

v∈V p v and let E _s ⁿ

−1

(V ) := {α ∈ E ⁿ | s(α) ∈ V }.

Theorem 4.2(ii) below shows that C ^∗ (E) ^γ and C ^∗ (E Z × E) (and C ^∗ (E N × E)) are all isomorphic if and only if C ^∗ (E) ^γ is stable. A vertex v ∈ E ⁰ is left-infinite if there is an infinite path α ending at v such that all edges of α are distinct (see [11, Lemma 2.11]) and E is left-infinite if every vertex of E is left-infinite.

It is known in [11, Lemma 2.13] that if E is a locally finite left-infinite graph,

C ^∗ (E) is stable. But the converse need not be true, in fact, E N × E is not

left invertible while C ^∗ (E N × E) is possibly stable (see Theorem 4.2(iii) and examples of this section).

Theorem 4.2. Let E be a locally finite infinite graph with no sinks and sources.

Then we have the following:

(i) Let c : E ¹ → G be a function. If E is left-infinite, so is E(c).

(ii) C ^∗ (E _Z × E) is stable.

(iii) If C ^∗ (E) ^γ is stable, then both C ^∗ (E) and C ^∗ (E N × E) are stable.

Proof. (i) Let E be left-infinite and (g 0 , v 0 ) ∈ E(c) ⁰ . Since v 0 ∈ E ⁰ is left- infinite, there is an infinite path α consisting of distinct edges, α = · · · α 3 α 2 α 1

with r(α 1 ) = v 0 . Then the infinite path

· · · (g 0 c(α 1 ) ⁻¹ c(α 2 ) ⁻¹ c(α 3 ) ⁻¹ , α 3 )(g 0 c(α 1 ) ⁻¹ c(α 2 ) ⁻¹ , α 2 )(g 0 c(α 1 ) ⁻¹ , α 1 ) ending at (g 0 , v 0 ) has distinct edges. Hence E(c) is left-infinite.

(ii) Note that E Z × E is left-infinite.

(iii) Suppose C ^∗ (E) ^γ is stable. Since C ^∗ (E) ^γ contains an approximate iden- tity { P _n

i=1 p v

i

| n = 1, 2, . . . } of C ^∗ (E) (Remark 2.2.(iii)), applying Lemma 4.1 we see that C ^∗ (E) is stable. For stability of C ^∗ (E N × E), let E N

n

× E (n ≥ 1) be the subgraph of E N × E with (E N

n

× E) ⁰ = {(k, v) | k ≥ n, v ∈ E ⁰ } and (E N

n

× E) ¹ = {(k, e) | k ≥ n, e ∈ E ¹ }. Clearly, ϕ n : C ^∗ (E N × E) → C ^∗ (E N

n

× E), ϕ n (p (i,v) ) = p (i+n,v) , ϕ n (s (j,e) ) = s (j+n,e) (i, j ≥ 1), is an iso- morphism. For each k ≥ 1 and a finite subset V ⊂ E ⁰ , set

[1, k] × V := {(i, v) ∈ (E N × E) ⁰ | 1 ≤ i ≤ k, v ∈ V }.

Then the corresponding projection p _[1,k]×V can be written as p [1,k]×V =

X k n=1

p {n}×V = X k n=1

¡ X

v∈V

p (n,v)

¢ .

For each n, consider the projection ϕ ⁻¹ _n (p {n}×V ) = p {1}×V in C ^∗ (E N × E).

Since p {1}×V belongs to A γ (∼ = C ^∗ (E) ^γ ) and we assume that C ^∗ (E) ^γ is stable, by Lemma 4.1.(i) there exists a finite vertex set W ⊂ E ⁰ with V ∩ W = ∅ and a partial isometry x ∈ C ^∗ (E) ^γ such that x ^∗ x = p {1}×V and xx ^∗ ≤ p {1}×W . Then x n := ϕ n (x) is a partial isometry in C ^∗ (E N

n

× E) satisfying x ^∗ _n x n = p {n}×V

and x n x ^∗ _n ≤ p _{n}×W . Now X := P _k

n=1 x n ∈ C ^∗ (E N × E) is a partial isometry such that X ^∗ X = p _[1,k]×V and XX ^∗ ≤ p _[1,k]×W . This completes the proof since every finite vertex subset of (E N × E) ⁰ is contained in [1, k] × V for some

k and V and ([1, k] × V ) ∩ ([1, k] × W ) = ∅. ¤

Proposition 4.3. Let E be a locally finite infinite graph without sinks or sources. Then we have the following:

(i) C ^∗ (E) ^γ is stable if for every finite subset V ⊂ E ⁰ , there is an l ∈ N

and a finite vertex subset W ⊂ E ⁰ with V ∩ W = ∅ such that for each

α ∈ E _s ^l

−1

(V ) , there is α ⁰ ∈ E _s ^l

−1

(W ) with r(α) = r(α ⁰ ) such that α 7→ α ⁰

is injective.

(ii) If every vertex of E receives at most one edge, then C ^∗ (E N × E) is stable.

Proof. (i) is obvious by Proposition 3.3 and Lemma 4.1.

(ii) Let {p (n,v) , s (n,e) } be the Cuntz-Krieger (E N × E)-family and V be a finite subset of (E N × E) ⁰ . Then there is k 0 ∈ N such that (n, v) ∈ V implies n < k 0 . For each (n, v) ∈ V , consider the following set of paths

S _(n,v) ^k

⁰

:= (E N × E) ^k _s

⁰−1

(n,v) = {(n, α) ∈ (E N × E) ^k

⁰

| s(n, α) = (n, v)}.

Then p _(n,v) = P

(n,α)∈S

^k0_(n,v)

s _(n,α) s ^∗ _(n,α) is equivalent to P

(n,α)∈S

_(n,v)^k0

s ^∗ _(n,α) s _(n,α) , note here that the projections s ^∗ _(n,α) s _(n,α) = p _r(n,α) are mutually orthogonal since there is only one path with range r(n, α) and with length k 0 . Moreover, if (m, w) ∈ W := {r(n, α) | (n, α) ∈ S _(n,v) ^k

⁰

}, then m ≥ k 0 and so (m, v) / ∈ V . Therefore we have V ∩ W = ∅ and P

(n,v)∈V p (n,v) ∼ P

(m,w)∈W p (m,w) . Thus

by Lemma 4.1 the assertion follows. ¤

Example 4.4. For the following graph E, C ^∗ (E N × E) is stable. But C ^∗ (E) ^γ is not.

• // • // • // • · · ·

>>

v 0 v 1 v 2 v 3

E :

By Proposition 4.3(ii), C ^∗ (E N × E) is stable. But C ^∗ (E) is not stable by [11, Lemma 2.16] since it has a quotient C ^∗ -algebra isomorphic to the nonstable algebra C(T) ∼ = C ^∗ (E)/I, where I is the ideal corresponding to the saturated hereditary vertex subset {v 1 , v 2 , . . . }. Thus C ^∗ (E) ^γ is not stable by Theo- rem 4.2.(iii).

Example 4.5. C ^∗ (F ) ^γ is stable if F is as follows:

• e 4 // • e 3 // • e 2 // • e 1 // •

cc v 5 v 4 v 3 v 2 v 1

e 0

· · · F :

In fact, the increasing sequence of projections p n := P _n

i=1 p v

i

, n ≥ 1, is an approximate identity for C ^∗ (E) ^γ such that each p n is equivalent to p m − p n for some m > n in C ^∗ (E) ^γ : The partial isometry

s := s e

2n−1

···e

1

s ^∗ _e

_n−1

_···e

₁

_e

ⁿ

0

+ · · · + s e

n

···e

1

s ^∗ _e

ⁿ

0

of C ^∗ (E) ^γ satisfies s ^∗ s = p n and ss ^∗ = p 2n − p n . Thus the stability of C ^∗ (E) ^γ follows from Lemma 4.1.

Example 4.6. C ^∗ (E) ^γ is stable for the following irreducible graph E:

• •

E : ÁÁ

ÁÁ •

v −2 v −1 ÁÁ •

^^ •

^^

^^ ÁÁ

^^ v 0 v 1 v 2

· · · · · ·

e ⁰ ₂ e ⁰ ₁ e 1 e 2 e 3

f ₂ ⁰ f ₁ ⁰ f 1 f 2 f 3

ÁÁ ^^

We show that A γ (of Proposition 3.3) is stable. Since E N × E has property (K) and there are only two nontrivial saturated hereditary vertex subsets V 0 , V 1

in (E N × E) ⁰ (V 0 ⊃ {(1, k) | k is even} and V 1 ⊃ {(1, k) | k is odd}) such that V 0 ∩ V 1 = ∅, we see that C ^∗ (E N × E) has only two nontrivial (proper) ideals I(V 0 ) and I(V 1 ). Moreover I(V 0 ) ∩ I(V 1 ) = {0} because V 0 ∩ V 1 = ∅. Since A γ is a full hereditary C ^∗ -subalgebra of C ^∗ (E N × E), I 7→ A γ ∩ I establishes a bijection between the sets of ideals of C ^∗ (E N × E) and A γ . Thus A γ has two nontrivial ideals A γ ∩ I(V 0 ) and A γ ∩ I(V 1 ). But actually these are isomorphic and A γ = (A γ ∩I(V 0 ))⊕(A γ ∩I(V 1 )). If A γ is not stable, there exists a nonzero bounded trace τ . Then τ | A

γ

∩I(V

i

) is nonzero for some i = 0, 1. Assume that τ | A

γ

∩I(V

0

) is nonzero. Note that the projections {p n := P _n

k=−n p (1,v

k

) } n forms an approximate identity for A γ ∩ I(V 0 ). Then τ (p _(1,v

_2k

₎ ) 6= 0 for some k. We may assume that τ (p _(1,v

₀

₎ ) = 1. Consider the following subgraph of E N × E.

• • • •

•

•

•

• • •

•

•

•

• • • •

•

•

•

• • •

•

•

•

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ÂÂ

(1, v 0 )

(5, v 0 )

(5, v −4 ) (5, v 4 )

(1, v 2 ) (1, v −2 )

(2, v 1 )

.. .

· · · · · ·

If (1, α), (1, β) ∈ (E N × E) ^2k are paths from (1, v 0 ) to (2k + 1, v 2i ), −k ≤ i ≤ k, then x := s _(1,α) s ^∗ _(1,β) ∈ A _γ satisfies

xx ^∗ = s (1,α) s ^∗ _(1,α) , x ^∗ x = s (1,β) s ^∗ _(1,β) . Thus τ (s _(1,α) s ^∗ _(1,α) ) = τ (s _(1,β) s ^∗ _(1,β) ), hence for each k ≥ 1,

1 = τ (p (1,v

₀

) ) = X

α∈E

^2k

s(α)=v

0

τ (s (1,α) s ^∗ _(1,α) ) = X

v

2i

X

α∈E

^2k

s(α)=v

0

r(α)=v

2i

τ (s (1,α) s ^∗ _(1,α) ),

where −k ≤ i ≤ k. If K 2i is the number of paths α ∈ E ^2k with s(α) = v 0 and r(α) = v 2i , then

K 2i = µ 2k

k − i

¶

= (2k)!

(k + i)!(k − i)! ≤ µ 2k

k

¶

= K 0 . Let t 2i := τ (s (1,α) s ^∗ _(1,α) ) for α ∈ E ^2k with s(α) = v 0 , r(α) = v 2i . Then

1 = τ (p _(1,v

₀

₎ ) = X k i=−k

K 2i t 2i = X k i=−k

µ 2k k − i

¶ t 2i ≤

X k i=−k

µ 2k k

¶

t 2i .

On the other hand, for each i, there are 2 ^2k paths µ ∈ E ^2k with r(µ) = v 2i . Thus we have that

kτ k ≥ X k i=−k

2 ^2k t 2i . Now we show by induction on k that 2 ^2k > k ^1/3 · ¡ _2k

k

¢ for all k ≥ 1. In fact, the inequality holds for k = 1. Suppose 2 ^2k > k ^1/3 · ¡ _2k

k

¢ , then

2 ^2k > k ^1/3 · µ 2k

k

¶

= k ^1/3 · (2k)!

(k!) ²

= k ^1/3 · (2k)!(2k + 1)(2k + 2)

(k!) ² (k + 1)(k + 1) · (k + 1)(k + 1) (2k + 1)(2(k + 1))

= (k + 1) ^1/3 · (2(k + 1))!

((k + 1)!) ² · k ^1/3

(k + 1) ^1/3 · k + 1 2(2k + 1) , from which we have

2 ^2k+2 > (k + 1) ^1/3 ·

µ 2(k + 1) k + 1

¶

· k ^1/3

(k + 1) ^1/3 · 4k + 4 4k + 2

> (k + 1) ^1/3 ·

µ 2(k + 1) k + 1

¶

since _(k+1) ^k

^1/31/3

· ^4k+4 _4k+2 > 1 for all k ≥ 1. Then

kτ k ≥ X k i=−k

2 ^2k t 2i >

X k i=−k

k ^1/3 µ 2k

k

¶

t 2i ≥ k ^1/3

which goes to ∞ as k → ∞, a contradiction to the boundedness of τ . 5. Nonstable case

In this section, we consider locally finite infinite graphs E for which C ^∗ (E N × E) have bounded traces (hence, not stable). Of course, then C ^∗ (E) ^γ is not stable by Theorem 4.2.

Recall ([11, Definition 2.7]) that τ : E ⁰ → [0, ∞) is a bounded graph-trace if τ (v) = X

{e|s(e)=v}

τ (r(e)) and X

v∈E

⁰

τ (v) < ∞.

If E has no loops, every bounded graph-trace on E extends to a bounded trace E ([11, Lemma 2.8]).

Theorem 5.1. Let E be a locally finite infinite graph with no sinks and sources.

Let E ⁰ := {1, 2, . . . } and A = (a ij ) be the vertex matrix of E, that is, A is an E ⁰ ×E ⁰ matrix with a ij edges from vertex i to vertex j. If there is an eigenvector ξ = (ξ 1 , ξ 2 , . . . ) of A with an eigenvalue λ (Aξ = λξ) such that

(i) λ > 1,

(ii) ξ i ≥ 0 for each i ∈ E ⁰ and 0 < P

i≥1 ξ i < ∞,

then C ^∗ (E N × E) admits a bounded trace. In particular, C ^∗ (E N × E) is not stable.

Proof. Since E N × E has no loops, it is enough to claim that there is a bounded graph-trace τ on E N × E. Define τ : (E N × E) ⁰ → [0, ∞) by

τ (p (n,i) ) = 1

λ ⁿ⁻¹ ξ i , n ≥ 1, i ≥ 1.

Then the sum P

(n,i) τ (p _(n,i) ) = P

n ( P

i 1

λ

ⁿ⁻¹

ξ i ) = P

n≥1 1 λ

ⁿ⁻¹

· P

i≥1 ξ i con- verges. Also Aξ = λξ (hence, ξ i = ¹ _λ P

j a ij ξ j ) implies that for each i ∈ E ⁰ , τ (p _(n,i) ) = 1

λ ⁿ⁻¹ ξ i = 1 λ ⁿ

X

j

a ij ξ j = X

j

a ij 1 λ ⁿ ξ j

= X

j

a ij τ (p (n+1,j) ) = X

{(n,e)|s(n,e)=(n,i)}

τ (p r(n,e) ).

¤ Example 5.2. C ^∗ (E) ^γ is not stable if E is an irreducible infinite graph as below:

• • • • •

ÁÁÂÂ "" ¿¿

^^ "" ¿¿

^^ "" ¿¿

^^ "" ¿¿

^^ v 1 v 2 v 3 v 4 v 5 · · ·

E :

The vertex matrix 

 

 

 

2 2 0 0 · · · 1 0 2 0 · · · 0 1 0 2 · · · 0 0 1 0 · · · .. . .. . .. . .. . . ..



 

 

 

has an eigenvector ξ = ( ¹ ₂ , ₂ ¹

2

, ₂ ¹

3

, ₂ ¹

4

, . . . ) with λ = 3. Thus C ^∗ (E N × E) is not stable by Theorem 5.1 and so C ^∗ (E) ^γ is not stable by Theorem 4.2.

Example 5.3. It is known in [27] that for a pair of positive real numbers 1 < p ≤ q, there exists an irreducible infinite graph E p,q with

h l (E p,q ) = log p and h b (E p,q ) = log q.

The following graph E := E 2,8 satisfies h l (E) = log 2 and h b (E) = log 8. There are 8 edges from the vertex v _n to the vertex v _n+1 for each n ≥ 0 (Example 3.3 of [14]).

•

•

•

• • • • • • • • •

• • •

;; yy yy yy oo oo oo oo oo oo oo oo OO OO OO

// // // //

v 0 v 1 v 2 v 3

8 8 8

v 01

v 02

v 10 v 11 v 12 v 30 v 31 · · ·

· · ·

E :

Note that if a vector ξ = (ξ v ) _v∈E

⁰

satisfies X

w∈E

⁰

a vw ξ w = 2ξ v , v ∈ E ⁰ , (2)

ξ is an eigenvector of the vertex matrix A such that Aξ = 2ξ. Let ξ = (ξ v ) _v∈E

⁰

be the vector with ξ v > 0 as follows:

•

•

•

• • • • • • • • •

• • •

;; yy yy yy oo oo oo oo oo oo oo oo OO OO OO

// // // //

8 8 8

1 2

1 2

²

1 2

³

1

2

⁴

1 2

⁵

1

2

⁶

1 2

⁷

1

2

³

1

2

⁶

1

2

⁹

1

· · ·

· · · E :

Then (2) can be shown at every vertex v, and by Theorem 5.1 with λ = 2, C ^∗ (E N × E) admits a bounded trace, hence is not stable. Hence C ^∗ (E) ^γ is not stable by Theorem 4.2 again.

Now we consider an AF subalgebra C ^∗ (E) ^γ _v of C ^∗ (E) ^γ for each v ∈ E ⁰ , C ^∗ (E) ^γ _v = span{s α s ^∗ _β ∈ C ^∗ (E) ^γ | r(α) = r(β) = v}.

The following example shows that C ^∗ (E) ^γ _u and C ^∗ (E) ^γ _v may not be isomorphic if u 6= v. The AF algebras C ^∗ (E) ^γ and C ^∗ (E) ^γ _v were denoted A E and A E (v), respectively, in [13, 15].

Example 5.4. Consider the following irreducible finite graph E:

• ÂÂ •

__

<<u v E :

e f

g

Let C ^∗ (E) be generated by a Cuntz-Krieger E-family {p u , p v , s e , s f , s g }. Then C ^∗ (E) ^γ _u = C ^∗ (E) ^γ and C ^∗ (E) ^γ _u  C ^∗ (E) ^γ _v .

In fact, if s α e ^∗ _β ∈ C ^∗ (E) ^γ _v , namely r(α) = r(β) = v (|α| = |β|), then s α s ^∗ _β = s α p v s ^∗ _β = s α (s g s ^∗ _g )s ^∗ _β ∈ C ^∗ (E) ^γ _u . Thus C ^∗ (E) ^γ _v ⊂ C ^∗ (E) ^γ _u and hence C ^∗ (E) ^γ _u = C ^∗ (E) ^γ . On the other hand, C ^∗ (E) ^γ _v has an approximate identity consisting of projections q n , where q n := p v + P _n

k=0 s _e

k

f s ^∗ _e

k

f . Since k1 − q n k = k(p u + p v ) − q n k = ks _e

ⁿ⁺¹

s ^∗ _e

n+1

k = 1,

it follows that C ^∗ (E) ^γ _v is nonunital while C ^∗ (E) ^γ _u is unital with unit p u + p v . Thus C ^∗ (E) ^γ _u is not isomorphic to C ^∗ (E) ^γ _v .

Theorem 5.5. Let E be a locally finite irreducible infinite graph and v ∈ E ⁰ .

Then C ^∗ (E) ^γ _v admits a nonzero bounded trace. In particular, C ^∗ (E) ^γ _v is not

stable.

Proof. For each n ≥ 0, put

C ^∗ (E) ^γ _v,n := span{s α s ^∗ _β ∈ C ^∗ (E) ^γ _v | |α| = |β| ≤ n}.

Then {C ^∗ (E) ^γ _v,n } n≥0 is an increasing sequence of finite dimensional C ^∗ -subalge- bras of C ^∗ (E) ^γ such that

C ^∗ (E) ^γ _v = ∪ ^∞ _n=0 C ^∗ (E) ^γ v,n . Since E is irreducible, the elements in the set

ω(v, n) := {s α s ^∗ _β ∈ C ^∗ (E) ^γ _v | |α| = |β| ≤ n}

are linearly independent by [14, Lemma 3.7], a linear map on C ^∗ (E) ^γ _v,n is determined by its values on s α s ^∗ _β ∈ ω(n, v). We define linear functionals

τ n : C ^∗ (E) ^γ _v,n → C, n ≥ 0

as follows. Let τ 0 (p v ) = ¹ ₂ . For n ≥ 1, define τ n : C ^∗ (E) ^γ _v,n → C by

τ n (s α s ^∗ _β ) =

 



1/2, if α = β = v,

1

N

k

2

^k+1

, if α = β ∈ E ^k , 1 ≤ k ≤ n, 0, otherwise,

where N k := |{α ∈ E ^k | r(α) = v}|. Extend τ n to a linear map on C ^∗ (E) ^γ _v,n . Then τ n | _C

^∗

_(E)

^γ

v,n−1

= τ n−1 is obvious. Now let τ : ∪ ^∞ _n=0 C ^∗ (E) ^γ _v,n → C

be the linear map given by τ (x) = τ n (x) if x ∈ C ^∗ (E) ^γ _v,n . To see that τ is a trace, it suffices to show τ ((s µ s ^∗ _ν )(s α s ^∗ _β )) = τ ((s α s ^∗ _β )(s µ s ^∗ _ν )) for s α s ^∗ _β , s µ s ^∗ _ν ∈ ω(n, v). But

τ ((s α s ^∗ _β )(s µ s ^∗ _ν )) =

 



1

N

k

2

^k+1

, if β = µδ and α = νδ ∈ E ^k ,

1

N

k

2

^k+1

, if µ = βδ and ν = αδ ∈ E ^k , 0, otherwise,

which implies that

τ ((s µ s ^∗ _ν )(s α s ^∗ _β )) = τ ((s α s ^∗ _β )(s µ s ^∗ _ν )).

Now we show that τ (X ^∗ X) ≥ 0 for any X ∈ span(ω(v, n)) = C ^∗ (E) ^γ _v,n . For this, choose s α s ^∗ _β with the smallest length |α| among the terms appearing in the expression of X. Then decompose X = Y + Z in a way that the terms in Y are of the form λ s αµ s ^∗ _βν (λ ∈ C and r(αµ) = r(βν) = v, hence µ, ν must be loops at v whenever |µ| = |ν| ≥ 1) and Z is the sum of the remainders. Then τ (X ^∗ X) = τ (Y ^∗ Y ) + τ (Z ^∗ Z). If we show τ (Y ^∗ Y ) ≥ 0, the same argument can be applied (to Z ^∗ Z) repeatedly to prove τ (X ^∗ X) ≥ 0 since X has only finite terms. Moreover, by decomposing Y if needed, it is enough to consider Y of the form

Y = λ 0 s α s ^∗ _β + λ 1 s αµ

1

s ^∗ _βν

₁

+ · · · + λ k s αµ

1

···µ

k

s ^∗ _βν

₁

_···ν

_k

(3)