ABELIAN CORE OF GRAPH ALGEBRAS
GABRIEL NAGY AND SARAH REZNIKOFF
Abstract. We introduce a certain natural abelian C*-subalgebra of a graph
C*-algebra, which has functorial properties that can be used for characterizing
injectivity of representations of the ambient C*-algebra. In particular, a short
proof of the Cuntz-Krieger Uniqueness Theorem, for graphs that may have
loops without entries, is given.
Graph C*-algebras are interesting objects with a structure that can often be
understood in terms of the combinatorics and geometry of the underlying graph.
A large class of examples include AF-algebras, Cuntz-Krieger algebras, and C*algebras that are built up from matrices over C(T).
Here we consider C*-algebras defined from countable directed graphs. In particular, such a graph is determined by a four-tuple E = (E 0 , E 1 , r, s) consisting of its
vertex set, edge set, and range and source maps. Given such a graph E we define a
Cuntz-Krieger E-system on a Hilbert space H to be a collection {Sµ | µ ∈ E 0 ∪ E 1 }
of operators on H, where Sµ is an orthogonal projection whenever µ ∈ E 0 and
a partial isometry whenever µ ∈ E 1 , and the usual Cuntz-Krieger conditions are
satisfied:
(CK 1) Se∗ Se = Ss(e) for every e ∈ E 1
X
(CK 2) Sv =
Se Se∗ whenever {e ∈ E 1 | r(e) = v} is nonempty and finite.
r(e)=v
In light of the restriction on the vertex v in condition (CK 2) we need also to require
that Se Se∗ ≤ Sr(e) for every e ∈ E 1 , and that the projections Se Se∗ are mutually
orthogonal. Let us call such a system nondegenerate if there is no vertex v ∈ E 0
with Pv = 0. Note that our convention for the initial and range spaces of the partial
isometries agrees with [R].
In the usual way, one can prove the existence of a universal C*-algebra generated
by a Cuntz-Krieger E system; we denote this algebra by C ∗ hEi. By a uniqueness
theorem one means a set of conditions on the graph E or on the map π guaranteeing
that a representation π : C ∗ hEi → A is injective. The first of the Cuntz-Krieger
uniqueness theorems was Coburn’s Theorem (see, e.g., [Mu]), which asserts that any
C*-algebra generated by a nonunitary partial isometry is isomorphic to the Toeplitz
Algebra T . Taking the graph E with E 0 = {v, w} and E 1 = {e, f }, where s(f ) = v
and s(e) = r(e) = r(f ) = w, one can verify that in any E-system the operator
Se + Sf is a nonunitary partial isometry, and thus C ∗ hEi = T . Ten years later, in
[C], Cuntz showed that when E consists of a single vertex and n edges (loops), any
algebra generated by an E-system is isomorphic to the Cuntz algebra On ([CK]).
Subsequently, Cuntz and Krieger proved that the graph with adjacency matrix A
1991 Mathematics Subject Classification. Primary 46L10; Secondary 46L30.
Key words and phrases. Graph algebra, abelian C*-algebra.
1
2
GABRIEL NAGY AND SARAH REZNIKOFF
gives rise to the C*-algebra OA . A number of people, including Pask, Laca, Fowler,
Raeburn, and others refined the theory in the 1990’s, culminating in the CuntzKrieger Uniqueness Theorem based on the following condition on the graph E:
(L) Every cycle has an entry.
The theorem states that if E is a countable row finite graph (for every vertex v,
the set r−1 ({v}) is finite) satisfying condition (L), and (P, S) is a nondegenerate
E-system in A, then the natural representation π(P,S) : C ∗ hEi → C ∗ (P, S) is injective (see, e.g. [KPR]).
The gauge invariant uniqueness theorem of an Huef and Raeburn ([aHR]) replaces the stringent condition on the graph with a condition on the C*-algebra.
Specifically, it states that if there is a continuous gauge action γ : T → Aut(A) that
intertwines with the gauge action on the universal Cuntz-Krieger algebra for the
same graph, then the representation is injective. Again, row-finiteness is assumed.
The statement of the Cuntz-Krieger Uniqueness Theorem was improved considerably by Szymański’s result of 2000, which replaced condition (L) with the
requirement that the spectrum of the image of any entry-less cycle contains the
circle ([S]). In 2005, Fang showed that the row-finiteness condition can be eliminated in both this and the gauge invariant theorems by introducing a condition on
the spectrum of the operators corresponding to the entry-less cycles of the graph
([F]). In this paper, we present a new proof of Szymański’s result, which we state
as follows:
Theorem
Let E be a countable directed graph and A a C*-algebra. For a *-homomorphism
φ : C ∗ (E) → A, the following are equivalent.
(i) φ is injective
(ii) φ(Sα Sα∗ ) 6= 0 for all α ∈ E ∗ , and T ⊂ σ(φ(Sα Sλ Sα∗ )) whenever λ is an
entry-less cycle and r(λ) = s(α) is the only vertex shared by α and λ.
Our proof involves a particular representation of the algebra C ∗ hEi on the `2 space
TE of all essentially aperiodic trails in the graph. This set consists of all (backwards)
infinite paths that are genuinely aperiodic, as well as those that are forced to be
periodic—those that “begin” in a cycle with no entry—along with all finite paths
that start at a source. The algebra C ∗ hEi acts on `2 (TE ) by composition (when
possible). In order to make the representation injective, we enhance the `2 space
with a copy of the integers.
There is a natural conditional expectation on this Hilbert space and a corresponding one on the universal algebra C ∗ hEi, which intertwine with the representation.
The image of the expectation on the universal algebra we call the abelian core of
C ∗ hEi and denote by M hEi. It turns out that M hEi is the commutant of the diagonal ∆hEi := {Sµ Sµ∗ | µ is a path in E}, and is thus a masa. In addition, we prove
that this subalgebra has a lifting property: a representation of C ∗ hEi is injective if
and only if it is injective on the abelian core. Thus we can add the third equivalent
statement to our main theorem:
(iii) The restriction of φ to M hEi is injective.
This subalgebra has other nice properties as well, which makes it a Cartan C*subalgebra, in the sense of Renault ([Re]).
ABELIAN CORE OF GRAPH ALGEBRAS
3
The paper is organized as follows. The first section is devoted to setting up
notation and reviewing background material. Specifically, we introduce CuntzKrieger E-systems via the notion of a presentation generator. In addition we extend
the Cuntz-Krieger relation (CK 2) on edges to a result on paths (1.1).
In Section 2 we describe the aperiodic representation on C ∗ E and discuss the
Gelfand spectrum of the diagonal; it turns out that the essentially aperiodic trails
naturally correspond to a dense subset of this spectrum. In the third section we
identify the abelian core and prove the existence of a conditional expectation onto
this subalgebra. In addition, we show how to break the composition πap ◦ EM into
“coordinate maps” Eτ indexed by the essentially aperiodic trails τ . These maps
form a jointly faithful collection of maps on C ∗ hEi; this observation provides a
critical step in the proof of our main theorem, which also appears in that section
of the paper.
Both authors wish to thank Iain Raeburn for valuable discussion and feedback
on this work.
1. Notations and Preliminaries
In this section we set up the notations and terminology, by adopting the conventions from [R].
Definitions. An oriented graph is a system E = (E 0 , E 1 , r, s), consisting of two
disjoint sets E 0 (whose elements are called vertices) and E 1 (whose elements are
called edges), together with two maps r, s : E 1 → E 0 , referred to as the range map
and the source map, respectively.
Given some integer n ≥ 0, a path of length n in E is either
(a) a vertex, if n = 0, or
(b) an edge, if n = 1, or
(c) an n-tuple µ = (e1 , . . . , en ) ∈ (E 1 )n , such that s(ej ) = r(ej+1 ), ∀ j =
1, . . . , n − 1, if n ≥ 2.
Given such a path µ, its length (that is, the non-negative integer n) is denoted
simply by |µ|. In case (a), when µ ∈ E 0 , we define the source and the range as
s(µ) = r(µ) = µ. In case (b), we use the source and the range from the definition.
Finally, in case (c), we define the source s(µ) = s(en ) and the range r(µ) = r(e1 ).
Being consistent with the notation already introduced, we denote by E n the set
of all paths of lengthSn in E, and we will understand that all sets E n , n ≥ 0 are
disjoint. The union n≥0 E n is denoted by E ∗ , and the source and range maps
will all be viewed as maps s, r : E ∗ → E 0 . In order to distinguish between these
maps and their restrictions to E 1 , we will denote the restrictions by r1 and s1 ,
respectively.
For a path π ∈ E ∗ , we denote by [π] the set of vertices visited by π. This
means that, if |π| = 0, we have [π] = {s(π)}, while in the case when |π| ≥ 1, say
π = (e1 , . . . , en ), we have [π] = {r(e1 ), s(e1 ), s(e2 ), . . . s(en )}.
A pair (µ, ν) ∈ E ∗ × E ∗ of paths is said to be composable, if s(µ) = r(ν). For
such a pair, we denote by µ ◦ ν the concatenated path, so:
• if |µ| = 0, we have µ ◦ ν = ν;
• if |ν| = 0, we have µ ◦ ν = µ;
4
GABRIEL NAGY AND SARAH REZNIKOFF
• if µ = (e1 , . . . , en ) and ν = (f1 , . . . , fm ), with m, n ≥ 1, then µ ◦ ν =
(e1 , . . . , en , f1 , . . . , fm ).
In any event, we have |µ ◦ ν| = |µ| + |ν|.
Given two paths µ and ν, we write µ ≺ ν if there exists a path α, such that
(µ, α) is composable and µ ◦ α = ν. The path α is, of course, unique, and it will be
denoted from now on as ν µ.
Given some field K and some K-algebra A, a presentation of E in A is a map
E ∗ 3 µ 7−→ Sµ ∈ R, which satisfies the relations:
Sµ◦ν , if (µ, ν) is composable
(1)
Sµ Sν =
0, otherwise
The universal K-algebra generated by symbols {Xµ : µ ∈ E ∗ }, subject to (1), is
called the path K-algebra of E, and is denoted by K[E]. It is not hard to see that
to give a presentation of E in A is equivalent to giving an algebra homomorphism
ρ : K[E] → A (by defining Sµ = ρ(Xµ ), µ ∈ E ∗ ). Furthermore, the path algebra
can be described solely in terms of the system Ξ = (Xµ )µ∈E 0 ∪E 1 , as the universal
algebra generated by Ξ, subject to the relations:
Xv , if v = w
0
(2)
if v, w ∈ E : Xv Xw =
0, otherwise
Xe , if v = r(e)
* Xv Xe =
0, otherwise
(3)
if v ∈ E 0 , e ∈ E 1 :
Xe , if v = s(e)
Xe Xv =
0, otherwise
(Given these relations, for a path µ = (e1 , . . . , en ) with n ≥ 2, one is forced to define
Xµ = Xe1 · · · Xen .) Thus, in order to give a presentation of E in some algebra A,
all we need is a system Σ = (Sµ )µ∈E 0 ∪E 1 ⊂ A, satisfying (2) and (3). Such a system
is referred to as a presentation generator.
Convention. All graphs considered in this paper are assumed to be countable, meaning that both E 0 and E 1 are countable.
Definition. Assume E is as above, and A is a C*-algebra. A Cuntz-Krieger Esystem in A is a presentation generator Σ = (Sµ )µ∈E 0 ∪E 1 of E in A, which in
addition to (2) and (3), satisfies the following relations:
(4) Sv∗ = Sv , ∀ v ∈ E 0 ;
(5) Se∗ Se = Ss(e) , ∀ e ∈ E 1 ;
(6) Se∗ Sf = 0, for any two distinct edges e, f ∈ E 1
(7) if v ∈ E 0 has r1−1 ({v}) non-empty and finite, then Sv =
X
Se Se∗ .
e∈r1−1 ({v})
(8) for all e ∈ E
1
, Se Se∗
≤ Sv , and e 6= f implies
Se Se∗
and Sf Sf∗ are orthogonal
In the case that r−1 ({v}) is empty we will call v a source, while we call v an
infinite receiver in the case that this set is infinite. For future convenience let us
denote by S and R, respectively, the set of sources and infinite receivers of E.
Convention. Since any Cuntz-Krieger system Σ ⊂ A is assumed to be a presentation
generator for E, we are going to extend it by defining all elements Sµ ∈ A, µ ∈ E ∗ ,
ABELIAN CORE OF GRAPH ALGEBRAS
5
which will automatically satisfy (1). So, at times, instead of simply indicating Σ
as a presentation generator (by specifying only the elements Sµ , µ ∈ E 0 ∪ E 1 ), we
will abuse the notation and write it as an extended system Σ = (Sµ )µ∈E ∗ , with the
understanding that, in addition to (2)-(7), the identities (1) are also satisfied.
The reader should be aware that, using the extra conditions (4), (5) and (6)—
hereafter referred to as the Cuntz-Krieger relations—all Cuntz-Krieger systems satisfy the following
(a) all elements of the form Sv , v ∈ E 0 are projections in A;
(b) one has Sµ∗ Sµ = Ss(µ) , ∀ µ ∈ E ∗ , so all Sµ are partial isometries; in
particular, all elements of the form Sµ Sµ∗ are also projections.
Comment. In the Graph Algebra literature (see for instance [?]) the notations are a
bit different: instead of using one uniform symbol Sµ , µ ∈ E 0 ∪ E 1 , for all elements
of Σ, the projections Sv , v ∈ E 0 are sometimes denoted using a different letter.
For instance, one writes Pv for Sv , if v ∈ E 0 , so the single symbol Σ is replaced by
(P, S).
The natural way to describe a “path version” of the Cuntz-Krieger relation (7)
is as follows.
1.1. Proposition. Let v ∈ E 0 and k ∈ N. If there are finitely many paths µ ∈ E
with r(µ) = v and |µ| ≤ k, then
X
X
Sv =
Sµ Sµ∗ +
Sµ Sµ∗ .
r(µ)=v
|µ|=k
r(µ)=v
|µ|<k and s(µ)∈S
Proof. We prove this by induction on k. The statement is trivial for k = 0. Assume
it holds for k = M ≥ 0 and that the hypothesis on v and k = M + 1 holds. Then
X
X
X
X
X
Sv =
Sµ Sµ∗ +
Sµ Sµ∗ =
Sµ Se Se∗ Sµ +
Sµ Sµ∗
r(µ)=v
|µ|=k and s(µ)∈S
/
=
X
r(µ)=v
|µ|≤k and s(µ)∈S
Sµ Sµ∗ +
r(µ)=v
|µ|=k+1
X
µ∈E r(e)=s(µ)
r(µ)=v
r(µ)=v
|µ|≤k and s(µ)∈S
Sµ Sµ∗
r(µ)=v
|µ|≤k and s(µ)∈S
For future reference, we record a set of important properties in the following.
1.2. Proposition. Given a Cuntz-Krieger E-system Σ = (Sµ )µ∈E ∗ in some C*algebra A, the collection
G(Σ) = {Sµ Sν∗ : µ, ν ∈ E ∗ , s(µ) = s(ν)} ⊂ C ∗ (Σ)
is a involutive semigroup under multiplication, which is total in C ∗ (Σ), i.e. span G(Σ)
is norm-dense ∗-subalgebra in C ∗ (Σ). Specifically:
(i) given paths µ, ν ∈ E ∗ , with s(µ) = s(ν), we have (Sµ Sν∗ )∗ = Sν Sµ∗ ;
(ii) given four paths α, β, µ, ν ∈ E ∗ with s(α) = s(β) and s(µ) = s(ν), one has
the equality

∗
 Sα◦(µβ) Sν , if β ≺ µ
∗
∗
∗
S S
, if µ ≺ β
(9)
(Sα Sβ )(Sµ Sν ) =
 α ν◦(βµ)
0, otherwise
6
GABRIEL NAGY AND SARAH REZNIKOFF
(Here, as well as in rest of the paper, for a subset Σ ⊂ A, the symbol C ∗ (Σ)
designates the C*-subalgebra of A generated by Σ.)
The collection G(Σ), defined above, will be referred to as the standard CuntzKrieger generator set associated with Σ.
Definition. Given a graph E as above, one defines the universal C*-algebra C ∗ hEi
generated by the universal Cuntz-Krieger E-system Σuniv
= (Sµ )µ∈E 0 ∪E 1 , as follows.
E
Consider first the free ∗-algebra U generated by the symbols (sµ )µ∈E 0 ∪E 1 , and define
for any Cuntz-Krieger E-system Σ = (Sµ )µ∈E 0 ∪E 1 in some C*-algebra A, the unique
∗-homomorphism ωΣ : U → A, given by ωΣ (sµ ) = Sµ . Using ωΣ one then defines a
C*-seminorm pΣ on U, by pΣ (x) = kωΣ (x)k, and the universal C*-seminorm puniv
on U, by
puniv (x) = sup pΣ (x) : Σ Cuntz-Krieger E-system .
Secondly one defines C ∗ hEi as the C*-algebra obtained by separate-completion,
that is, the completion of the algebra U/N in the C*-norm k . kuniv defined by puniv ,
where N = {x ∈ U : puniv (x) = 0}. If we denote by θ : U → C ∗ hEi the quotient
is given by Sµ = θ(sµ ). If we
map, then the universal Cuntz-Krieger E-system Σuniv
E
specialize Proposition 1.2 to C ∗ hEi, we obtain the standard (universal) generator
set G(Σuniv
)—hereafter denoted simply by GE —for C ∗ (Σuniv
)(= C ∗ hEi).
E
E
By construction, the C*-algebra C ∗ hEi—hereafter referred to as the graph C*algebra of E—has the following universal property:
(u) for any Cuntz-Krieger E-system Σ = (Sµ )µ∈E ∗ in a C*-algebra A, there
exists a unique ∗-homomorphism πΣ : C ∗ hEi → A, such that πΣ (Sµ ) = Sµ ,
∀ µ ∈ E ∗.
Of course, the representation πΣ has Range πΣ = C ∗ (Σ), and it maps GE onto
G(Σ).
1.3. Remark. One can show that, in the graph algebra C ∗ hEi, whose norm will be
from now on denoted without the subscript, all projections Sv , v ∈ E 0 , thus all
partial isometries Sµ , µ ∈ E ∗ and the projections Sµ S∗µ , hereafter denoted by Rµ ,
are non-zero (thus they all have norm 1). This statement will be in fact proved
a little later, by exhibiting a certain particular representation of C ∗ hEi with the
above properties.
Definition. Using the universal property, it follows that for any z ∈ T (the
unit circle), there exists a unique automorphism γz ∈ Aut(C ∗ hEi), such that
γz (Sµ ) = z |µ| Sµ , ∀ µ ∈ E ∗ . One can in fact easily show that the correspondence
T 3 z 7−→ γz ∈ Aut(C ∗ hEi) defines a continuous (in the point-norm topology)
group of automorphisms, which is referred to as the gauge action.
With this terminology, one has the following fundamental result, which was
proved in the row-finite case by an Huef and Raeburn ([aHR]) and extended to the
general situation by Fang ([F]).
1.4. Theorem (an Huef, Fang). If Σ = (Sµ )µ∈E 0 ∪E 1 is a Cuntz-Krieger E-system
in some C*-algebra A, which admits a continuous group action T 3 z 7−→ αz ∈
Aut(A), such that αz (Sµ ) = z |µ| Sµ , ∀ µ ∈ E 0 ∪ E 1 , z ∈ T, then the following
conditions are equivalent:
(i) the associated representation πΣ : C ∗ hEi → A is injective;
(ii) Sv 6= 0, ∀ v ∈ E 0 .
ABELIAN CORE OF GRAPH ALGEBRAS
7
Thus, if the hypothesis and condition (ii) are satisfied, πΣ establishes a ∗-isomorphism
C ∗ hEi ' C ∗ (Σ).
2. The Diagonal and the Aperiodic Representation
Definitions. Assume E is a countable, and Σ = (Sµ )µ∈E ∗ is a Cuntz-Krieger Esystem (in some C*-algebra A). Consider the subset
G∆ (Σ) = {Sµ Sµ∗ : µ ∈ E ∗ }.
By Proposition 1.2 it is obvious that G∆ (Σ) is a commuting set of projections in
C ∗ (Σ), which is referred to as the standard diagonal generator set.
The C*-subalgebra C ∗ (G∆ (Σ)) ⊂ A, hereafter denoted simply by ∆(Σ), is referred to as the diagonal algebra associated with Σ.
When we specialize to the case of the graph algebra C ∗ hEi, the diagonal generator
set will be denoted simply by G∆
E and the C*-subalgebra generated by it will be
∗
∆
) = G∆
denoted simply by ∆hEi; i.e., G∆ (Σuniv
E and C (GE ) = ∆hEi.
E
2.1. Remark. The diagonal algebras ∆(Σ) and their generator sets G∆ (Σ) are functorial, in the following sense. Whenever Σ = (Sµ )µ∈E ∗ is a Cuntz-Krieger E-system
in some C*-algebra A, and if π : A → B is some ∗-homomorphism (into a C*algebra B), then π(Σ) = (π(Sµ ))µ∈E ∗ is a Cuntz-Krieger E-system in B, and π
maps G∆ (Σ) onto G∆ (π(Σ)) and ∆(Σ) onto ∆(π(Σ)).
In particular, when we specialize to the case when C ∗ hEi, and we start with
a Cuntz-Krieger E-system Σ in some C*-algebra A, the associated representation
∆
πΣ : C ∗ (E) → A maps G∆
E onto G (Σ), thus it maps the (universal) diagonal ∆hEi
onto ∆(Σ).
Comment. The diagonal ∆(Σ) associated with a Cuntz-Krieger E-system Σ (in
some C*-algebra A) is a commutative AF-algebra. Thus it has totally disconnected
Gelfand spectrum. Note that for any two projections P, Q ∈ G∆ (Σ), one either has
P Q = 0 or P and Q are comparable (i.e. either P ≥ Q or Q ≥ P ); in particular,
P = Sµ∗ Sµ and Q = Sν∗ Sν are comparable if and only if µ ≺ ν or ν ≺ µ. Thus
any infinite totally ordered subset of G∆ (Σ) is of the form {τ(n) | n ∈ N∗ } for some
infinite path τ .
Thus each point in the spectrum corresponds either to a maximal totally ordered
subset of G∆ (Σ) or to a set of the form {ν ∈ E | ν ≺ µ} where µ ∈ E is a finite
path such that s(µ) r−1 (s(µ)) is infinite. (In the latter case, we say that s(µ) is an
infinite receiver.) In either case, the corresponding element of the spectrum is the
characteristic function of the set.
2.2. Remark. If we specialize to the universal diagonal ∆hEi, whose standard gen∗
erator set is G∆
E = {Rα : α ∈ E }, is pretty obvious that, when we think of
∆hEi as C0 (Ω), where Ω is the Gelfand spectrum, then for every positive element
X ∈ ∆hEi r {0}, there exists some scalar t > 0 and some α ∈ E ∗ , such that
X ≥ tRα . Therefore, for a ∗-homomorphism Φ : ∆hEi → A, the following conditions are equivalent:
• Φ is injective;
• Φ maps all generators in G∆
E to non-zero elements (projections) in A.
In turn, since each generator Rα = Sα S∗α is Murray-von Neumann equivalent to
S∗α Sα = Ss(α) , it follows that, for a ∗-homomorphism Φ : C ∗ hEi → A, the condition
8
GABRIEL NAGY AND SARAH REZNIKOFF
that Φ∆hEi is injective is equivalent to the condition that
(10)
Φ(Sv ) 6= 0, ∀ v ∈ E 0 ,
If we use Cuntz-Krieger systems, this is of course equivalent to condition (ii) from
Theorem 1.4.
In order to better understand the points in the Gelfand spectrum of the (universal) diagonal, it will be helpful to introduce the following terminology.
Definition. Given a countable graph E, a trail in E is either
(i) a finite path τ (possibly of length zero) whose start s(τ ) has no entries, i.e.
r−1 (s(τ )) = ∅, or
(ii) an infinite backward path, i.e. an infinite sequence τ = (e1 , e2 , . . . ) of edges,
such that s(en ) = r(en+1 ), ∀ n ∈ N.
Given a trail τ and some integer n ≥ 0, we define the tail of length n to be the
finite path

r(τ ), if n = 0 and τ is finite



r1 (e1 ), if n = 0 and τ is infinite
τ(n) =
(e1 , . . . , en ), if n > 0 and τ is either infinite, or finite with |τ | > n



τ , if n > 0 and τ is finite, with |τ | ≤ n
Of course, all tails of τ have the same range, i.e. r(τ(n) ) = r(τ(0) ), and this special
vertex will be denoted simply by r(τ ), and will be referred to as the range of τ .
Given a trail τ , and a path µ ∈ E ∗ , we write µ ≺ τ , if µ = τ(n) for some n ≥ 0.
It is pretty obvious that, if µ ≺ τ , then removing µ from τ still yields a trail, which
we denote by τ µ.
Likewise, if we have trail τ and a path µ ∈ E ∗ with s(µ) = r(τ ), the obvious
concatenation µ ◦ τ is again a trail.
With this terminology, it is quite obvious that, if we work with the universal
diagonal generator set G∆
E (whose elements are all non-zero projections), the maximal totally ordered subsets of G∆
E are in one-to-one correspondence with the trails
in E. Specifically, every maximal totally ordered subset P ⊂ G∆
E can be uniquely
presented as Pτ = {Rτ(n) : n ≥ 0} (with possible repetitions, of course, allowed)
for some trail τ .
In what follows we are going to exhibit a dense subset in the Gelfand spectrum
of the universal diagonal G∆
E , by selecting a set of special trails, for which we need
a bit more terminology.
Definitions. Assume as above, E is a countable graph.
A. A loop in E is a finite path µ = (e1 , . . . , en ), of length n ≥ 1, with r(µ) =
s(µ). If the set [µ] of vertices visited by µ has exactly n elements, we call
µ simple.
B. A loop µ, as above, is said to entry-less, if for every v ∈ [µ], there exists
exactly one edge e ∈ E 1 with r1 (e) = v. (If, for instance, v = r1 (ej ), then
this unique edge is e = ej .) Remark that every entry-less loop µ can be
uniquely written as
µ = ν| ◦ . {z
. . . . . ν},
k times
for some integer k ≥ 1 and some simple entry-less loop ν, which has length
equal to the cardinality of the set [µ].
ABELIAN CORE OF GRAPH ALGEBRAS
9
C. An infinite trail τ = (e1 , e2 , . . . ) in E is said to be periodic, if there exist
integers j, k ≥ 1, such that
(11)
en+k = en , ∀ n ≥ j.
It is obvious that this forces the path ρ = (ej , . . . , ej+k−1 ) to be a loop. If
we choose j and k with smallest possible value of j + k, so that (11) holds,
and if we consider the paths α = (e1 , . . . , ej−1 ) and λ = (ej , . . . , ej+k−1 ),
we will call the pair (α, λ) the seed of τ . Of course, α may have length zero.
In any case, λ is a loop, which will be called the period of τ .
D. A trail τ is said to be essentially aperiodic, if either
(i) τ is finite, or
(ii) τ is periodic, but its period is an entry-less loop (which automatically
has to be simple), or
(iii) τ is not periodic.
The trails of type (i) and (ii) will be called discrete, while the ones of
type (iii) will be called continuous. A discrete essentially aperiodic trail
τ is completely determined by a certain path, which we denote by τ(ess) ,
hereafter referred to as the essential tail of τ , defined accordingly as follows:
(i) if τ is finite, we let τ(ess) = τ ;
(ii) if τ is periodic, with seed (α, λ), we let τ(ess) = α.
2.3. Lemma. For every vertex v ∈ E 0 , there exists at least one essentially aperiodic
trail τ with v = r(τ ).
Proof. Call a vertex w ∈ E 0 a trap, if either
(a) r1−1 (w) = ∅, i.e. w is a source, or
(b) w is a vertex visited by an entry-less loop, or
(c) there exist two loops λ = (e1 , . . . , em ) and µ = (f1 . . . . , fn ) with s(λ) =
r(λ) = s(µ) = r(µ) = w, such that e1 6= f1 .
The Lemma will be proved according to the following two cases (with the second
case split into three sub-cases)
(i) no path α ∈ E ∗ with r(α) = v has s(α) a trap;
(ii) there exists a path α ∈ E ∗ with r(α) = v which has s(α) a trap.
∗
In case (i) we construct a sequence (αn )∞
n=1 ⊂ E , with r(α1 ) = v, such that
(12)
(13)
s(αn−1 ) = r(αn ),
[αn ] 6⊂ [α1 ◦ · · · ◦ αn−1 ]
for all n ≥ 2. We start off with an arbitrary length one path α1 with s(α1 ) = v.
Assume now the paths α1 , . . . , αN have been constructed, so that (12) and (13) hold
for all n ≤ N , and let us indicate how αN +1 is constructed. Assume α1 ◦ · · · ◦ αN =
(e1 , . . . , ek ). Since the vertex s(ek ) is not a trap, there exists at least one edge,
which we denote by ek+1 , such that r(ek+1 ) = s(ek ). If s(ek+1 ) 6∈ [α1 ◦ · · · ◦ αN ],
then we let αN +1 be the length one path ek+1 , and we are done. Assume now that
s(ek+1 ) ∈ [α1 ◦ · · · ◦ αN ], which means that we have a loop λ, which is of the form
λ = (ek+1 , ep , ep+1 , . . . , ek ), for some p ∈ {1, 2, . . . , k + 1}. (If p = k + 1, the loop
λ consists of one edge ek+1 .) By assumption (no point in [λ] is a trap), the loop
λ must have an entry, which we denote by ek+2 , which means that there exists
j ∈ {p, p + 1, . . . , k, k + 1} with r(ek+2 ) = r(ej ), but ek+2 6= ej . Notice now that
s(ek+2 ) does not belong to [α1 ◦ · · · ◦ αN ]. Indeed, if the vertex w = r(ek+2 ) were
10
GABRIEL NAGY AND SARAH REZNIKOFF
in [α1 ◦ · · · ◦ αN ], then ek+2 would be part of a loop µ = (ek+2 , eq , . . . ) which starts
and ends at w, where another loop ν = (ej , ej+1 , . . . , ek+1 , ep , ep+1 , . . . , ej−1 ) starts
and ends, satisfying condition (c) and thus contradicting the fact that the vertex
w is not a trap. Since s(ek+2 ) 6∈ [α1 ◦ · · · ◦ αN ], we can construct the next path
as αN +1 = (ek+1 , ep , ep+1 , . . . , ej−1 , ek+2 ), so the set [αN +1 ] contains the vertex
s(ek+2 ), which is not in [α1 ◦ · · · ◦ αN ]. Having constructed the (infinite) sequence
(αn )∞
n=1 , satisfying (12) and (13), we can now form the infinite trail τ = α1 ◦α2 ◦. . . ,
which clearly visits infinitely many vertices, thus it cannot be periodic.
Having proved the Lemma in case (i), let us analyze now case (ii), by which we
assume we have a path α ∈ E ∗ , with r(α) = v and s(α) a trap, and let us consider
the three possibilities for s(α).
(a) If s(α) is a source, we can simply take τ = α.
(b) If s(α) sits on an entry-less loop λ, which starts and ends at s(α), then we
can take τ to be the periodic trail with seed (α, λ).
(c) Assume now we have two loops λ = (e1 , . . . , em ) and µ = (f1 , . . . , fn )
which both start and end at s(α), but such that e1 6= f1 . By replacing λ
with λ
· · ◦ λ} and µ with µ ◦ · · · ◦ µ, we can assume that λ and µ have
| ◦ ·{z
| {z }
n times
m times
equal lengths. The trail τ is now constructed as an infinite concatenation
τ = α ◦ α1 ◦ α2 ◦ . . . , where
αn = λ ◦ (µ ◦ · · · ◦ µ).
| {z }
n times
It is pretty obvious that τ is not periodic. (Its period is not k, since αk is
a subpath.)
2.4. Proposition-Definition. Assume, as before, E is a countable graph, and let
TE denote the set of all essentially aperiodic trails in E. If we consider the standard
orthonormal basis (ξτn )τ ∈TE for `2 (TE × Z), there exists a unique Cuntz-Krieger
n∈Z
E-system Σ = (Sα )α∈E ∗ ⊂ B `2 (TE × Z) , satisfying:
 n+|α|
 ξα◦τ , if r(τ ) = s(α)
(14)
Sα ξτn =

0, otherwise
Furthermore, the associated ∗-representation πΣ : C ∗ hEi → B `2 (TE × Z) is injective.
The above representation will be referred to as the (twisted essentially) aperiodic
representation, and will be denoted by πap .
Proof. Fix for the moment α ∈ E ∗ , consider the given orthonormal basis as a set
M = {ξτn : τ ∈ TE , n ∈ Z}, and let Sα0 : M → M ∪ {0} be the map defined
by (14). It should be pointed out here that, if τ is an essentially aperiodic trail,
and α ∈ E ∗ has s(α) = r(τ ), then clearly α ◦ τ is an essentially aperiodic trail. Of
course, when we consider the subset
M0α = {ξτn ∈ M : τ ∈ TE , n ∈ Z, s(α) 6= r(τ )},
by construction we know that Sα0 M0 is constantly zero, while Sα0 MrM0 is inα
α
jective, so clearly Sα0 extends to a partial isometry Sα ∈ B `2 (TE × Z) with
ABELIAN CORE OF GRAPH ALGEBRAS
11
Ker Sα = span M0α . An easy calculation shows that, if we start with a vertex
v ∈ E 0 (a path in E ∗ of length zero), then Sv ξ = ξ, ∀ ξ ∈ M r M0v , so Sv is a
projection, with
Range Sv = span (M r M0v ) = span {ξτn : τ ∈ TE , n ∈ Z, r(τ ) = v},
so in particular the first Cuntz-Krieger relation (4) is satisfied.
Likewise, if we start with some arbitrary path α of positive length, then Sα is a
partial isometry, with
(15)
Range Sα = span {ξτn : τ ∈ TE , n ∈ Z, α ≺ τ }.
The fact that the system Σ = (Sα )α∈E ∗ is a presentation of E in B `2 (TE × Z) ,
i.e. the relations (1) are satisfied, is pretty obvious.
To check the remaining three Cuntz-Krieger relations (5)-(7), we simply notice
that, for an edge e ∈ E ∗ , the adjoint of Se is simply given by
n−1
ξτ e , if e ≺ τ
∗
Se ξτ,n =
0, otherwise
and then the identities (5) and (6) are obvious. Finally, to check (7), we simply
notice that, if v ∈ E 0 has r1−1 (v) finite and non-empty, then
X
Sv −
Se Se∗ ξ = 0, ∀ ξ ∈ M,
e∈r1−1 (v)
P
which forces e∈r−1 (v) Se Se∗ = Sv .
1
To prove the injectivity of πΣ , we first notice that by Lemma 2.3 it follows
immediately that, for every v ∈ E 0 there exists at least one τ ∈ TE , such that
Sv ξτn = ξτn , ∀ n ∈ Z, and therefore Sv 6= 0. This means that, in order to obtain
injectivity, one may employ Theorem 1.4, which can be easily accomplished,
by
considering the unitary representation T 3 z 7−→ Uz ∈ B `2 (TE × Z) , given on
the orthonormal basis by
Uz ξτn = z n ξτn ,
which clearly satisfies Uz Sα Uz∗ = z |α| Sα , ∀ α ∈ E ∗ , z ∈ T.
2.5. Remark. The Cuntz-Krieger E-system (Sα )α∈E ∗ from Proposition-Definition 2.4
satisfies a slightly more general condition that generalizes the Cuntz-Krieger relation (7), namely
X
(16)
if v ∈ E 0 has r1−1 ({v}) 6= ∅, then: soSe Se∗ = Sv .
e∈r1−1 ({v})
The above equality can be checked immediately on the orthonormal basis, since for
any trail τ ∈ TE and any integer n, one has the equivalences
Sv ξτn 6= 0 ⇔ Sv ξτn = ξτn ⇔ r(τ ) = v,
and in the case when these conditions are satisfied, we have
X
Se Se∗ ξτn = Sτ(1) Sτ∗(1) ξτn = Sv ξτn .
e∈r1−1 ({v})
12
GABRIEL NAGY AND SARAH REZNIKOFF
Notation. For every essentially aperiodic trail τ ∈ TE , we denote by Qτ the orthogonal projection onto span {ξτn : n ∈ Z}. Given a path α ∈ E ∗ , we denote by Rα
the orthogonal projection onto the space
span {ξτn : τ ∈ TE , n ∈ Z, α ≺ τ }.
Note that, using the notations from Proposition-Definition 2.4, we have Rα = Sα Sα∗ .
2.6. Remark. When we restrict the aperiodic representation πap to the universal
diagonal ∆hEi, it is, of course, still injective. Furthermore, for every X ∈ ∆hEi and
every τ ∈ TE , there exists a unique scalar ετ (X) ∈ C, such that
Qτ πap (X) = πap (X)Qτ = ετ (X)Qτ .
Specifically, if X is one of the standard generators Rα (= Sα S∗α ), α ∈ E ∗ , for ∆hEi,
then πap (Rα ) = Rα , so
Qτ , if α ≺ τ
Qτ πap (Rα ) = πap (Rα )Qτ =
0, otherwise,
so τ (Rα ) = 1 if α ≺ τ and 0 otherwise. Of course, this means that we have a whole
collection (ετ )τ ∈TE of characters of the commutative C*-algebra ∆hEi. Furthermore, since clearly all projections (Qτ )τ ∈TE are mutually orthogonal, and satisfy
X
(17)
soQτ = I,
τ ∈TE
it follows that
(18)
πap (X) = so-
X
ετ (X)Qτ , ∀ X ∈ ∆hEi.
τ ∈TE
On the one hand, the above identity justifies the term “diagonal” used when
describing ∆hEi. On the other hand, Lemma 2.3 shows that the set {ετ }τ ∈TE is
dense in the Gelfand spectrum of ∆hEi.
2.7. Remark. Using the notations and assumptions as above, for any trail τ ∈ TE ,
one has
(19)
so- lim Rτ(n) = Qτ .
n→∞
Moreover, if τ is discrete, then the sequence (Rτ(n) )∞
n=0 becomes stationary, namely
we have
(20)
Rτ(n) = Rτ(ess) = Qτ , ∀ n ≥ |τ(ess) |,
so by the injectivity of πap the same holds in C ∗ hEi, that is,
(21)
Rτ(n) = Rτ(ess) , ∀ n ≥ |τ(ess) |,
3. The Abelian Core
3.1. Proposition-Definition. Assume, as before, E is a countable graph. Given
two paths α, β ∈ E ∗ , with s(α) = s(β) the standard generator Sα S∗β ∈ GE is a
normal element in C ∗ hEi, if and only if one of the following holds:
(a) α = β;
(b) β ≺ α and α β is an entry-less loop;
(c) α ≺ β and β α is an entry-less loop.
ABELIAN CORE OF GRAPH ALGEBRAS
13
Furthermore, if we denote by GM
E the set of all such (universally) normal generators,
∗
then the C*-subalgebra MhEi = C ∗ (GM
E ) ⊂ C hEi is abelian.
The C*-subalgebra MhEi is referred to as the abelian core of C ∗ hEi.
Proof. We first prove a lemma that will be useful in the sequel.
3.2. Lemma. A loop λ is entry-less if and only if Sλ S∗λ = Sr(λ) .
Proof. Let k = |λ|. It is easy to see that λ has an entry if and only if there is a
path µ 6= λ such that r(µ) = r(λ), and either
(i) |µ| < k and s(µ) a source, or
(ii) |µ| = k.
Since Sµ S∗µ 6= 0, the result follows from Proposition 1.1.
Now we prove the Proposition-Definition. Let X = Sα S∗β . Note that X ∗ X =
Sβ S∗β . If α = β then clearly X is normal. Suppose β ≺ α and λ := α β is an
entry-less loop. Then Sα = Sβ Sλ and by the lemma we have Sλ S∗λ = Ss(β) , so
XX ∗ = Sβ Sλ S∗λ S∗β = X ∗ X.
Conversely, assume that X = Sα S∗β is nonzero and normal. It follows from
X 6= 0 that s(α) = s(β) and from XX ∗ = X ∗ X that r(α) = r(β). We have
(X ∗ X)2 = X ∗ XXX ∗ nonzero and so Sα S∗β Sα S∗β = X 2 is nonzero, and thus either
β ≺ α or α ≺ β, by Prop. 1.2. Without loss of generality assume β ≺ α. These
facts together imply that λ := α β is a loop. Suppose λ has an entry. Then by
Lemma 3.2 Sλ S∗λ < Sr(λ) = Ss(β) , so
XX ∗ = Sβ Sλ S∗λ S∗β < Sβ S∗β = X ∗ X
In what follows, we are going to describe the geometry of the inclusion MhEi ⊂
C ∗ hEi using the inclusion GM
E ⊂ GE of the generator sets, combined with the
aperiodic representation. Let us recall first some standard terminology.
Definitions. Given a C*-algebra A and a C*-subalgebra B ⊂ A, a linear map
E : A → A is said to be a conditional expectation of A onto B, if
• E is positive, i.e. a ≥ 0 ⇒ E(a) ≥ 0;
• E is an idempotent (i.e. E 2 = E) with Range E = B;
• E(ba) = bE(a), ∀ a ∈ A, b ∈ B.
(Taking adjoints, the third condition also implies: E(ab) = E(a)b, ∀ a ∈ A, b ∈ B.)
Given such a conditional expectation E, we say that it is faithful, if whenever
a ≥ 0 satisfies E(a) = 0, it follows that a = 0.
Comment. According to Tomiyama’s Theorem ([T]), any idempotent linear continuous map E : A → A of norm 1, with Range E = B, is automatically a conditional
expectation of A onto B.
3.3. Example. Assume H is a Hilbert
P space, and Q = {Qi }i∈I ⊂ B(H) is a collection
of non-zero projections, with so- i∈I Qi = I. Then the map
X
E Q : B(H) 3 T 7−→ soQi T Qi ∈ B(H)
i∈I
is a faithful conditional expectation of B(H) onto the commutant
Q0 = {T ∈ B(H) : T Qi = Qi T, ∀ i ∈ I}.
14
GABRIEL NAGY AND SARAH REZNIKOFF
3.4. Remark. When
we specialize the above Example to the system Q = {Qτ }τ ∈TE ⊂
B `2 (TE × Z) , the above conditional expectation will be denoted by Eap .
On the one hand, if we consider the system R = {Rα }α∈E ∗ = {πap (Rα )}α∈E ∗ ,
then by Remark 2.6 we clearly have the inclusion R ⊂ Q0 . On the other hand, we
also have the inclusion
R0 ⊂ Q0 .
(22)
Indeed, if T commutes with all Rα ’s then, for every τ ∈ TE we have T Rτ(n) =
Rτ(n) T , ∀ n ≥ 0, so taking so-limits, by Remark 2.7 we get T Qτ = Qτ T .
Using the inclusion (22) it follows that
Eap (T ) = T, ∀ T ∈ R0 .
(23)
Of course, when we consider the abelian C*-subalgebra MhEi, which contains all
the projections Rα , α ∈ E ∗ (i.e. the generators of the diagonal), using the aperiodic
representation it follows that πap (MhEi) ⊂ R0 , so by (23) we immediately get
(24)
Eap πap (X) = πap (X), ∀ X ∈ MhEi.
3.5. Theorem. There exists a unique conditional expectation EM of C ∗ hEi onto
MhEi, such that
(25)
EM (X) = X, ∀ X ∈ GM
E ;
(26)
EM (X) = 0, ∀ X ∈ GE r GM
E .
Furthermore, if we consider the aperiodic representation πap : C ∗ hEi → B `2 (TE × Z)
and the conditional expectation Eap described above, then we have the identity
πap ◦ EM = Eap ◦ πap ,
(27)
so in particular EM is faithful.
Proof. Let us first prove that
(28)
Eap (πap (X)) = 0, ∀ X ∈ GE r GM
E
Start with an arbitrary generator X = Sα S∗β ∈ GE , defined by two paths α, β ∈ E ∗
with s(α) = s(β). Let τ ∈ T. Clearly, Qτ Sα Sβ∗ Qτ = 0 unless both β ≺ τ and
α ≺ τ . In this case, either α ≺ β or β ≺ α; without loss of generality, let us assume
the latter holds. Since α and β have the same source, it follows that α = β ◦ λ
for a loop λ. Now Qτ Sα Sβ∗ Qτ 6= 0 implies that β ◦ λ ◦ (τ β) = α ◦ (τ β) = τ ,
i.e., λ ◦ (τ β) = τ β, which implies that τ is periodic with period λ. Since τ is
aperiodic, it follows that λ is entryless. Thus we have shown that Eq. (26) holds.
We now see that if we also take into account (24), it follows that Eap maps
πap (GE ) into itself, so by continuity it maps πap (C ∗ hEi) into itself, and then using
the injectivity of πap , there exists a unique linear continuous map EM : C ∗ hEi →
C ∗ hEi, satisfying (27), which combined with the injectivity of πap and the identities
(28) and (24) yields
X, if X ∈ GM
E
EM (X) =
0, if X ∈ GE r GM
E
so clearly EM is a conditional expectation of C ∗ hEi onto MhEi.
Finally, the faithfulness of EM is immediate from the faithfulness of Eap , combined with (27) and the injectivity of πap .
Theorem 3.5 has the following important consequence.
ABELIAN CORE OF GRAPH ALGEBRAS
15
3.6. Theorem. The C*-subalgebra MhEi coincides with the commutant of ∆hEi in
C ∗ hEi, that is,
(29)
MhEi = {X ∈ C ∗ hEi : XD = DX, ∀ D ∈ ∆hEi}.
In particular, MhEi is a maximal abelian C*-subalgebra in C ∗ hEi.
Proof. Since we already have the inclusion ∆hEi ⊂ MhEi and MhEi is abelian, all
we have to do is to prove that the right-hand side of (29) is contained in MhEi.
Start with some element X in the commutant. Then, using the generators for ∆hEi
it follows that
πap (X)Rα = πap (XRα ) = πap (Rα X) = Rα πap (X), ∀ α ∈ E ∗ ,
so using the notations from Remark 3.4 it follows that πap (X) belongs to R0 , and
then by (23) combined with (27) we have
πap (X) = Eap πap (X) = πap EM (X) ,
which by the injectivity of πap forces X = EM (X) ∈ MhEi.
Comment. In what follows, we examine the structure of the conditional expectation
EM , by taking a closer look at its “coordinate” maps
(30)
Eτ : C ∗ hEi 3 X 7−→ Qτ πap (X)Qτ ∈ Qτ B `2 (TE × Z) Qτ , τ ∈ TE .
By construction, the maps Eτ can be also written as
(31)
Eτ (X) = Eap (Qτ πap (X)Qτ ) = Qτ πap EM (X) Qτ
and this allows us to represent the restriction of πap to the abelian core as
X
(32)
πap (X) = soEτ (X), ∀ X ∈ MhEi.
τ ∈TE
In particular, the restrictions of Eτ to MhEi, define ∗-homomorphisms, hereafter
denoted by
πτM : MhEi → Qτ B `2 (TE × Z) Qτ ,
which can also interpreted as representations on the Hilbert space Hτ = Range Qτ .
Of course, the right-hand side of (32) makes sense for arbitrary elements in
C ∗ hEi, but then the correct identity is:
X
(33)
πap EM (X) = soEτ (X), ∀ X ∈ C ∗ hEi.
τ ∈TE
In any event, the maps Eτ are completely positive, since they factor as Eτ =
πτM ◦ EM . Furthermore, the system (Eτ )τ ∈TE is jointly faithful, in the sense that
(jf) whenever X ∈ C ∗ hEi is a positive element, such that Eτ (X) = 0, ∀ τ ∈ TE ,
it follows that X = 0.
3.7. Remark. If τ ∈ TE is discrete, then for every X ∈ C ∗ hEi, we have the equalities
(34)
(35)
Rτ(ess) XRτ(ess) = EM (X)Rτ(ess) ,
πap (Rτ(ess) XRτ(ess) ) = Eτ (X).
To prove the second equality, note that in the discrete case we have Rτ(ess) = Qτ ,
and so
(36)
πap (Rτ(ess) XRτ(ess) ) = Rτ(ess) πap (X)Rτ(ess) = Qτ πap (X)Qτ = Eτ (X).
16
GABRIEL NAGY AND SARAH REZNIKOFF
In order to get (34) we apply the (injective) map πap , and observe that
(37)
πap (EM (X)Rτ(ess) ) =πap (EM (X))Qτ = Eap ◦ πap (X)Qτ = so−
X
Qγ πap (X)Qγ Qτ
γ
=Qτ πap (X)Qτ = Eτ (X).
In the case when τ is finite (which forces it to be discrete), we shall see that the
representation πτM has one-dimensional range, thus it is given by a character. We
treat this case, together with the continuous case, in the following result.
3.8. Proposition. If τ ∈ TE is either finite or continuous, then there exists a
unique state ωτ on C ∗ hEi, such that
(38)
Eτ (X) = ωτ (X)Qτ , ∀ X ∈ C ∗ hEi.
Furthermore,
(i) ωτ = ωτ ◦ EM ;
(ii) the restriction ε̃τ = ωτ MhEi is a character, which when further restricted
to the diagonal ∆hEi, coincides with the character ετ , defined in 2.6.
Proof. First note that by Eq. (31) Eτ EM (X) = Eτ , so to prove the first statement
we need only consider the case when X ∈ GM
E , i.e. X is of the form described
in Proposition-Definition 3.1; moreover, (i) will following immediately once the
existence of ωτ satisfying Eq. (38) is established.
On the one hand, if X = Sα S∗β , with α, β ∈ E ∗ with either α or β beginning
with an entry-less loop, then (because of the assumption on τ ) either
(a) α 6≺ τ , in which case it follows that S∗α Rτ(n) = 0, ∀ n ≥ |α|,
(b) β ≺
6 τ , in which case it follows that S∗β Rτ(n) = 0, ∀ n ≥ |β|,
so it follows immediately that
(39)
Rτ(n) Sα S∗β Rτ(n) = 0, ∀ n ≥ max{|α|, |β|},
so applying πap and taking so-limits yields
(40)
Eτ (X) = Qτ Sα Sβ∗ Qτ = 0.
This means that the only possibility for a generator X = Sα S∗β ∈ GM
E to have
Eτ (X)(= Qτ Sα Sβ∗ Qτ ) non-zero, is when α = β, in which case everything in clear,
since
Qτ , if α ≺ τ
∗
Eτ (X) = Qτ Sα Sα Qτ = Qτ Rα Qτ =
0, otherwise.
Of course, this proves that, given two paths α, β ∈ E ∗ , with s(α) = s(β), we have
1, if α = β ≺ τ
∗
(41)
ωτ (Sα Sβ ) =
0, otherwise
and we are done.
Statements (i) and (ii) are immediate.
The identities (34) and (35) have some approximate counterparts in the continuous case, as discussed in Proposition 3.10, for which we need the following result.
ABELIAN CORE OF GRAPH ALGEBRAS
17
3.9. Lemma. If X = Sα S∗β is a generator in GE , i.e. α, β ∈ E ∗ have s(α) = s(β),
then for every continuous trail τ ∈ TE , there exists some integer Nτ ≥ 0, such that
(42)
Rτ(n) XRτ(n) = ωτ (X)Rτ(n) , ∀ n ≥ Nτ .
Proof. Let τ be a continuous trail. Without any loss of generality (by taking
adjoints, if necessary), we can assume that |α| ≥ |β|.
Let n ≥ |α|. Suppose Rτ (n) XRτ (n) 6= 0. It follows that both S∗τ (n) Sα and
S∗β Sτ (n) are nonzero, and so β α ≺ τ. Furthermore, since s(α) = s(β), we have
α = β ◦ λ, where λ is a (possibly non-simple) loop. Assume λ has length at least
one (i.e. α 6= β). Let k be the maximum integer such that β ◦ λk ≺ τ . (Such a k
exists because τ is continuous.) Let Nτ = |β| + (k + 1)|λ|, and n ≥ Nτ , so that
τ(n) = β ◦ λk ◦ γ for some γ of length at least |λ|. We have
S∗τ (n) Sα S∗β Sτ (n) =S∗γ Sλ∗k S∗β Sβ Sλ S∗β Sβ Sλk Sγ
=S∗γ S∗λk Sλk+1 Sγ
(43)
=S∗γ Sλ Sγ ,
which can only be non-zero if λ γ, a contradiction to the choice of k.
Thus α = β and we can take Nτ = |α| and we note that if n ≥ Nτ we have
Rτ(n) Sα S∗α Rτ(n) = Rα Rτ(n) = Rτ(n) ,
since α ≺ τ and Rα and Rτ (n) commute, and we are done.
3.10. Proposition. Assume τ ∈ TE is continuous, and X ∈ C ∗ hEi. For a complex
number ω, the following are equivalent
(i) limn→∞ kRτ(n) XRτ(n) − ωRτ(n) k = 0;
(ii) limn→∞ kEM (X)Rτ(n) − ωRτ(n) k = 0;
(iii) ω = ωτ (X).
Proof. (iii) ⇒ (i). For this implication, all we must prove is the equality
(44)
lim kRτ(n) XRτ(n) − ωτ (X)Rτ(n) k = 0.
n→∞
Since the left-hand side is no greater than 2kXk, it suffices to prove (44) for X in
a dense subset. Furthermore, by linearity, it suffices to prove (44) for X ∈ GE ; this
is immediate from Lemma 3.9.
(i) ⇒ (ii). Assuming ω satisfies condition (i), we can apply EM , from which we
get
lim kEM (Rτ(n) XRτ(n) ) − ωEM (Rτ(n) )k = 0,
n→∞
and everything follows from the fact that EM (Rτ(n) ) = Rτ(n) (since Rτ (n) =
Sτ (n) S∗τ (n) is in GM
E ) and the properties of conditional expectations, which give
(45)
EM (Rτ(n) XRτ(n) ) = Rτ(n) EM (X) = EM (X)Rτ(n) .
(The last equality due to the fact that Range EM is MhEi, which is abelian.)
(ii) ⇒ (iii). If ω satisfies (ii), then by applying πap we have
lim kEap πap (X) Rτ(n) − ωRτ(n) k = 0,
n→∞
so using so-limits, we get Eap πap (X) Qτ = ωQτ , which forces ω = ωτ (X), because
Eap πap (X)Qτ = Eτ (X) = ωτ (X)Qτ , by Proposition 3.8 and Equation (37).
18
GABRIEL NAGY AND SARAH REZNIKOFF
Comment. Based on Proposition 3.8, up to this point we can conclude that the
finite trails and the continuous trails in TE each contribute a point in the Gelfand
spectrum of the abelian core MhEi. We can also state that these points can be
“seen from within MhEi” either directly, as indicated in Remark 3.7 (in the finite
case), or implicitly by the approximation shown in Proposition 3.9.
What remains to be treated is the contribution of the infinite discrete essentially
aperiodic trails: those that contain (simple) entry-less loops. The first observation is
that these trails are completely parameterized by their essential tails, so the objects
to consider are (finite) paths α ∈ E ∗ , such that s(α) is visited by a (necessarily
unique) simple entry-less loop, which we denote by λα , with the seed property
(s) (α, λα ) is a seed of a periodic trail (which is automatically discrete infinite
essentially aperiodic).
This means that, if |α| = 0, then α is simply a vertex v for which the loop λα starts
and ends at v, while if |α| ≥ 1, say α = (e1 , . . . , en ), then λα is a loop that starts
and ends at s(α) = s1 (en ), but does not visit any of the vertices r1 (e1 ), . . . , r1 (en ).
(This is equivalent to requiring that only r1 (en ) is not visited by λα .)
Conventions. Let us agree to call a path α, as above, distinguished. In case when
|α| = 0, we will call it a distinguished vertex. Of course, for any distinguished path
α, its source s(α) is distinguished vertex.
For a distinguished path α, we take the (unique) simple entry-less loop λα =
(e1 , . . . , en ) that starts and ends at s(α), and we denote the element Sα Sλα S∗α
simply by Wα . Of course, Wα is one of the generators in GM
E , so it is a normal
partial isometry. As it turns out, the C*-subalgebra C ∗ (Wα ) ⊂ MhEi generated by
Wα is unital, and Wα is unitary in C ∗ (Wα ). The gauge group acts on C ∗ (Wα )
as γz (Wα ) = z n Wα , so Wα has full spectrum in C ∗ (Wα ), i.e.
SpecC ∗ (Wα ) (Wα ) = T.
(Of course, unless in a very trivial case, when we compute the spectrum of Wα in
the ambient C*-algebras, we have SpecC ∗ hEi (Wα ) = SpecMhEi (Wα ) = T ∪ {0}.)
n
Since we view Wα as a unitary, we can define its integer powers Wα
, with the
∗ −n
n
convention that, when n < 0 we let Wα = (Wα ) . Note that the generator set
GM
E can now be written as a disjoint union
∆
n
GM
E = GE ∪ {Wα : α distinguished, n 6= 0}.
Because each Wα has full spectrum, there is a unique ∗-isomorphism Γα :
C ∗ (Wα ) → C(T) which maps Wα to the function ζ : T 3 z 7−→ z ∈ T.
3.11. Remark. We can see now that every discrete trail τ ∈ TE gives rise to a corner
C*-subalgebra
Mτ hEi = Rτ(ess) C ∗ hEiRτ(ess)
in C ∗ hEi, which turns out (by Remark 3.7) to be contained in MhEi, more specifially:
(i) if τ is finite, then Mτ hEi = CRτ(ess) ;
(ii) if τ is infinite, then Mτ hEi = C ∗ (Wτ(ess) ).
Note that the C*-subalgebras Mτ hEi, for τ discrete, are pairwise orthogonal, i.e. if
τ1 6= τ2 , then
Mτ1 hEi · Mτ2 hEi = {0},
ABELIAN CORE OF GRAPH ALGEBRAS
19
so the C*-subalgebra
Mdisc hEi = Span
[
Mτ hEi ⊂ MhEi
τ ∈TE
τ discrete
is ∗-isomorphic to the C*-algebraic direct sum of the Mτ hEi, which is (∗-isomorphic
to) a direct sum of copies of C and C(T).
This way, the “discrete coordinates” of the conditional expectation EM can be
“seen in Mdisc hEi” as the compressions maps
Fτ : C ∗ hEi 3 X 7−→ Rτ(ess) XRτ(ess) ∈ Mτ hEi.
Of course, we have πap ◦ Fτ = Eτ , so the system of completely positive maps
{Fτ : τ ∈ TE , τ discrete } ∪ {Eτ : τ ∈ TE , τ continuous }
is still jointly faithful.
We are in position now to prove the following uniqueness result.
3.12. Theorem. For a ∗-homomorphism Φ : C ∗ hEi → A, the following conditions
are equivalent
(i) Φ is injective;
(ii) the restriction of Φ to MhEi is injective;
(iii) both conditions below are satisfied
(a) Φ(Rα ) 6= 0, ∀ α ∈ E ∗ ;
(b) for every distinguished path α ∈ E ∗ , the normal
partial isometry Φ(Wα )
has full spectrum in A, i.e. SpecA Φ(Wα ) ⊃ T.
Proof. Of course, we only need to be concerned with the implication (iii) ⇒ (i).
Assume both conditions (a) and (b) are satisfied, and fix some element X ∈ Ker Φ.
If τ is discrete, by Remark 3.11, we know that the element Rτ(ess) X ∗ XRτ(ess) =
Fτ (X ∗ X) belongs to Ker Φ ∩ Mτ hEi, and since by conditions (a) and (b) we know
that Φ is injective on Mτ hEi, it follows that Fτ (X ∗ X) = 0.
If τ is continuous, then by Proposition 3.9 we have
lim kRτ(n) X ∗ XRτ(n) − ωτ (X ∗ X)Rτ(n) k = 0,
n→∞
so by applying Φ, it follows that
lim kωτ (X ∗ X)Φ(Rτ(n) )k = 0.
(46)
n→∞
Of course, by condition (a), we know that kΦ(Rτ(n) )k = 1, ∀ n ≥ 1, so by (46) it
follows that ωτ (X ∗ X) = 0, which means that Eτ (X ∗ X) = 0.
Now by Remark 3.11 this forces X ∗ X = 0, thus proving that Ker Φ = {0}. 3.13. Remark. Using the same proof, we see that condition (iii) from Theorem 3.12
also characterizes injectivity for a ∗-homomorphisms Φ solely defined on MhEi → A.
Comment. As pointed out earlier, in 2.2, condition (a) from Theorem 3.12 is equivalent to the condition that Φ(Rv ) 6= 0, ∀ v ∈ E 0 .
Since every for every distinguished path α we have the identities
Φ(Wα ) = Φ(Sα Ws(α) )Φ(S∗α ) and Φ(Ws(α) ) = Φ(S∗α )Φ(Sα Ws(α) ),
it follows that
{0} ∪ SpecA Φ(Wα ) = {0} ∪ SpecA Φ(Ws(α) ) .
20
GABRIEL NAGY AND SARAH REZNIKOFF
This means that condition (b) in Theorem 3.12 simply reduces to the condition that
(b0 ) all Φ(Wv )—with v a distinguished vertex—have full spectrum in A.
The pair of conditions (a) and (b) can in fact be optimized further, as follows.
0
Start of by considering the set Edisc
of all sources and all distinguished vertices, and
divide it into connected components. (If v is a source, its connected component is
{v}. If v is distinguished, its connected component is the set [λv ] of all vertices
0
visited by λv .) Select one vertex from each connected component in Edisc
and call
0
the resulting set representative for Edisc .
0
Consider then the set E00 of all vertices that are visited by paths starting in Edisc
,
0
0
0
0
0
and let E1 = E r E0 . The vertices in E1 (if any) are precisely those v ∈ E , for
which all trails τ ∈ TE with r(τ ) = v are continuous. We call a subset V1 ⊂ E10
Murray-von Neumann dominant for E10 , if for any w ∈ E10 , there exists some v ∈ V1 ,
such that Rv is Murray-von Neumann equivalent (in C ∗ hEi) to a sub-projection
of Rw . (An example of such a set is one for which for every w ∈ E10 , there exists
α ∈ E ∗ with r(α) = v and s(α) ∈ V1 .)
With this terminology, condition (iii) from Theorem 3.12 is equivalent to the
following:
0
0
, and a set
, which is representative for Edisc
(iii’) There exist a set V0 ⊂ Edisc
0
0
V1 ⊂ E1 , which is representative for E1 , such that
(a) Φ(Rv ) 6= 0, ∀ v ∈ V0 ∪ V1 ;
(b) for each distinguished v ∈ V0 , the normal partial isometry Φ(Wv ) has
full spectrum in A.
References
[aHR] Astrid an Huef, Iain Raeburn, The ideal structure of Cuntz-Krieger algebras, Ergodic
Theory Dynam. Sys. 17 (1997) 611–624.
[C]
Joachim Cuntz, Simple C*-algebras generated by isometries, Comm. Math. Phys. 57 (1977)
173–185.
[CK] Joachim Cuntz and Wolfgang Krieger, A class of C*-algebras and topological Markov
chains, Invent. Math. 56 (1980), 251–268. MR 82f:46073a
[F]
Xiaochun Fang, Graph C*-Algebras and Their Ideals Defined by Cuntz-Krieger Family of
Possibly Row-Infinite Directed Graphs, Integra Equations and Operator Theory 54 (2006)
301–316.
[KPR] Alexander Kumjian, David Pask, and Iain Raeburn, Cuntz-Krieger Algebras of directed
graphs, Pacific J. Math, 184, No. 1 (1998) 161–174.
[Mu] Gerard Murphy, C*-Algebras and Operator Theory, Academic Press, San Diego, 1990.
[R]
Iain Raeburn, Graph Algebras, CBMS Regional Conference Series in Mathematics 103
Published for the Conference Board of the Mathematical Sciences, Washington D.C. by the
AMS, Providence, RI (2005).
[Re] Jean Renault, Cartan subalgebras in C*-algebras, Irish Math. Soc. Bulletin 61 (2008) 29–
63.
[RS] Iain Raeburn and Wojciech Szymanski, Cuntz-Krieger algebras of infinite graphs and matrices, Transactions of the American Math. Society 356 (2003) 39–59.
[S]
Wojciech Szymański, Generalized Cuntz-Krieger Uniqueness Theorem, International Journal of Mathematics, 13 No. 5 (2002) 549–555.
[T]
Tomiyama, On the projection of norm one in W ∗ -algebras, Proc. Japan Acad. 3 (1957),
608–612.
Department of Mathematics, Kansas State university, Manhattan KS 66506, U.S.A.
E-mail address: nagy@math.ksu.edu
Department of Mathematics, Kansas State university, Manhattan KS 66506, U.S.A.
E-mail address: sararez@math.ksu.edu