CIMPA RESEARCH SCHOOL 2015: EQUILIBRIUM STATES OF THE
C ∗ -ALGEBRAS OF FINITE DIRECTED GRAPHS
ASTRID AN HUEF
Let A be a C ∗ algebra with identity 1. A state f : A → C of A is a positive linear
functional of norm 1. Positive means that f (a∗ a) ≥ 0 for all a ∈ A. The norm is kf k =
sup{|f (a)| : kak = 1}. Here f (1) = 1 implies kf k = 1.
Example 1. Let π : A → B(H) be a unital representation. Fix h ∈ H of norm 1. Then
f (a) = (π(a)h | h) is a state of A.
Iain mentioned the Gelfand-Naimark theorem: every C ∗ -algebra has a faithful (injective)
representation. The rough idea is that if f is a state, then (a, b) 7→ f (b∗ a) is close to an
inner product on A. Let N = {a ∈ A : f (a∗ a) = 0}. Then N is a closed left ideal of A
and the quotient space A/N is an inner product space which carries a left action of A by
multiplication. Completing A/N gives a Hilbert space Hf . The action of A extends to the
completion to give a unital representation πf : A → B(Hf ). There is a vector hf (the vector
1 + N in the completion) such that
f (a) = (πf (a)hf | hf ).
(This gives a surjection from the set S(A) of states of A to the unital cyclic representation
of A: πf (A)hf is dense in Hf and hf is called a cyclic vector. The pure states are the
extreme points of S(A) and give irreducibleL
representations.) For each a ∈ A, there exists
a state fa of A such that fa (a) 6= 0. Then a∈A πfa is a faithful representation of A. So
states are important in the study of C ∗ -algebras.
Equilibrium states. Let α : R → Aut A be an action of the real line R on a C ∗ -algebra A
with identity. In physical models, observables of the system are represented by self-adjoint
elements of A, and states of the system by states of φ of A. Then φ(a) is the expected value
of the observable a in the state φ (which is real because a = a∗ and φ ≥ 0). Since C ∗ (A) is
commutative, it is isomorphic to the continuous function on the spectrum σ(a), and φ|C ∗ (a)
corresponds to a probability measure µ on σ(a).
The action α represents the time evolution of the system: the observable a at time 0
moves to αt (a) at time t. In statistical physics, an important role is played by equilibrium
states, which are in particular invariant under the time evolution. In C ∗ -algebraic models
equilibrium states are called KMS states, after Kubo, Martin and Schwinger.
A state φ on A is a KMS state at inverse temperature β if
(1)
φ(ab) = φ(bαiβ (a)) for all analytic a, b.
(Here i2 = −1.) Here are some facts:
• KMS states are α-invariant. (If β = 0, then (1) just says φ is a trace, and we impose
the α-invariance.)
• It suffices to check the KMSβ condition on a set of analytic elements which span a
dense subspace of A. In practice, these are easy to find.
Date: 6 July 2015.
1
2
ASTRID AN HUEF
• The KMSβ states always form a simplex. (This is where we need A to have an
identity.)
General facts about KMS states can be gleaned from [2, 10].
Equilibrium states of algebras associated to graphs. Let E = (E 0 , E 1 , r, s) be a finite
directed graph. I’ll assume no sources for convenience1. Both the graph algebra C ∗ (E) and
its Toeplitz extension T C ∗ (E) will come up below. I will use the “southern convention” to
be consistent with work on algebras of higher-rank graphs. If you don’t like this, just apply
your conventions to the opposite graph E op obtained by reversing the direction of the edges
of E. The main reference for what follows is [6].
A Cuntz-Krieger E-family consists of mutually orthogonal projections {Pv : v ∈ E 0 } and
partial isometries {Se : e ∈ E 1 } satisfying
(CK1) Se∗ Se = Ps(e) and
X
(CK2) Pv =
Se Se∗ .
r(e)=v
Remember there are lots of consequences. Iain showed that (CK2) implies {Se Se∗ : r(e) = v}
is a set of mutually orthogonal projections. Let f ∈ r−1 (v) and multiply (CK2) by Sf Sf∗ to
get Pr(f ) Sf Sf∗ = Sf Sf∗ ; then multiply on the right by Sf to get Pr(f ) Sf = Sf . Multiplying
(CK1) on the left by Se gives Se = Se Ps(e) . If e, f have r(e) 6= r(f ) then
Se Se∗ Sf Sf∗ = Se Se∗ Pr(e) Pr(f ) Sf Sf∗ = 0.
Now {Se Se∗ : r ∈ E 1 } is a set of mutually orthogonal projections. If e, f ∈ E 1 , then
Se∗ Sf = Se∗ (Se Se∗ Sf Sf∗ )Sf = δe,f Se∗ Se = δe,f Ps(e) ,
using (CK1) at the end. It follows that C ∗ (P, S) = span{Sµ Sν∗ : µ, ν ∈ E ∗ }. The graph
algebra C ∗ (E) is universal for Cuntz-Krieger E-families and is generated by a universal
E-family (p, s); and C ∗ (E) = span{sµ s∗ν : µ, ν ∈ E ∗ }. There is an action, called the gauge
action, γ : T → Aut C ∗ (E) characterised by
γz (pv ) = pv and γz (se ) = zse .
Then γz (s∗e ) = zs∗e and γz (sµ s∗ν ) = z |µ|−|ν| sµ s∗ν . We can lift γ to an action of α : R →
Aut C ∗ (E) using t 7→ eit .
Does (C ∗ (E), α) have KMS states? First notice that
t 7→ αt (sµ s∗ν ) = eit(|µ|−|ν|) sµ s∗ν
extends to an analytic function (just replace t by z). So it suffices to check the KMS
condition on the spanning elements of the dense subspace span{sµ s∗ν : µ, ν ∈ E ∗ } of C ∗ (E).
Let φ be a KMSβ state on (C ∗ (E), α). We have
φ(sµ s∗ν ) = φ(s∗ν αiβ (sµ )) = e−β|µ| φ(s∗ν sµ ) = e−β(|µ|−|ν|) φ(sµ s∗ν ).
Thus φ(sµ s∗ν ) 6= 0 implies |µ| = |ν|. So suppose |µ| = |ν|. Then s∗ν sµ = δµ,ν ps(µ) . From
above we get
φ(sµ s∗ν ) = e−β|µ| φ(s∗ν sµ ) = e−β|µ| φ(ps(µ) ).
So we have proved half of the following lemma:
1In
fact, sources give rise to KMS states, as first noted by Kajiwara and Watatani in [7].
EQUILIBRIUM STATES
3
Lemma 2. A state φ on C ∗ (E) is KMSβ for α if and only if
(
0
when ν 6= µ
φ(sµ s∗ν ) =
e−β|µ| φ(ps(µ) ) when ν = µ.
Notice that the state depends only on its values on the vertex projections. The proof of
the converse is a tedious calculation, based on cases, showing
φ(sµ s∗ν sσ s∗τ ) = e−β(|µ|−|ν|) φ(sσ s∗τ sµ s∗ν );
see [6, Proposition 2.1(a)]. A lemma of this type is always the first thing we try to prove:
we need easy ways to recognise KMS states.
We haven’t used the Cuntz-Krieger relation (CK2) yet: Suppose again that φ is a KMSβ
state on (C ∗ (E), α). Then
X
X
φ(pv ) =
φ(se s∗e ) =
e−β φ(ps(e) ).
r(e)=v
r(e)=v
The vertex matrix of E is the E 0 × E 0 integer matrix A with entry A(v, w) equal to the
number of edges from w to v. We can rearrange the above sum as
X
X
X
(2)
eβ φ(pv ) =
φ(pw ) =
A(v, w)φ(pw ).
w∈E 0 r(e)=v, s(e)=w
w∈E 0
Consider the vector m = (mv ) := (φ(pv )). Then m ≥ 0 since φ is positive, m 6= 0 since φ
has norm 1, and Am = eβ m.
If A were an irreducible matrix, in the sense that A has non-negative entries and for each
pair of vertices (v, w) there exists n such that An (v, w) > 0, then Perron-Frobenius Theory
would give us lots of information (see, for example [11, Chapter 1] or my version of those
theorems below). Note that A is irreducible if and only of E is strongly connected.
Theorem 3 (Perron-Frobenius Theorem). Let A be an irreducible matrix. Then there exists
a real eigenvalue of A, called the Perron-Frobenius eigenvalue, such that
(a) 0 < r = ρ(A) := max{|λ| : λ is an eigenvalue of A};
(b) the eigenspace of r is one-dimensional, and it contains an eigenvector m > 0.
Theorem 4 (Perron-Frobenius Subinvariance Theorem). Let A be an irreducible matrix and
r its Perron-Frobenius eigenvalue. Let s > 0 and let m be a vector such that m ≥ 0, 6= 0
and Am ≤ sm. Then
(a) m > 0;
(b) s ≥ r;
(c) s = r if and only if Am = sm.
Now we have:
Proposition 5. Suppose E is a finite, strongly connected directed graph. Let A be the vertex
matrix of A such that A(v, w) equals the number of edges from w to v. Then (C ∗ (E), α) has
at most one KMS state. If so, the KMS state must have inverse temperature β = ln ρ(A).
Proof. Suppose that φ is a KMSβ state of (C ∗ (E), α). Let m = (mv ) := (φ(pv )) as above.
Then m ≥ 0, 6= 0 and Am = eβ m. Theorem 4 implies that eβ = ρ(A) is the Perron-Frobenius
eigenvalue and m > 0. Thus β = ln ρ(A). Notice that
X
X
1 = φ(1) =
φ(pv ) =
mv ,
v∈E 0
v∈E 0
4
ASTRID AN HUEF
that is, m is the unique Perron-Frobenius eigenvector of `1 norm 1. Now Lemma 2 tells us
(
0
when ν 6= µ
φ(sµ s∗ν ) =
−β|µ|
e
ms(µ) when ν = µ.
Thus there is at most one KMSln ρ(A) state of (C ∗ (E), α).
In [3], Enomomto, Fuji and Watatani proved that (OA , α) has a unique KMS state which
occurs at β = ln ρ(A), and in [9], Olesen and Pedersen proved that (On , α) has a unique
KMS state which occurs at β = ln n.
To prove the existence of a KMSln ρ(A) state of (C ∗ (E), α) we will use the philosophy of
Exel-Laca [4] and Laca-Neshveyev [8]: the Toeplitz algebra T C ∗ (E) of E has a much richer
supply of KMS states.
The Toeplitz algebra T C ∗ (E) is universal for Toeplitz-Cuntz-Krieger families (Q, T )
where {Qv : v ∈ E 0 } is a set of mutually orthogonal projections and {Te : e ∈ E 1 } are
partial isometries such that
(CK1) Te∗ Te = Qs(e) and
X
(TCK) Qv ≥
Te Te∗ .
r(e)=v
Again we use lower-case letters to denote the generating universal Toeplitz-Cuntz-Krieger
family (q, t); the point of the relations is (as always) that T C ∗ (E) = span{tµ t∗ν : µ, ν ∈ E ∗ }.
The Toeplitz algebra also has a gauge action γ̃ : T → Aut T C ∗ (E) such that γ̃z (qv ) = qv
and γ̃z (te ) = zte . (See [5] for the existence of T C ∗ (E) and the gauge action.)
∗
it
Again, we lift this action to an action
P α̃ : R ∗→ Aut T0C (E) via t 7→ e . Let I ∗be the
∗
ideal of T C (E) generated by {qv − r(e)=v te te : v ∈ E }. Then the quotient T C (E)/I
is isomorphic to C ∗ (E). The quotient map ρ is γ̃z –γz equivariant, and hence is α̃z –αz
equivariant as well. It follows that if φ is a KMSβ state of (C ∗ (E), α), then φ ◦ ρ is a a
KMSβ state of (T C ∗ (E), α̃).
Let ψ be a KMSβ state of (T C ∗ (E), α̃). It is again easy to recognise a KMSβ state ψ
on (T C ∗ (E), α), and the formulas look the same as those in Lemma 2. But now m =
(mv ) := (ψ(qv )) is the `1 unit vector satisfying the subinvariance relation Am ≤ eβ m. The
Perron-Frobenius Subinvariance Theorem gives guidance when E is strongly connected:
• Am = eβ m =⇒ eβ = ρ(A) =⇒ β = ln ρ(A);
• Am ≤ eβ m and β = ln ρ(A) =⇒ m is the PF eigenvector;
• Am ≤ eβ m and Am 6= eβ m =⇒ β > ln ρ(A).
So we consider β > ln ρ(A) (whether E is strongly connected or not); the import is:
P
−βn n
Lemma 6. Let β > ln ρ(A). Then ∞
A converges in the operator norm to (1 −
n=0 e
−β
−1
e A) .
P
−βn n
Proof. We use the nth root test on ∞
A k:
n=0 ke
lim (e−βn kAn k)1/n = e−β lim kAn k1/n = e−β ρ(A)
n→∞
n→∞
using
spectral radius formula. Since β > ln ρ(A) we have e−β ρ(A) < 1 and hence
P∞ the
−βn n
A converges absolutely. An argument with partial sums shows the limit is
n=0 e
−β
(1 − e A)−1 .
EQUILIBRIUM STATES
5
Our question now is: what sort of unit vectors m ≥ 0, 6= 0 satisfy Am ≤ eβ m? Notice
that Am ≤ eβ m if and only if := (1 − eβ A)m ≥ 0 (we think of the difference as small, so
we named it epsilon). Compute:
∞
X
X
X X
−β
−1
mv =
((I − e A) )v =
e−βn An v∈E 0
v∈E 0
=
v∈E 0
∞ X
XX
n=0
v
e−βn An (v, w)w
v∈E 0 n=0 w∈E 0
=
X
w
∞
XX
=
w∈E 0
v∈E 0 n=0
w∈E 0
X
e−βn # of path of length n from w to v
w
e−β|µ| .
X
µ∈E ∗ ,s(µ)=w
Let y = (yv ) be the vector with entries
yv :=
X
e−β|µ| .
µ∈E ∗ ,s(µ)=w
Then m is a unit vector if and only if · y = 1. So there are constraints on m = (ψ(qv )).
We still need to construct KMS states. For this we’ll use the finite-path representation
of T C ∗ (E) on `2 (E ∗ ). Let {hλ : λ ∈ E ∗ } be the orthonormal basis of point masses. There
are projections Qv and partial isometries Te on `2 (E ∗ ) such that Then (Q, T ) is a ToeplitzCK family (exercise), and the universal property of T C ∗ (E) gives a representation πQ,T of
T C ∗ (E) on `2 (E ∗ ) (in fact injective) such that πQ,T (qv ) = Qv and πQ,T (te ) = Te .
(
hµ if r(µ) = v
Qv hµ =
0
else;
(
heµ if r(µ) = s(e)
Te hµ =
0
else.
Theorem 7 (an Huef-Laca-Raeburn-Sims, 2012). Suppose E is a finite graph with vertex
matrix A, and β > ln ρ(A). Take y = (yv ) as above, and suppose · y = 1.
(a) Then there is a KMSβ state φ of T C ∗ (E) such that
X
(3)
ψ (a) =
e−β|λ| s(λ) (πQ,T (a)hλ | hλ ).
λ∈E ∗
and
ψ (tµ t∗ν ) = δµ,nu e−β|µ| ms(µ) .
(b) The map 7→ ψ is an affine isomorphism of ∆β = { = (v ) : · y = 1} onto the
simplex of KMSβ states.
Note that Ψ is a weighted sum of states, and that there is no irreducibility hypothesis
on the graph. The simplex ∆β has dimension |E 0 | − 1.
Sketch of proof of (1). First, we have to check that ψ is P
well-defined, that is, (3) converges.
Since |(πQ,T (a)hλ | hλ )| ≤ kak, it suffices to show that λ∈E ∗ e−β|λ| s(λ) = 1, which boils
P
down to showing that λ∈E ∗ ,r(λ)=v e−β|λ| s(λ) = mv and using that m is a unit vector.
6
ASTRID AN HUEF
To verify that ψ is a KMS state, we use a version of Lemma 2 for (T C ∗ (E), α̃). Let
µ, ν ∈ E ∗ . Then
(
1
(πQ,T (tµ t∗ν )hλ | hλ ) = (Tν∗ hλ | Tµ∗ hλ ) =
0
if λ = µλ0 = νλ0
otherwise.
Since µλ0 = νλ0 forces µ = ν, we have ψ (tµ t∗ν ) = 0 if µ 6= ν. So suppose µ = ν. Then
ψ (tµ t∗µ ) =
X
0
e−β|µλ | s(λ0 ) = e−β|µ|
λ=µλ0
0
X
e−β|λ | s(λ0 ) = e−β|µ| ms(µ)
λ0 ∈E ∗ ,r(λ0 )=s(µ)
and
ψ (qs(µ) ) =
X
λ∈E ∗
X
e−β|λ| s(λ) (Qs(µ) hλ | hλ ) =
e−β|λ| s(λ)
λ∈E ∗ ,r(λ)=s(µ)
Thus ψ (tµ t∗µ ) = δµ,ν e−β|µ| ψ (qs(µ) ), and ψ is a KMSβ state.
Corollary 8. There exists a ln ρ(A) state of (T C ∗ (E), α̃).
Proof. Choose a sequence {βn } ⊆ (ln ρ(A), ∞) converging to ln ρ(A). For each βn , there
exists a KMSβn state ψn by Theorem 7. The unit ball in the dual of T C ∗ (E) is weak*
compact, and by passing to a subsequence we may assume that ψn → ψ for some state ψ.
By general arguements, ψ is a KMSln ρ(A) state.
Corollary 9. Let E be a strongly connected finite graph. Let x be the unital PerronFrobenius eigenvector of A. Then
(a) (T C ∗ (E), α̃) has a unique KMSln ρ(A) state ψ with formula
ψ(tµ t∗ν ) = δµ,nu ρ(A)−|µ| xs(µ) .
Moreover, ψ factors through a state ψ of (C ∗ (E), α), and ψ is the unique KMSln ρ(A)
state of (C ∗ (E), α).
(b) If β < ln ρ(A), then there are no KMSβ states of (T C ∗ (E), α̃).
We will need the following lemma ([1, Lemma 6.2]) for the proof:
Lemma 10. Suppose I is an ideal in a C ∗ -algebra B generated by a set P of positive
elements fixed by β : R → Aut B. If ψ is a KMSβ state of (B, β) and ψ(p) = 0 for all
p ∈ P , then ψ factors through a state of (B/I, β).
Proof of Corollary 9. There exists a KMSln ρ(A) state ψ by Corollary 8. Let x = (xv ) :=
(ψ(qv )). Then x ≥ 0, 6= 0 and Ax ≤ ρ(A)x. The Perron-Frobenius Subinvariance Theorem
implies Am = ρ(A)x, that is, x is the Perron-Frobenius eigenvector of norm 1. Uniqueness
and the formula for ψ now follow from a version of Lemma 2 for (T C ∗ (E), α̃).
EQUILIBRIUM STATES
7
P
We have C ∗ (E) ∼
= T C ∗ (E)/I where I is generated by elements of the form qv − r(e)=v te t∗e .
We compute:




X
X
ρ(A) φ qv −
te t∗e  = ρ(A) xv −
ρ(A)−1 φ(qs(e) ) 
r(e0=v
r(e)=v
= ρ(A)xv −
X
xs(e)
r(e)=v
= ρ(A)xv −
X
A(v, w)xw
w∈E 0
= ρ(A)xv − (Ax)v = 0
because x is the Perron-Frobenius eigenvector. Thus ψ factors through a state ψ of C ∗ (E)
by Lemma 10. This is now the unique KMSln ρ(A) state of (C ∗ (E), α) state by Proposition 5.
That there are no KMSβ states of (T C ∗ (E), α̃) for β < ln ρ(A) also follows from the
Perron-Frobenius Subinvariance Theorem.
Example
11. Let E be the strongly connected graph with two vertices and vertex matrix
1 1
. Then for β > ln 2, the KMSβ simplex of (T ∗ (E), α̃) has dimension 2 − 1 = 1, and
1 1
at β = ln 2 there is a unique KMS state (the simplex is 0-dimensional). For this reason we
say there is a phase transition at β = ln 2.
References
[1] Z. Afsar, A. an Huef and I. Raeburn, KMS states on C ∗ -algebras associated to local homeomorphisms,
Internat. J. Math. 25 (2014).
[2] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics II, second ed.,
Springer-Verlag, Berlin, 1997.
[3] M. Enomoto, M. Fujii and Y. Watatani, KMS states for gauge action on OA , Math. Japon. 29 (1984),
607–619.
[4] R. Exel and M. Laca, Partial dynamical systems and the KMS condition, Comm. Math. Phys. 232
(2003), 223–277.
[5] N.J. Fowler and I. Raeburn, The Toeplitz algebra of a Hilbert bimodule, Indiana Univ. Math. J. 48
(1999), 155–181.
[6] A. an Huef, M. Laca, I. Raeburn and A. Sims, KMS states on C ∗ -algebras of finite graphs, J. Math.
Anal. Appl. 405 (2013), 388–399.
[7] T. Kajiwara and Y. Watatani, KMS states on finite-graph C ∗ -algebras, Kyushu J. Math. 67 (2013),
83–104.
[8] M. Laca and S. Neshveyev, KMS states of quasi-free dynamics on Pimsner algebras, J. Funct. Anal.
211 (2004), 457–482.
[9] D. Olesen and G.K. Pedersen, Some C ∗ -dynamical systems with a single KMS state, Math. Scand. 42
(1978), 111–118.
[10] G.K. Pedersen, C ∗ -Algebras and their Automorphism Groups, London Math. Soc. Monographs, vol.
14, Academic Press, London, 1979.
[11] E. Seneta, Non-Negative Matrices and Markov Chains, second ed., Springer-Verlag, New York, 1981.