Table Of ContentKMS STATES ON C∗-ALGEBRAS ASSOCIATED TO
A FAMILY OF ∗-COMMUTING LOCAL HOMEOMORPHISMS
ZAHRA AFSAR, ASTRID AN HUEF, AND IAIN RAEBURN
Abstract. We considera family of -commutinglocalhomeomorphismsona com-
∗
pact space, and build a compactly aligned product system of Hilbert bimodules (in
the senseofFowler). This productsystemhas a Nica-Toeplitz algebraanda Cuntz–
Pimsner algebra. Both algebras carry a gauge action of a higher-dimensional torus,
7 and there are many possible dynamics obtainedby composing with different embed-
1 dings of the real line in this torus. We study the KMS states of these dynamics.
0 For large inverse temperatures, we describe the simplex of KMS states on the Nica-
2
Toeplitz algebra. To study KMS states for smaller inverse temperature, we consider
n a preferred dynamics for which there is a single critical inverse temperature. We
a find a KMS state on the Nica-Toeplitz algebra at this critical inverse temperature
J
which factors through the Cuntz–Pimsner algebra. We then illustrate our results by
5
considering backward shifts on the infinite-path spaces of a class of k-graphs.
2
]
A
O
1. Introduction
.
h
Suppose that α is an action of the real line R by automorphisms of a C∗-algebra
t
a
A. Operator-algebraic dynamical systems such as (A,R,α) provide operator-algebraic
m
models for physical systems in quantum statistical physics [4]. The equilibrium states
[
of the physical system are then the states on A that satisfy a commutation relation
1
calledtheKMScondition. This relationinvolves aparameter β, whichisa realnumber
v
3 interpreted as the inverse temperature of the physical system. The KMS condition
β
8
makes sense forabstract dynamical systems aswell asphysical ones, andmany authors
1
7 have studied KMS states in other purely mathematical contexts [9, 3, 36, 10, 26, 23,
0 22, 24]. The results have often been fascinating.
.
1 Suppose that A is a C∗-algebra. Fowler [16] defined product systems of Hilbert
0
A–A bimodules over semigroups P. Following the construction for individual bimod-
7
1 ules in [17], Fowler associated to each product system X a Toeplitz algebra (X)
T
: which is universal for a family of Toeplitz representations of X, and a quotient (X)
v
O
i which is universal for a family of Cuntz–Pimsner covariant representations. When P is
X
the positive cone in a quasi-lattice ordered group (G,P) as in [32], Fowler considered
r
a also a smaller family of Nica-covariant Toeplitz representations, and the Nica-Toeplitz
algebra (X) is the quotient of (X) which is universal for Nica-covariant represen-
NT T
tations. The algebra (X) is only tractable for a class of compactly aligned product
NT
systems, and for such systems, (X) is a quotient of (X) (see Lemma 2.2 below).
O NT
Therearemanyinteresting examples ofproductsystems over theadditive semigroup
Nk, including ones associated to the k-graphs of Kumjian-Pask [25] (see [18, page
1492]). For systems over Nk, the universal properties of (X) and (X) give
NT O
strongly continuous gauge actions of the torus Tk. Composing with an embedding
Date: 25 January 2017.
2010 Mathematics Subject Classification. 46L35.
Key words and phrases. Product system; -commuting homeomorphisms; Toeplitz algebra; Nica–
∗
Toeplitz algebra, Cuntz–Pimsner algebra; KMS state.
This research was supported by the Marsden Fund of the Royal Society of New Zealand.
1
2 ZAHRAAFSAR,ASTRIDAN HUEF, ANDIAINRAEBURN
of R in Tk gives actions of R on (X) and (X), and we are interested in the
NT O
equilibrium states of the resulting dynamical systems.
Our approach is informed by our previous work [1], where we studied the KMS
states on the Toeplitz algebra and the Cuntz–Pimsner algebra of a Hilbert bimodule
associated to a local homeomorphism on a compact space Z. Here we consider a
family of k -commuting local homeomorphisms on the same space Z, use them to
∗
construct a product system X of Hilbert C(Z)–C(Z) bimodules over Nk, and study
the KMS states of dynamical systems based on (X) and (X) as discussed above.
NT O
Our approach extends the results of [1] to product systems in a way parallel to the
extension of results about the algebras of finite graphs in [21] to the algebras of k-
graphs in [22]. For large inverse temperatures we find an infinite-dimensional simplex
of KMS states on (X). At a critical inverse temperature β determined by the
β c
NT
dynamical properties of the local homeomorphisms, there is at least one KMS state
βc
that factors though (X).
O
Outline. We begin with a section on background material. We first discuss some gen-
eral propertiesof Hilbert bimodules andtheir representations. Onetechnical difference
between our analysis and that of Hong-Larsen-Szyman´ski [19] is that our bimodules
do not have orthonormal bases, but do have Parseval frames. So we review basic
properties of such frames. Then in 2.2, we review Fowler’s definition of product sys-
§
tems and the family of compactly aligned product systems. In 2.3, we discuss the
§
Nica-Toeplitz algebra, and establish some basic properties which are hard to point to
in the literature. In particular, we show that the Cuntz–Pimsner algebra, which was
defined in [16] as a quotient of (X), can also be viewed as a quotient of (X). We
T NT
then have some short notes about various other topics, including a discussion of the
-commuting hypothesis.
∗
In 3, we start with k commuting local homeomorphisms on a compact Hausdorff
§
space Z, and construct a compactly aligned product system X over Nk. This is based
on constructions of Larsen [29] and Brownlowe [5, Proposition 3.2].
We next choose a vector r = (r ) (0, )k and construct a dynamics αr : R
j
∈ ∞ →
Aut (X) by composing the gauge action of Tk with the map t eitrj from
NT 7→
R Tk. Every KMS functional restricts to a positive functional on the coefficient
→ (cid:0) (cid:1)
algebra C(Z), which is implemented by a measure µ on Z. The KMS condition
then imposes constraints on the measure µ, which are analogues of the subinvariance
relations arising in the analysis of KMS states on k-graph algebras [22, 4 and The-
§
orem 6.1(a)]. In 5, we describe the subinvariance relations arising here and their
§
solutions (Proposition 4.3). To find these solutions we need to restrict β to a range
β > β , and we describe the critical inverse temperature β in Proposition 4.3. It is a
c c
several-variable analogue of the critical inverse temperature in [1, Proposition 4.2].
The second step in our analysis is to find a way of recognising KMS states in terms
β
of their behaviour on a spanning family (Proposition 5.1). For this calculation, we
need to assume that our local homeomorphisms -commute, and we therefore assume
∗
this for the rest of the paper. The proof of Proposition 5.1 is long and involved, and
occupies the whole of 5.
§
In 6 we prove our main results about the KMS states of ( (X),αr) for β
β
§ NT
larger than the critical value β (Theorem 6.1). The proof of that theorem occupies
c
most of 6, but we also discuss the implications for the KMS states on ( (X),αr) in
§ O
Corollary 6.2.
Our main result about the existence of KMS states is Theorem 7.1, which is an
βc
analogue of [1, Theorem 6.1]. At least one such state factors through a state of the
KMS STATES 3
Cuntz–Pimsner algebra (X). In 8, we discuss the ground states of ( (X),αr):
O § T
Proposition 8.1 says that they are parametrised by the entire simplex of probability
measures on Z. For this system, every ground state is a KMS state, and hence there
∞
is no further phase transition at , as occurs, for example, in [27].
∞
Important examples of local homeomorphisms on compact spaces are the shifts
on the path spaces for finite directed graphs. In [1, 7], we showed that there are
§
interesting relations between the Toeplitz and Cuntz-Krieger algebras of a finite graph
E and the Toeplitz and Cuntz–Pimsner algebras of the associated shifts. In 9, we
§
consider shift maps on the path spaces of k-graphs. There are now k different shifts
to consider, and the factorisation property of the k-graph implies that these shifts
commute. They do not always -commute, but there is a family of 1-coaligned graphs
∗
for which they do, and then we can apply our results. We show that in this case we
obtain relations between the graphs algebras C∗(Λ) and C∗(Λ) rather like those of
T
[1, 7].
§
Toeplitz algebras and Cuntz–Pimsner algebras of product systems over Nk have
previously appeared in work of Solel [42]. We show in an appendix that the canonical
representations of our product systems in (X) are “doubly commuting” in the
NT
sense of [42].
2. Notation and conventions
2.1. Hilbert bimodules. Let A be a C∗-algebra, and let X be a right Hilbert A–A
bimodule. This means that X is a right Hilbert A-module with a left action of A
implemented by a homomorphism ϕ : A (X) from A into the C∗-algebra (X)
→ L L
of adjointable operators on X. In other words, X is a correspondence over A. We
say that X is essential if X = span ϕ(a)x : a A,x X . If A is unital with
{ ∈ ∈ }
identity 1 and ϕ(1 )x = x for all x X, then X is essential. We write A for the
A A A A
∈
standard bimodule with the inner product given by a,b = ab∗ and the actions given
h i
by multiplication in A. For x,y X, we write Θ for the adjointable operator on X
x,y
∈
given by Θ (z) = x y,z . We call (X) := span Θ : x,y X the algebra of
x,y x,y
· h i K { ∈ }
compact operators on X.
A representation (ψ,π) of X in a C∗-algebra B consists of a linear map ψ : X B
→
and a homomorphism π : A B such that
→
ψ(a x b) = π(a)ψ(x)π(b) and π( x,y ) = ψ(x)∗ψ(y)
· · h i
for all x,y X and a,b A. Such a representation induces a homomorphism
∈ ∈
ψ(1) : (X) B
K →
such that ψ(1)(Θ ) = ψ (x)ψ (y)∗ (see [35, page 202]).
x,y p p
Following [13, 12], a sequence x d in X is called a Parseval frame for X if
{ i}i=1
d
(2.1) x x ,x = x for all x X.
i i A
·h i ∈
i=1
X
The formula (2.1) is known as the reconstruction formula. Let x d be a Parseval
{ i}i=1
frame for X and write 1 for the identity operator on X. The reconstruction formula
X
implies that d Θ = 1 . In particular 1 is compact.
i=1 xi,xi X X
If (ψ,π) is a representation of X on a Hilbert space , then [16, Proposition 4.1(1)]
P H
gives a unique endomorphism αψ,π of π(A)′ such that
αψ,π(T)ψ(x) = ψ(x)T for all T π(A)′, x X, and
∈ ∈
αψ,π(1)r = 0 for r (ψ(X) )⊥.
∈ H
4 ZAHRAAFSAR,ASTRIDAN HUEF, ANDIAINRAEBURN
The next lemma shows that if there is a Parseval frame for X, then we can formulate
the endomorphism αψ,π easily.
Lemma 2.1. Let X be a right Hilbert A–A bimodule. Suppose that x d is a Par-
{ i}i=1
seval frame for X. Let (ψ,π) be a representation of X on a Hilbert space and let
H
αψ,π be as above. Then
d
(2.2) αψ,π(1) = ψ(1)(1 ) = ψ (x )ψ (x )∗.
X p i p i
i=1
X
Proof. The second equality follows from the reconstruction formula and the formula
for ψ(1). To see the first equality, by uniqueness of αψ,π, it suffices to prove that
(a) ψ(1)(1 )ψ(x) = ψ(x) for all x X, and
X
∈
(b) ψ(1)(1 )r = 0 for all r (ψ(X) )⊥ = 0.
X
∈ H
To see (a), let x X. The reconstruction formula for x implies that
∈
d d
ψ(x) = ψ x x ,x = ψ(x )π( x ,x ).
j j j j
·h i h i
(cid:16)Xj=1 (cid:17) Xj=1
Using this and the second equality in (2.2), we have
d d
ψ(1)(1 )ψ(x) = ψ(x )ψ(x )∗ψ(x )π( x ,x ) = ψ(x )π( x ,x )π( x ,x )
X i i j j i i j j
h i h i h i
i,j=1 i,j=1
X X
d
= ψ(x x ,x )π( x ,x ).
i i j j
·h i h i
i,j=1
X
Rearranging this and two applications of the reconstruction formula give
d d d
ψ(1)(1 )ψ(x) = ψ x x ,x π( x ,x ) = ψ(x )π( x ,x )
X i i j j j j
·h i h i h i
Xj=1 (cid:16)Xi=1 (cid:17) Xj=1
d
= ψ(x x ,x ) = ψ(x).
j j
·h i
j=1
X
This is precisely (a).
For (b), fix r (ψ(X) )⊥. Notice that for all r′ we have
∈ H ∈ H
d d
ψ (x )ψ (x )∗ r r′ = r ψ (x )ψ (x )∗r′ = 0 for r′ .
p i p i p i p i
∈ H
(cid:16)(cid:16)Xi=1 (cid:17) (cid:12) (cid:17) Xi=1 (cid:0) (cid:12) (cid:1)
(cid:12)
It follows that ψ(1)(1 )r = 0 an(cid:12)d we have pro(cid:12)ved (b). (cid:3)
X
Given two right Hilbert A–A bimodules X and Y, we can form a balanced tensor
product of X and Y as follows: Let X Y be the algebraic tensor product of X and
⊙
Y and suppose that X Y is the quotient of X Y by the subspace
A
⊙ ⊙
(2.3) N := span (x a) y x (a y) : x X,y Y,a A .
{ · ⊙ − ⊙ · ∈ ∈ ∈ }
There is a well-defined right action of A on X Y such that (x y) a = x y a
A A A
⊙ ⊙ · ⊙ ·
for x y X Y,a A. By [28, Proposition 4.5], we can equip X Y with a
A A A
⊙ ∈ ⊙ ∈ ⊙
right A-valued inner product characterised by
(2.4) x y,z w = y,ϕ x,z w for x y,z w X Y.
A A Y A A A
⊙ ⊙ h i ⊙ ⊙ ∈ ⊙
(cid:10) (cid:11) (cid:10) (cid:0) (cid:1) (cid:11)
KMS STATES 5
Let X Y be the completion of X Y with respect to this inner product (2.4).
A A
⊗ ⊙
Lemma 2.16 of [39] implies that (2.4) extends to a right A-valued inner product on
X Y as well. Thus X Y is a right Hilbert A-module.
A A
⊗ ⊗
For every S (X), [40, Lemma I.3] gives a unique operator S 1 (X Y)
Y A
∈ L ⊗ ∈ L ⊗
such that S 1 (x y) = S(x) y for x y X Y. Thus the map a ϕ (a) 1
Y A X Y
⊗ ⊗ ⊗ ⊗ ∈ ⊗ 7→ ⊗
gives a left action of A by adjointable operators on X Y. Thus X Y is a
A A
⊗ ⊗
right Hilbert A–A bimodule which we call the balanced tensor product of X and Y.
Throughout, we use x y for elements of X Y and we write x y for elements of
⊙ ⊙ ⊗
both X Y and X Y.
A A
⊙ ⊗
2.2. Product systems of Hilbert bimodules. We use the conventions of [16] for
the basics of product systems of Hilbert bimodules. For convenience, we use the
following equivalent formulation from ([41, page 6]).
Suppose that P is a multiplicative semigroup with identity e, and let A be a C∗-
algebra. For each p P let X be a right Hilbert A–A bimodule and suppose that
p
∈
ϕ : A (X ) is the homomorphism which defines the left action of A on X . A
p p p
→ L
product system of right Hilbert A–A bimodules over P (or a product system with fibres
X over P) is the disjoint union X := X such that:
p p∈P p
(P1) The identity fibre X is the standard bimodule A .
e F A A
(P2) X is a semigroup and for each p,q P e the map (x,y) xy : X X
p q
∈ \{ } 7→ × →
X , extends to an isomorphism σ : X X X .
pq p,q p A q pq
⊗ →
(P3) The multiplications X X X and X X X satisfy
e p p p e p
× → × →
ax = ϕ (a)z, xa = x a for a X and x X .
p e p
· ∈ ∈
If each fibre X is essential, then we call X a product system of essential right Hilbert
p
A–A bimodules over P. We write 1 for the identity operator on the fibre X .
p p
The associativity of the multiplication in X implies that for p,q,r P,x X ,y
p
∈ ∈ ∈
X ,s X , we have
q r
∈
σ σ (x y) s = σ x σ (y s)
pq,r p,q p,qr q,r
⊗ ⊗ ⊗ ⊗
Let p,q P e and(cid:0)S (Xp). Th(cid:1)e isomor(cid:0)phism σp,q : Xp(cid:1) A Xq Xpq gives a
∈ \{ } ∈ L ⊗ →
homomorphism ιpq : (X ) (X ) defined by
p L p → L pq
ιpq(S) = σ (S 1 ) σ−1.
p p,q ◦ ⊗ Xq ◦ p,q
Suppose that P is a subsemigroup of a group G such that P P−1 = e . Then
∩ { }
p q p−1q P
≤ ⇔ ∈
defines a partial order on G. Following [32, Definition 2.1] and [8, Definition 6], we
say (G,P) is a quasi-lattice ordered group if every pair p,q G with a common upper
∈
boundinP hasaleast upperboundinP, which wedenoteby p q. Wewritep q =
∨ ∨ ∞
when p,q G have no common upper bound. A well-known example of a quasi-lattice
∈
ordered group is (Zk,Nk): for all m,n Nk, there is a least upper bound m n with
∈ ∨
i-th coordinate (m n) := max m ,n .
i i i
∨ { }
Let (G,P) be a quasi-lattice ordered group. A product system of right Hilbert A–A
bimodules over P is compactly aligned, if for all p,q P with p q < , S (X )
p
∈ ∨ ∞ ∈ K
and T (X ), we have ιp∨q(S)ιp∨q(T) (X ).
∈ K q p q ∈ K p∨q
2.3. C∗-algebras associated to product systems of Hilbert bimodules. Let P
be a semigroup with identity e, and let X be a product system of right Hilbert A–A
bimodules over P. Let ψ be a function from X to a C∗-algebra B. Write ψ for the
p
restriction of ψ to X . We call ψ a Toeplitz representation of X if:
p
6 ZAHRAAFSAR,ASTRIDAN HUEF, ANDIAINRAEBURN
(T1) Foreachp P e ,ψ : X B islinear, andψ : A B isahomomorphism,
p p e
∈ \{ } → →
(T2) ψ (x)∗ψ (y) = ψ ( x,y ) for p P, and x,y X ,
p p e p
h i ∈ ∈
(T3) ψ (xy) = ψ (x)ψ (y) for p,q P, x X , and y X .
pq p q p q
∈ ∈ ∈
Conditions (T1) and (T2) imply that (ψ ,ψ ) is a representation of the fibre X . Then
p e p
there is a homomorphism ψ(p) : (X ) B such that ψ(p)(Θ ) = ψ (x)ψ (y)∗.
p x,y p p
K →
The Toeplitz algebra (X) is generated by a universal Toeplitz representation of
T
X, say ω. Proposition 2.8 of [15] says that there is such an algebra (X). If T is a
T
Toeplitz representation of X in a C∗-algebraB, then we write T for the representation
∗
of (X) in B such that T ω = T.
∗
T ◦
A Toeplitz representation ψ of X is Cuntz–Pimsner covariant if
(2.5) ψ (a) = ψ(p)(ϕ (a)) for all p P,a ϕ−1( (X )).
e p ∈ ∈ p K p
The Cuntz–Pimsner algebra (X) is the quotient of (X) by the ideal
O T
(2.6) ω(a) ω(p)(ϕ (a)) : p P,a ϕ−1( (X )) .
− p ∈ ∈ p K p
Let (G,P) be a q(cid:10)uasi-lattice ordered group and suppose that(cid:11) X is a product sys-
tem of essential right Hilbert A–A bimodules over P. Suppose that ψ is a Toeplitz
representation of X on a Hilbert space . For each p P, let αψ be the map αψp,ψ0
H ∈ p
associated to the representation (ψ ,ψ ) of the fibre X as in Lemma 2.1. A Toeplitz
p 0 p
representation ψ of X on a Hilbert space is Nica covariant if for every p,q P, we
H ∈
have
αψ (1) if p q <
αψ(1)αψ(1) = p∨q ∨ ∞
p q 0 otherwise.
(
It follows from Lemma 2.1 that if each fibre in X has a Parseval frame, then a Toeplitz
representation ψ is Nica covariant if and only if for every p,q P, we have
∈
ψ(p∨q)(1 ) if p q <
(2.7) ψ(p)(1 )ψ(q)(1 ) = p∨q ∨ ∞
p q
0 otherwise.
(
Fowler showed in [16, Proposition 5.6] that we can extend the notion of Nica co-
variance to representations in C∗-algebras. A Toeplitz representation ψ of X in a
C∗-algebra B is Nica covariant if and only for every p,q P, S (X ), and
p
∈ ∈ K
T (X ), we have
q
∈ K
ψ(p∨q) ιp∨q(S)ιp∨q(T) if p q <
ψ(p)(S)ψ(q)(T) = p q ∨ ∞
0 otherwise.
( (cid:0) (cid:1)
The Nica-Toeplitz algebra (X)1 is the C∗-algebra generated by a universal Nica
NT
covariant representation of X, which in this paper we denote by ψ. It follows from
[16, Theorem 6.3] that
(X) = span ψ (x)ψ (y)∗ : p,q P,x X ,y X .
p q p q
NT { ∈ ∈ ∈ }
The Cuntz–Pimsner algebra (X) is by definition a quotient of (X). There is
O T
another Cuntz–Pimsner algebra (X) in [41], which is directly defined as a quotient
NO
of (X). In the product systems considered here, the left action of A on each fibre
NT
is by compact operators, and hence Cuntz–Pimsner covariance implies Nica covariance
[16, Proposition 5.4]. Also since the left action is injective, by [41, Proposition 5.1], the
Cuntz–Pimsner covariance considered in [41] is equivalent to Fowler’s Cuntz–Pimsner
covariance given at (2.5). Therefore by [41, Remark 3.14] the two Cuntz–Pimsner
1In Fowler’s paper the Nica-Toeplitz algebra is denoted by cov(X).
T
KMS STATES 7
algebras coincide. But we found it easier to work with (X). The next lemma shows
O
that we can still express (X) as a quotient of (X).
O NT
Lemma 2.2. Let (G,P) be a quasi-lattice ordered group, and let X be a compactly
aligned product system of right Hilbert A–A bimodules over P. Suppose that ev-
ery Cuntz–Pimsner-covariant representation of X is a Nica-covariant representation.
Then (X) is the quotient of (X) by the ideal
O NT
(2.8) ψ (a) ψ(p)(ϕ (a)) : p P,a ϕ−1( (X )) .
e − p ∈ ∈ p K p
(cid:10) (cid:11)
Proof. Let := kerπ : π is a Cuntz–Pimsner-covariant representation of X and
∗
I
let := kerθ : θ is a Nica-covariant representation of X . Let q be the quo-
∗ NT
J { T(cid:8) } (cid:9)
tient map (X) (X)/ . A standard argument shows that ( (X)/ ,q ω) is
NT
TT → T J T J ◦
universal for Nica-covariant representations, and hence it is canonically isomorphic to
( (X),ψ). Similarly, (X) is isomorphic to (X)/ . In particular, is the ideal
NT O T I I
of (2.6). Since every Cuntz–Pimsner-covariant representation of X is also a Nica-
covariant representation, . An application of the third isomorphism theorem in
J ⊆ I
algebra gives a quotient map q : (X) (X) such that kerq = / . We have
NT → O I J
(2.9) / = i+ : i = q (i) : i .
NT
I J J ∈ I ∈ I
Now plugging the generating(cid:8)elements ω(a(cid:9)) (cid:8)ω(p)(ϕ (a)) of (cid:9) into (2.9) and using
p
− I
q ω = ψ, we obtain / = ψ (a) ψ(p)(ϕ (a)) : p P,a ϕ−1( (X )) . Thus
N(TX◦) is the quotient of I J(X) bye the−ideal (2.p8) ∈ ∈ p K p (cid:3)
O NT (cid:8) (cid:9)
Let X be a product system of right Hilbert A–A bimodules over Nk. A standard
argument using the universal property shows that there is a strongly continuous gauge
action γ : Tk (X) such that for each x X we have γ (x) = zmx (in multi-
m z
→ NT ∈
index notation, zm = k zmi for z = (z ,...,z ) Tk and m Zk). Since γ fixes
i=1 i 1 k ∈ ∈
the elements of the set in (2.8), it induces a natural gauge action of Tk on (X). Now
Q O
we can lift these actions to actions of the real line R2.
2.4. The Fock representation. Let P be a semigroup with identity e and suppose
that X is a product system of right Hilbert A–A bimodules over P. Define r : X P
→
by r(x) := p for x X . Following [16], we write F(X) = X and call it the
∈ p p∈P p
Fock module. Fowler shows in [16, page 340] that for x X there is an adjointable
operator T(x) on F(X) determined by T(x)( x ) = ∈(xxL). The adjoint T(x)∗ is
p p
zero on any summand X for which p / r(x)P. If p r(x)P, say p = r(x)q for some
p
q P, then there is an isomorphism σ ∈ : XL ∈XL X , and the adjoint T(x)∗
r(x),q r(x) A q p
∈ ⊗ →
is given by
(2.10) T(x)∗ σ (y z) = x,y z.
r(x),q
⊗ h i·
Furthermore, T is a Toeplitz rep(cid:0)resentation of(cid:1)X called the Fock representation.
Let X be a compactly aligned product system of right Hilbert A–A bimodules over
Nk and suppose that the left action of A on each fibre is by compact operators. Then
the homomorphism T : (X) (F(X)) induced from the Fock representation is
∗
NT → L
faithful (see [19, Remark 4.8]).
2Togetthe actionofR on (X),we couldalsoapply the argumentofthe paragraphbefore[19,
Lemma 3.2] to the homomorpNhiTsm N :Zk (0, ) defined by N(n)= ki=1niri
→ ∞
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8 ZAHRAAFSAR,ASTRIDAN HUEF, ANDIAINRAEBURN
2.5. KMS states. Let (A,R,α) be a C∗-algebraic dynamical system. An element
a A is analytic if t α (a) is the restriction of an entire function z α (a) on C.
t z
∈ 7→ 7→
Following recent conventions [27, 21, 1, 22], we say a state φ of (A,R,α)is a KMS state
with inverse temperature β (or a KMS state) if φ(ab) = φ(bα (a)) for all analytic
β iβ
elements a,b. A state φ is a KMS state if it is the weak∗ limit of a sequence of KMS
∞ βi
states as β (see [7]). We distinguish between the ground states and the KMS
i ∞
→ ∞
states. But in older literature (for example in [4, 34]), there was no such a distinction.
For us, ground states are those for which z φ(aα (b)) is bounded in the upper-half
z
7→
plane for all analytic elements a,b. The argument of [27, page 19] shows that it is
enough to check both the ground state condition and the KMS condition on a set of
analytic elements which span a dense subalgebra of A.
2.6. Topological graphs. A topological graph E = (E0,E1,r,s) consists of locally
compact Hausdorff spaces E0 and E1, a continuous map r : E1 E0 and a lo-
→
cal homeomorphism s : E1 E0. Here E0 and E1 will always be compact. We
→
use the convention of [37] for paths in E: for example, if s(e) = r(f), then we
think of ef as a path of length 2. Since E0 and E1 are compact, the associated
graph correspondence X(E) = C(E1) is the right-Hilbert C(E0)–C(E0) bimodule
with module actions and inner product given by (a x b)(z) = a(r(z))x(z)b(s(z))
· ·
and x,y (z) = x(w)y(w) for a,b C(E0), x,y X(E) and z E1.
h i s(w)=z ∈ ∈ ∈
P
2.7. -commuting local homeomorphisms. Let f,g be commuting maps on a set
∗
Z. Following [2] and [11, 10] we say f,g -commute, if for every x,y Z satisfying
§ ∗ ∈
f(x) = g(y), there exists a unique z Z such that x = g(z) and y = f(z). The
∈
following diagram illustrates this property:
z
f g
y x
g f
f(x) = g(y)
We also say that a family of maps -commute if every two of them -commute.
∗ ∗
Given -commuting maps f,g,h on a space Z, Proposition 10.2 of [11] implies that fi
∗
and gj -commute for i,j N. Lemma 1.3 of [43] shows that f and g h -commute.
∗ ∈ ◦ ∗
Remark 2.3. Let h ,...,h be -commuting local homeomorphisms ona space Z and
1 k
∗
take m,n Nk. Throughout we write m n for the element of Nk with (m n) :=
i
∈ ∧ ∧
min m ,n . We also define
i i
{ }
hm := hm1 hmk.
1 ◦···◦ k
Observe that if m n = 0, then the local homeomorphisms appearing in hm1 hmk
∧ 1 ◦···◦ k
do not appear in hn1 hnk. Then the argument of the previous paragraph shows
1 ◦···◦ k
thathm andhn -commute. Theconditionm n = 0hereisvery crucial. Inparticular,
∗ ∧
-commuting local homeomorphisms need not -commute with themselves. Thus hm
∗ ∗
and hn may not -commute in general.
∗
KMS STATES 9
2.8. Measures. All the measures we consider here are positive in the sense that they
take values in [0, ). We write M(Z) for the set of finite Borel measures on Z.
+
∞
Some of the measures used here are defined by linear functionals on C(Z). Then by
the Riesz representation theorem (see [15, Corollary 7.6] for example) these measures
are automatically regular. A probability measure is a Borel measure with total mass
1.
3. A product system associated to a family of local homeomorphisms
In [1, Lemma 5.2] we proved that for a local homeomorphism f and the associated
graph correspondence X(E), there is an isomorphism from X(E) X(E) onto the
A
⊗
graph correspondence associated to f f. The next lemma generalises this to graph
◦
correspondences of two different local homeomorphisms. There is also a similar result
in the dynamics arising from graph algebras (see [5, Proposition 3.2]).
Lemma 3.1. Let f,g be surjective local homeomorphisms on a compact Hausdorff
space Z. Let A := C(Z) and suppose that X(E ), X(E ) and X(F) are the graph
1 2
correspondences related to the topological graphs E = (Z,Z,id,f), E = (Z,Z,id,g),
1 2
and F = (Z,Z,id,g f). Then there is an isomorphism σ from X(E ) X(E )
f,g 1 A 2
◦ ⊗
onto X(F) such that
(3.1) σ (x y)(z) = x(z)y(f(z)) for all z Z.
f,g
⊗ ∈
Proof. Defineσ : C(Z) C(Z) C(Z)byσ(x,y)(z) = x(z)y(f(z))forallx,y C(Z).
× → ∈
Clearly σ is bilinear. Taking y = 1 implies that σ is surjective. Then the universal
property of the algebraic tensor product gives a unique surjective linear map σ˜ :
C(Z) C(Z) C(Z)satisfying σ˜(x y)(z) = x(z)y(f(z)) forall x y C(Z) C(Z).
⊙ → ⊙ ⊙ ∈ ⊙
Since σ˜ vanishes on the elements of the form (2.3), it induces a surjective linear map
σ : C(Z) C(Z) C(Z) satisfying (3.1).
f,g A
⊙ →
To show that σ preserves the actions, let x y C(Z) C(Z), a C(Z) and
f,g A
⊗ ∈ ⊙ ∈
z Z. It follows from (3.1) that
∈
σ x y a (z) = x(z)(y a)(f(z)) = x(z)y(f(z))a(g f(z))
f,g
⊗ · · ◦
= σ (x y)(z)a(g f(z)) = σ (x y) a (z).
(cid:0) (cid:1) f,g f,g
⊗ ◦ ⊗ ·
Similarly for the left action, we have σ a (x y) (z) =(cid:0) a σ (x (cid:1)y) (z).
f,g f,g
· ⊗ · ⊗
To see that σ preserves the inner products, let x y,x′ y′ C(Z) C(Z).
f,g A
(cid:0) (cid:1) ⊗ (cid:0) ⊗ ∈ (cid:1) ⊙
Since all the range maps are the identity and the source maps are f,g, and g f,
◦
respectively, we have
σ (x y),σ (x′ y′) (z) = σ (x y)(w)σ (x′ y′)(w)
f,g f,g f,g f,g
⊗ ⊗ ⊗ ⊗
g◦f(w)=z
(cid:10) (cid:11) X
= x(w)y(f(w))x′(w)y′(f(w))
g◦f(w)=z
X
= x(w)x′(w) y(v)y′(v)
g(Xv)=z(cid:16)f(Xw)=v (cid:17)
= x,x′ (v)y(v)y′(v)
h i
g(v)=z
X
= y(v) x,x′ y′ (v)
h i·
g(v)=z
X (cid:0) (cid:1)
= y, x,x′ y′ (z)
h i·
(cid:10) (cid:11)
10 ZAHRAAFSAR,ASTRIDAN HUEF, ANDIAINRAEBURN
= x y,x′ y′ (z).
⊗ ⊗
It follows that σf,g is an isometry on C(cid:10) (Z) A C(Z(cid:11)), and hence it extends to an
isomorphism σ of X(E ) X(E ) onto X(⊙F) which satisfies (3.1). (cid:3)
f,g 1 A 2
⊗
Now we can prove an analogue of [5, Proposition 3.7] to build a product system.
Proposition 3.2. Let h ,...,h be surjective commuting local homeomorphisms on a
1 k
compact Hausdorff space Z. For each m Nk, let X be the graph correspondence
m
∈
associated to the topological graph (Z,Z,id,hm). Suppose that X := X and
m∈Nk m
A := C(Z). Let σ : X X X be the isomorphism obtained by applying
m,n m A n m+n
⊗ → F
Lemma 3.1 with the local homeomorphisms hm,hn. Then X is a compactly aligned
product system of essential right Hilbert A–A bimodules over Nk with the multiplication
given by xy := σ (x y) for x X ,y Y , so that
m,n m n
⊗ ∈ ∈
(3.2) (xy)(z) = x(z)y(hm(z)) for z Z.
∈
The left action of A on each fibre X is by compact operators.
m
Proof. An easy computation using (3.2) shows that X is a semigroup. Since the source
map in the fibre X is the identity, X = A and therefore (P1) holds. Lemma 3.1
0 0 A A
gives (P2). Let a A and x X . Then (P3) follows from:
m
∈ ∈
ax(z) = a(z)x(z) = (a x)(z) and xa(z) = x(z)a(hm(z)) = (x a)(z).
· ·
To show that the fibre X is essential, notice that A = C(Z) is unital with the
m
identity 1 : Z C defined by 1 (z) = 1 for all z Z. Since the left action is
C(Z) C(Z)
→ ∈
by pointwise multiplication, ϕ (1 )x = x for all x X . Thus X is essential.
m C(Z) m m
∈
To see that the left action of A on the fibre X is by compact operators, choose an
m
open cover U : 1 j d of Z such that hm is injective and choose a partition
{ j ≤ ≤ } |Uj
of unity ρ subordinate to U . Define ξ := √ρ . We aim to show that for each
j j j j
{ } { }
a A, the left action of a on the fibre X is by d Θ .
∈ m j=1 a·ξj,ξj
Take x X and z Z, we have
m
∈ ∈ P
d d d
Θ (x) (z) = (a ξ ) ξ ,x (z) = (a ξ )(z) ξ ,x (hm(z))
a·ξj,ξj · j ·h j i · j h j i
(cid:16)Xj=1 (cid:17) Xj=1 (cid:0) (cid:1) Xj=1
d
= a(z)ξ (z) ξ (w)x(w).
j j
j=1 hm(w)=hm(z)
X X
Since hm is injective on each suppξ ,
j
d d d
Θ (x) (z) = a(z)ξ (z)ξ (z)x(z) = a(z)x(z) ξ (z) 2 = a(z)x(z).
a·ξj,ξj j j | j |
(cid:16)Xj=1 (cid:17) Xj=1 Xj=1
Thus d Θ = ϕ (a). Sincetheleftactiononeachfibreisbycompactoperators,
j=1 a·ξj,ξj m
[16, Proposition5.8]impliesthatX isacompactlyalignedproductsystemover Nk. (cid:3)
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4. KMS states and the subinvariance relation
Suppose that φ is a KMS state on the Toeplitz algebra of a directed graph or a
k-graph, and that p are the projections associated to the vertices of the graph. We
v
{ }
know from [21, 22] that the KMS condition implies that the vector φ(p ) satisfies
v
an inequality, which is known in Perron-Frobenius theory as a subinvariance relation.
(cid:0) (cid:1)
The constructions of KMS states in [21, 22] depend crucially on being able to find
the general solution of this relation. By drawing an analogy between the continuous
