GIBBS STATES FOR AF-ALGEBRAS
Valentin Ya. Golodets
Sergey V. Neshveyev
Institute for Low Temperature Physics & Engineering, Lenin Ave 47, Kharkov 310164, Ukraine,
tel. 38(0572) 30-85-85, fax (0572) 322370, e-mail: golodets@ilt.kharkov.ua
Abstract
We consider a special class of C ∗ -systems containing asymptotically abelian binary
shifts and shifts of Temperley-Lieb algebras. We study Gibbs states for these systems
corresponding to potentials with finite range interaction, and obtain the same results
as well-known Araki’s results for a one-dimensional quantum lattice. In particular, it
is proved that a Gibbs state in the infinite volume is a translation invariant KMS-state
having the exponential uniform clustering property. Entropic properties of the Gibbs
states are also discussed. This allows, in particular, to construct new examples of
quantum K-systems.
I
Introduction
In [1] Araki studied a one-dimensional infinite quantum spin lattice system with a finite
range interaction. He showed that the Gibbs state φ of the system in the infinite volume
is a limit of local Gibbs states, is a factor-state of the UHF-algebra A corresponding to the
quantum spin lattice system, is invariant under time and lattice translation, satisfies the
KMS boundary condition and has the exponential uniform clustering property.
Developing the transfer matrix technique in a fashion analogous to [2] Araki introduced
an auxiliary operator L acting on A, and found an eigenstate ν for L. It turned out that
the Gibbs state φ is a perturbation of the state ν. To realize this approach Araki developed
the Tomonaga-Shwinger-Dyson perturbation theory to the case of UHF-algebras (see [3],[4]
for a more detailed exposition and applications).
It is a natural problem to extend Araki’s theory to a more general class of AF-algebras.
We would like to include in this class such important algebras as Temperley-Lieb algebras
and asymptotically abelian binary shifts. So we will consider quadruples (A, {A[n,m] }, τ, γ),
where A is a unital C ∗ -algebra, A[n,m] , n, m ∈ Z, n ≤ m, is a finite-dimensional subalgebra
of A such that A[n,m] ⊂ A[n0 ,m0 ] for n0 ≤ n ≤ m ≤ m0 , ∪n A[−n,n] is dense in A, τ is a faithful
trace on A, and γ is a τ -preserving automorphism of A such that γ(A[n,m] ) = A[n+1,m+1] .
Moreover, we suppose that the trace τ has some multiplicativity property, and the local
algebras satisfy Popa’s commuting square condition (see Section II below). Nevertheless
there are difficulties on this way. For example, we cannot define an analogue of Araki’s
1
operator L on the whole algebra A, since in general the algebra cannot be ”cut” into two
commuting pieces as in [1]. Fortunately, it is possible to define such an operator on the
subalgebra A[0,∞) as in Ruelle’s paper [2]. Then basing on Araki’s perturbation technique
we can prove the existence of an eigenstate ν on A[0,∞) . The Gibbs state φ on A can be
restored from this state ν. Again, as in [1], the Gibbs state is the limit of local Gibbs states,
invariant under γ, satisfies the KMS boundary condition and has the exponential uniform
clustering property. These results are proved in Section III.
The constructed Gibbs state is a σt -KMS state (β = 1), where σt is the time evolution.
For a one-dimensional quantum lattice it is a unique σt -KMS state (see [5], [4]). The same
result holds for binary shifts, or for the C ∗ -tensor product of a binary shift and a quantum
spin lattice system. To prove this one can apply the same arguments as in [5], or [4] (see
Remark III.13), but in general this problem apparently requires new arguments.
The present research has been undertaken to construct new examples of quantum Kolmogorov systems in the sense of Narnhofer and Thirring [6]. Indeed, let φ be a Gibbs state,
πφ the GNS-representation of A corresponding to φ. Then (M, φ, γ), where M = πφ (A)00 ,
is a quantum K-system. We deduce this result in Section IV from the proved clustering
property and from the sufficient condition for the K-property in our paper [7].
In Section IV we also compute the mean entropy of a Gibbs state for certain C ∗ -dynamical
systems. In particular, for a binary shift and for a Temperley-Lieb algebra of index≤ 4. It is
a very interesting problem to investigate when this mean entropy is equal to the dynamical
entropy hφ (γ). But this problem has not been solved even for a quantum spin lattice system
[8] (see also Remark IV.7 below).
It is important to note that Hiai and Petz [9] considered quantum statistical thermodynamics in AF C ∗ -systems. They studied, in particular, such problems as the Gibbs conditions, the variational principle for states, their relations with the KMS-conditions, and
presented a lot of interesting examples. But Araki’s approach to Gibbs states was not considered.
II
Notations and Examples
We consider quadruples (A, {A[n,m] }n≤m , τ, γ), n, m ∈ Z, where A is a unital C ∗ -algebra,
A[n,m] is a finite-dimensional C ∗ -subalgebra of A such that A[n0 ,m0 ] ⊂ A[n00 ,m00 ] for n00 ≤ n0 ≤
m0 ≤ m00 and ∪n A[−n,n] is dense in A, τ a faithful trace of A, γ a τ -preserving automorphism
of A such that γ(A[n,m] ) = A[n+1,m+1] . For any subset Λ of Z, we denote by AΛ the C ∗ subalgebra of A generated by A[k,n] , [k, n] ⊂ Λ, and write An instead of A[n,n] .
Throughout the paper we suppose that the following conditions are satisfied.
Assumptions II.1
(i) There exists n0 ≥ 0 such that A(−∞,0] and A[n0 +1,∞) commute.
(ii) τ (xy) = τ (x)τ (y) for x ∈ A(−∞,0] , y ∈ A[n0 +1,∞) .
2
(iii) There exists a τ -preserving conditional expectation E: A[0,∞) → A[1,∞) , and
E(A[0,n] ) = A[1,n] for any n ≥ 1. Equivalently,
A[0,n]
∪
A[1,n]
⊂ A[0,n+1]
∪
⊂ A[1,n+1]
is a commuting square [10], [11] for any n.
Besides a one-dimensional quantum spin lattice system we have the following standard
examples.
Example II.2 Automorphism θλ [11].
Let λ ∈ {(4 cos2 πn )−1 }n>3 ∪ (0, 14 ], A the C ∗ -algebra generated by projections en , n ∈ Z,
satisfying ei ej = ej ei for |i − j| ≥ 2, and ei ei±1 ei = λei , A[n,m] = C ∗ (en , en+1 , . . . , em ), τ the
λ-Markov trace on A, i. e. τ (wen ) = λτ (w) for any w ∈ A(−∞,n−1] , γ(en ) = en+1 .
Example II.3 Canonical shift on the tower of relative commutants [12].
Let M be a II1 -factor, N ⊂ M its subfactor of finite index, N ⊂ M ⊂e1 M1 ⊂e2 M2 ⊂ . . .
0
the Jones tower, . . . ⊂e−2 M−2 ⊂e−1 M−1 = N ⊂e0 M0 = M a tunnel, A[n,m] = M2n
∩ M2m ,
A = ∪n A[−n,n] , τ the restriction of the trace on ∪n Mn . For any n ∈ N, the triple M−n ⊂
Mn ⊂ M3n is a basic construction in a canonical way. In particular, a (not necessarily
0
τ -preserving) antiautomorphism γn (x) = Jn x∗ Jn of M−n
∩ M3n is defined, where Jn is the
2
0
canonical conjugation on L (Mn ). Then γ(x) = γn+1 (γn (x)) for x ∈ M−n
∩ M3n .
Example II.4 Asymptotically abelian binary shift [13], [14].
Let X be a non-empty finite subset of N, a(x) the characteristic function of X, A the
∗
C -algebra with unit generated by symmetries sn , n ∈ Z, satisfying si sj = (−1)a(|i−j|) sj si ,
A[n,m] = C ∗ (sn , sn+1 , . . . , sm ), τ (w) = 0 for any non-empty word w in sn ’s, γ(sn ) = sn+1 .
It is worth to note that the C ∗ -tensor product of systems satisfying Assumptions II.1
also satisfies these assumptions.
By an interaction potential we mean a mapping X 7→ Φ(X) ∈ AX defined on finite subsets
X of Z such that Φ(X) = Φ(X)∗ , Φ(X + n) = γ n (Φ(X)). We suppose that Φ(X) = 0 if X is
not within an interval of length r > 0. Following Araki, we denote by U (Λ) the Hamiltonian
for a finite interval Λ,
X
U (Λ) =
Φ(X).
X⊂Λ
We write U (a, b) instead of U ([a, b]).
For a finite interval Λ, the local Gibbs state is
φΛ (Q) =
τ (Qe−U (Λ) )
.
τ (e−U (Λ) )
3
III
Existence and clustering properties of Gibbs states
Our main results in this section are as follows.
Theorem III.1
(i) The limit P (Φ) = lim
1
log τ (e−U (a,b) ) exists and is finite.
b−a→∞ b − a
(ii) For any Q ∈ A, there exists the limit φ(Q) = a→−∞
lim φ[a,b] (Q).
b→+∞
Moreover, there exist C, q > 0 such that
|φ[−n+a,b+n] (Q) − φ(Q)| ≤ Ce−qn ||Q|| f or Q ∈ A[a,b] , n ∈ N.
Theorem III.2 The state φ is uniformly exponentially clustering, i. e. there exist C, q > 0
such that
|φ(Q1 Q2 ) − φ(Q1 )φ(Q2 )| ≤ Ce−qn ||Q1 || ||Q2 ||
for Q1 ∈ A(−∞,−n] , Q2 ∈ A[n,∞) , n ∈ N.
We shall call the state φ the Gibbs state corresponding to the interaction potential Φ(X).
The proofs of Theorems follow closely the work by Araki [1]. We introduce an auxiliary
operator L and found an eigenstate ν for this operator. Then the Gibbs state can be restored
from ν. The main difference of our work from Araki’s one is that the operator L can not
be defined on the whole algebra A, but only on A[0,∞) . So ν is a state on A[0,∞) , and the
perturbation argument by Araki can’t be applied. Our arguments are closer to Ruelle’s work
[2] (see also [15]).
Following Araki, for Q ∈ A[0,∞) , we define
||Q||l =
inf
Ql ∈A[0,l]
|||Q|||N,x = ||Q|| +
||Q − Ql ||,
∞
X
||Q||n xn (x > 1, N ∈ N),
n=N
A(x) = {Q ∈ A[0,∞) | |||Q|||1,x < ∞}.
Then A(x) is a Banach space with respect to the norm ||| |||1,x . Every ||| |||N,x is an equivalent
norm on A(x). For Q > 0, we also define
αl (Q) =
inf
Ql ∈A[0,l] , Ql >0
||Q − Ql || ||Q−1
l ||,
α(Q) = ||Q|| ||Q−1 || =
4
l.u.b. Spec Q
.
g.l.b Spec Q
Lemma III.3 For Q > 0, we have
α(Q)−1
(i) αl (Q) ≤ α(Q)+1
<1;
(ii) there exists Ql ∈ A[0,l] , Ql > 0, such that αl (Q) = ||Q − Ql || ||Q−1
l ||, then
(1 − αl (Q))α(Q) ≤ ||Q|| ||Q−1
l || ≤ (1 + αl (Q))α(Q) ;
(iii) if ||Q||l ||Q−1 || < 1, then αl (Q) ≤
(iv) ||Q||l ||Q−1 || ≤
||Q||l ||Q−1 ||
1−||Q||l ||Q−1 ||
;
αl (Q)
.
1−αl (Q)
Proof. Cf. [1, Lemma 3.9]. It is worth only to note that if ||Q||l ||Q−1 || < 1, then, for
Ql = Q∗l ∈ A[0,l] , ||Q − Ql || = ||Q||l , we have Ql > 0. Indeed,
|| ||Q||1 − Ql || ≤ || ||Q||1 − Q|| + ||Q||l = ||Q|| − ||Q−1 ||−1 + ||Q||l < ||Q||.
2
Let us also define
X Φ(I)
Φ=
I⊂[0,r]
n(I)
X
and H(I) =
γ n (Φ),
n : [n,n+r]⊂I
where n(I) is the number of translates I +a of I that are still in [0, r]. H(a, b) = H([a, b]) and
U (a, b) differ only near the two ends. More precisely, there exist ∆− ∈ A[0,r] and ∆+ ∈ A[−r,0]
such that, for b − a > 2r, we have
U (a, b) = H(a, b) + γ a (∆− ) + γ b (∆+ ).
For any subset I ⊂ Z, the sequence Ad exp(itH(I ∩ [−n, n])) pointwise converges to a
strongly continuous one-parameter automorphism group σtI on A, and every Q ∈ ∪n A[−n,n]
is a σtI -analytic element (see [1], Theorem 4.2, or [4], Theorem 6.2.4). We write σt instead
of σtZ . For Q ∈ ∪n A[−n,n] , λ ∈ R, let us also consider the perturbed dynamics (σtI )Q (see [4],
Section 5.4.1). These dynamics are related via a σtI -cocycle ΓQ
t . Following Araki, an analytic
continuation of this cocycle at the point t = −iλ is denoted by E(λQ; λH(I)),
E(λQ; λH(I)) = 1 +
∞ Z
X
n=1 0
1
ds1
Z
0
s1
ds2 . . .
Z
sn−1
0
I
I
dsn σ−iλs
(λQ) . . . σ−iλs
(λQ)
n
1
(see [1], Definition 5.1, or [4], p.149). If I is finite, then
E(λQ; λH(I)) = eλQ+λH(I) e−λH(I) .
Lemma III.4 There exist constants q > 0 and Cn > 0, n ∈ N, such that
||E(Q; λH(I))|| ≤ eCn ||Q||
and
h
||E(Q; λH(I)) − E(Q; λH(I ∩ [−N, n + N ]))|| ≤ Cn q
1+
N
r+n0
h
i
N
r+n0
i eCn ||Q||
!
for Q ∈ A[0,n] , n ≥ r, N > 0, |λ| ≤ 1, I ⊂ Z, where n0 is defined in Assumptions II.1.
5
Proof. Cf. [1], Theorem 4.2 and Lemma 5.2. We would only mention that if
[γ jm (Φ), [. . . , [γ j1 (Φ), Q] . . .]] 6= ∅ then [jk , jk + r] has non-empty intersection with
[−n0 , n + n0 ] ∪ ∪ [jl − n0 , jl + r + n0 ]
1≤l<k
for k = 1, . . . , m. But [jk , jk + r] ∩ [a − n0 , b + n0 ] 6= ∅ iff [jk , jk + r + n0 ] ∩ [a, b + n0 ] 6= ∅.
So the estimates by Araki hold with the replacement of r by r + n0 .
2
We define the transfer operator L: A[0,∞) → A[0,∞) as follows: L(Q) = γ −1 E(K ∗ QK),
where K = E(− 21 Φ; − 12 H(1, ∞)) and E: A[0,∞) → A[1,∞) is the τ -preserving conditional
expectation.
Lemma III.5
(i) L is a faithful completely positive mapping. Moreover, if Q ≥ 0 is invertible, L(Q) is
invertible too.
(ii) Ln (Q) = γ −n En (Kn∗ QKn e−H(0,n−n0 −1) ), where En : A[0,∞) → A[n,∞) is the τ -preserving
conditional expectation,
 P
 E − 1 n−1 γ i (Φ); − 1 H(n, ∞) , n ≤ r + n0
2
2 j=0
;
Kn =
 E − 1 γ n−1 (Ψ); − 1 [H(0, n − n − 1) + H(n, ∞)] , n > r + n
0
0
2
2
Ψ=
0
X
γ j (Φ).
j=1−r−n0
Proof. We follow the proof of Lemma 6.3 in [1].
(i) is trivial, since E is faithful and K is invertible (K −1 = E( 21 Φ; 12 H(1, ∞))∗ ).
(ii) Define Kn0 = Kγ(K) . . . γ n−1 (K). Then Ln (Q) = γ −n En (Kn0∗ QKn0 ). Indeed, for n = 1
it is true by definition. By induction,
0∗
0
Ln (Q) = γ −n+1 En−1 (Kn−1
L(Q)Kn−1
)
0
0
= γ −n+1 En−1 γ −1 E (Kγ(Kn−1
))∗ QKγ(Kn−1
)
= γ −n En (Kn0∗ QKn0 ),
since En |A[1,∞) = γEn−1 γ −1 and En E = En .
Using the identity E(Q1 ; Q2 + R)E(Q2 ; R) = E(Q1 + Q2 ; R), we obtain

Kn0

X
1 n−1
1
= E −
γ j (Φ); − H(n, ∞)
2 j=0
2



X
X0 −1
1 n−1
1 n−r−n
(for n > r + n0 ) = E −
γ j (Φ); − 
γ j (Φ) + H(n, ∞)
2 j=n−r−n0
2
j=0
6


X0 −1
1
1 n−r−n
×E −
γ j (Φ); − H(n, ∞)
2 j=0
2
1
1
= E − γ n−1 (Ψ); − [H(0, n − n0 − 1) + H(n, ∞)]
2
2
1
1
×E − H(0, n − n0 − 1); − H(n, ∞)
2
2
1
1 n−1
= E − γ (Ψ); − [H(0, n − n0 − 1) + H(n, ∞)]
2
2
− 12 H(0,n−n0 −1)
×e
,
since H(0, n − n0 − 1) commute with A[n,∞) . It remains to note that En (xy) = En (yx) for
any x ∈ A[0,n−n0 −1] and y ∈ A[0,∞) by Assumption II.1(i).
2
Lemma III.6 There exist constants C, q > 0 such that, for Q > 0,
(i) α(Ln (Q)) ≤ Cα(Q) ;
n
(ii) αl (L (Q)) ≤ C (α(Q)δl−r + αn+l (Q)) f or l ≥ r, where δl−r = q
1+
[ l−r+1
]
n +r
0
h
l−r+1
n0 +r
i
!
(see
Lemma III.4) ;
(iii) if |||Q|||M,x ||Q−1 || ≤ a, then α(Ln (Q)) ≤ C for any n ≥ N (a, M, x) for a constant
N (a, M, x) depending on a > 0, x > 1, M ∈ N.
Proof. Cf. [1, Lemma 6.4]. We give a proof to demonstrate that all the Assumptions II.1
are applied.
(i) Let Kn be as in Lemma III.5. By Lemma III.4, there exist a constant C such that
||Kn ||, ||Kn−1 || ≤ C.
Let pn = τ (e−H(0,n−n0 −1) ). Then by Assumption II.1(ii), En (e−H(0,n−n0 −1) ) = pn 1, so that
||(Kn∗ QKn )−1 ||−1 ≤
1 n
L (Q) ≤ ||Kn∗ QKn ||,
pn
hence α(Ln (Q)) ≤ C 4 α(Q).
(ii) Let Qn+l ∈ A[0,n+l] , Qn+l > 0, αn+l (Q) = ||Q − Qn+l || ||Q−1
n+l || (Lemma III.3(ii)),
1
1
K(n,l) = E − γ n−1 (Ψ); − [H(0, n − n0 − 1) + H(n, n + l)] ,
2
2
∗
Q0l = γ −n En (K(n,l)
Qn+l K(n,l) e−H(0,n−n0 −1) ). By Assumption II.1(iii), Q0l ∈ A[0,l] . Let us
compute ||Ln (Q) − Q0l || ||(Q0l )−1 ||.
By Lemma III.4,
−1
||K(n,l) ||, ||K(n,l)
|| ≤ C and ||Kn − K(n,l) || ≤ Cδl−r .
7
Hence
∗
∗
||Kn∗ QKn − K(n,l)
QK(n,l) || ≤ 2C 2 δl−r ||Q|| and ||K(n,l)
(Q − Qn+l )K(n,l) || ≤ C 2 ||Q − Qn+l ||.
Being a completely positive unital mapping, A[0,∞) 3 x 7→ p1n En (xe−H(0,n−n0 −1) ) has norm
one, so that
||Ln (Q) − Q0l || ≤ pn C 2 (2δl−r ||Q|| + ||Q − Qn+l ||).
(1)
∗
∗
−1
On the other hand, K(n,l)
Qn+l K(n,l) ≥ ||(K(n,l)
Qn+l K(n,l) )−1 ||−1 ≥ C −2 ||Q−1
n+l || . Hence
Q0l ≥
pn
−1
||Q−1
n+l || .
2
C
(2)
Using (1) and (2) we obtain
αl (Ln (Q)) ≤ ||Ln (Q) − Q0l || ||(Q0l )−1 || ≤ C 4 (2δl−r ||Q|| ||Q−1
n+l || + αn+l (Q)).
Since ||Q|| ||Q−1
n+l || ≤ 2α(Q) by Lemma III.3(i),(ii), the proof of (ii) is complete.
(iii) Let Qk ∈ A[0,k] , Qk > 0, ||Q − Qk || ||Q−1
k || = αk (Q),
1
1
Kn,l = E − γ n−1 (Ψ); − [H(n − l, n − n0 − 1) + H(n, ∞)] ,
2
2
∗
Q0 = γ −n En (Kn,l
Qk Kn,l e−H(0,n−n0 −1) ). Then as in the proof of (ii) we conclude that, for a
−1
constant C depending only on the potential, we have ||Kn,l ||, ||Kn,l
|| ≤ C and
||Ln (Q) − Q0 || ||(Q0 )−1 || ≤ C(α(Q)δl−r−n0 + αk (Q)).
(3)
We denote the right hand part of (3) by ∆. Then
Ln (Q) ≤ Q0 + ||Ln (Q) − Q0 || ≤ Q0 + ||Ln (Q) − Q0 || ||(Q0 )−1 ||Q0 ≤ (1 + ∆)Q0 ,
Ln (Q) ≥ Q0 − ||Ln (Q) − Q0 || ≥ Q0 − ||Ln (Q) − Q0 || ||(Q0 )−1 ||Q0 ≥ (1 − ∆)Q0 ,
so that
1+∆
α(Q0 ) whenever ∆ < 1.
(4)
1−∆
Since |||Q|||M,x ||Q−1 || ≤ a, we have α(Q) ≤ a and ||Q||k ||Q−1 || ≤ ax−k for k ≥ M ,
ax−k
hence (Lemma III.3(iii)) αk (Q) ≤ 1−ax
−k . We see that we can choose l ≥ r + n0 and k such
1
that ∆ ≤ 2 independently of Q. Let N (a, M, x) = k + l + n0 + 1. Then, for n ≥ N , Kn,l
α(Ln (Q)) ≤
1
1
and Qk commute. If p = τ (Qk e−H(0,n−n0 −1) ), the mapping y 7→ p1 En (Qk2 yQk2 e−H(0,n−n0 −1) ) is
completely positive and unital, so that
1
1
∗
∗
∗
≤ ||(Kn,l
Kn,l )−1 ||−1 ≤ En (Kn,l
Kn,l Qk e−H(0,n−n0 −1) ) ≤ ||Kn,l
Kn,l || ≤ C 2 ,
2
C
p
hence α(Q0 ) ≤ C 4 . By virtue of this inequality, the inequality (4) and the choice of N , we
obtain α(Ln (Q)) ≤ 3C 4 .
2
The operator ψ 7→ ψ(L(1))−1 ψL is defined on the state space of A[0,∞) . By the Schauder
theorem, there exists a state ν of A[0,∞) such that νL = λν with λ = ν(L(1)) > 0.
Let L = λ−1 L.
8
Corollary III.7 The sequence {Ln }∞
n=1 is bounded.
Proof. First let Q > 0 and α(Q) ≤ 2. By Lemma III.6(i), α(Ln (Q)) = α(Ln (Q)) ≤ 2C for a
constant C independent of Q. Then
||Ln (Q)|| = α(Ln (Q))||Ln (Q)−1 ||−1 ≤ 2Cν(Ln (Q)) = 2Cν(Q) ≤ 2C||Q||.
For arbitrary Q ≥ 0 let Q0 = Q + ||Q||1. Then α(Q0 ) ≤ 2. Hence
||Ln (Q)|| ≤ ||Ln (Q0 )|| + ||Q|| ||Ln (1)|| ≤ 6C||Q||.
So for any Q ∈ A[0,∞) , we have ||Ln (Q)|| ≤ 24C||Q||.
2
Proposition III.8 There exists an element h ∈ ∩ A(x) such that h > 0, ν(h) = 1, Lh = h.
x>1
Then
|||Ln (Q) − ν(Q)h|||1,x ≤ Cx e−qx n |||Q|||1,x f or any Q ∈ A(x)
for constants Cx , qx > 0 depending on x.
Proof. See Lemmas 6.5, 7.5, 7.6 in [1].
Step 1. Existence of h.
Let C be chosen as in the formulation of Lemma III.6, and δl be as defined there. Let X
be the norm-closure of the convex hull of {Ln (1)}∞
n=1 . The set X is L-invariant. We prove
that X is compact.
Cδl−r
, l ≥ r}. The set Y is compact, since
Let Y = {Q ∈ A[0,∞) | ||Q|| ≤ C, ||Q||l ≤ 1−Cδ
l−r
2Cδl−r
the compact ball of radius C in A[0,l] is a 1−Cδ
-net for Y . X is a subset of Y . Indeed, it
l−r
n
is enough to show that L (1) ∈ Y for any n. We have
||Ln (1)|| = α(Ln (1))||Ln (1)−1 ||−1 ≤ α(Ln (1))ν(Ln (1)) = α(Ln (1)) ≤ C and
||Ln (1)||l ≤
αl (Ln (1))
αl (Ln (1))
Cδl−r
n
−1 −1
||L
(1)
||
≤
≤
n
n
1 − αl (L (1))
1 − αl (L (1))
1 − Cδl−r
by Lemma III.6(i),(ii) and Lemma III.3(iv).
By the Schauder theorem, there exists h ∈ X such that Lh = h. We have h ≥ 0,
ν(h) = 1, and h ∈ ∩x>1 A(x), since Y ⊂ ∩x>1 A(x). Since
Ln (1) ≥
||Ln (1)||
ν(Ln (1))
1
≥
= ,
n
α(L (1))
C
C
we have h ≥ C −1 .
Step 2. Continuity of L|A(x) .
Consider Q = Q∗ , |||Q|||1,x ≤ 1. Let Q0 = 2 + Q. Then 1 ≤ Q0 ≤ 3 and ||Q0 ||l = ||Q||l for
any l. By Lemma III.6(i) we obtain α(L(Q0 )) ≤ 3C, and as in Step 1 we have
αl (L(Q0 ))
3
||L(Q )||l ≤
ν(Q0 ) ≤ (3C + 1)αl (L(Q0 )),
0
1 − αl (L(Q ))
2
0
9
(5)
where the latter inequality follows from Lemma III.3(i). By Lemma III.6(ii), for l ≥ r,
αl (L(Q0 )) ≤ C(3δl−r + αl+1 (Q0 )),
and by Lemma III.3(iii),
αl+1 (Q0 ) ≤
||Q0 ||l+1 ||(Q0 )−1 ||
||Q0 ||l+1
||Q0 ||l+1
≤
≤
,
1 − ||Q0 ||l+1 ||(Q0 )−1 ||
1 − ||Q0 ||l+1
1 − x−1
so that
||Q0 ||l+1
αl (L(Q )) ≤ C 3δl−r +
.
1 − x−1
!
0
Using (5), (6) and
P
l
(6)
||Q0 ||l xl ≤ 1 we obtain
|||L(Q0 )|||r,x = ||L(Q0 )|| +
∞
X
||L(Q0 )||l xl
l=r
∞
X
3
||Q0 ||l+1
≤ 3||L(1)|| + (3C + 1)
C 3δl−r +
xl
−1
2
1
−
x
l=r
!
∞
9C(3C + 1) X
3C(3C + 1)
≤ 3||L(1)|| +
δl−r xl +
.
2
2(x − 1)
l=r
In particular, |||L(1)|||r,x < ∞. Since |||L(Q)|||r,x ≤ 2|||L(1)|||r,x +|||L(Q0 )|||r,x , the continuity
of L|A(x) is proved.
Step 3. Convergence proof.
Let C be chosen as in the formulation of Lemma III.6. Then we take
a ≥ 4C
and chose M ∈ N such that
2Cax
−M
∞
X
1
1
1
≤ , Caδl−r ≤ for l ≥ M, 4C
δl−r xl ≤ .
4
4
4
l=M
Finally, we fix N ∈ N such that
8Cx−N ≤
1
and N ≥ N (a, M, x),
4
where N (a, M, x) is as in the formulation of Lemma III.6(iii).
We define an auxiliary operator Λ on A(x),
Λ(Q) = LN (Q) −
ν(Q)
.
2C
We state that
if Q > 0, |||Q|||M,x ||Q−1 || ≤ a, then Λ(Q) > 0, |||Λ(Q)|||M,x ||Λ(Q)−1 || ≤ a.
10
(7)
1
Suppose the statement is proved. Let qx = − N1 log(1 − 2C
) > 0. Then ν(Λ(Q)) =
1
−qx N
(1 − 2C )ν(Q) = e
ν(Q). In particular, for ν(Q) = 0, we have Λn (Q) = LN n (Q).
Consider Q = Q∗ ∈ A(x), |||Q|||M,x ≤ 1, ν(Q) = 0. Define Q1 = 2 + Q, Q2 = 2. Then
1 ≤ Q1 ≤ 3, |||Q1 |||M,x ||Q−1
1 || ≤ |||Q1 |||M,x ≤ 3 ≤ a. By (7),
|||Λn (Q1 )|||M,x ≤ a||Λn (Q1 )−1 ||−1 ≤ aν(Λn (Q1 )) = ae−qx N n ν(Q1 ) = 2ae−qx N n ,
analogously |||Λn (2)|||M,x ≤ 2ae−qx N n , so that
|||LN n (Q)|||M,x = |||Λn (Q)|||M,x = |||Λn (Q1 ) − Λn (Q2 )|||M,x ≤ 4ae−qx N n .
For arbitrary k ∈ N we obtain
k
k−[ k ]N
|||Lk (Q)|||M,x ≤ |||L|||M,xN 4ae−qx [ N ]N ≤ 4aeqx N (1 + |||L|||M,x )N e−qx k ,
where |||L|||M,x is the norm of the operator L on the Banach space (A(x), ||| |||M,x ) that is
finite by Step 2. Let C̃x = 4aeqx N (1 + |||L|||M,x )N . Then, for Q = Q∗ ∈ A(x), we have
|||Lk (Q) − ν(Q)h|||M,x = |||Lk (Q − ν(Q)h)|||M,x ≤ C̃x e−qx k |||Q − ν(Q)h|||M,x
≤ C̃x (1 + |||h|||M,x )e−qx k |||Q|||M,x ,
so that, for arbitrary Q ∈ A(x),
|||Lk (Q) − ν(Q)h|||M,x ≤ 2C̃x (1 + |||h|||M,x )e−qx k |||Q|||M,x .
It remains to prove (7). We have α(LN (Q)) ≤ C by Lemma III.6(iii) and the choice of
N . Hence
||LN (Q)||
ν(Q)
||LN (Q)||
≥
LN (Q) ≥
≥
.
N
α(L (Q))
C
C
In particular, Λ(Q) ≥
ν(Q)
2C
α(Λ(Q)) =
> 0 and
ν(Q)
||LN (Q)−1 ||
C
ν(Q)
2C
||LN (Q)−1 ||−1 − ν(Q)
2C
||LN (Q)|| −
≤
=
≤ 1. Further,
ν(Q)
||LN (Q)−1 ||
2C
ν(Q)
||LN (Q)−1 ||
2C
α(LN (Q)) −
1−
α(LN (Q))
a
= 2α(LN (Q)) ≤ 2C ≤
1
2
1− 2
(8)
and
−1
N
−1
N
N
−1 −1
||Λ(Q)||l ||Λ(Q) || = ||L (Q)||l ||Λ(Q) || = ||L (Q)||l ||L (Q) ||
≤ 2||LN (Q)||l ||LN (Q)−1 || ≤
11
2αl (LN (Q))
,
1 − αl (LN (Q))
ν(Q)
−
2C
!−1
(9)
where we have used Lemma III.3(iv) in the latter inequality. By Lemma III.6(ii),
αl (LN (Q)) ≤ C(α(Q)δl−r + αl+N (Q)).
||Q||l+N ||Q−1 ||
1−||Q||l+N ||Q−1 ||
Since ||Q||l+N ||Q−1 || ≤ ax−l−N and αl+N (Q) ≤
ax−M
Cαl+N (Q) ≤ C 1−ax−M ≤ 14 . We also have Cα(Q)δl−r ≤ Caδl−r ≤
αl (LN (Q)) ≤ 12 for any l ≥ M . Together with (9) this gives
(10)
by Lemma III.3(iii),
1
4
for l ≥ M , so that
||Λ(Q)||l ||Λ(Q)−1 || ≤ 4αl (LN (Q)).
Now it follows from (10) again that
||Q||l+N ||Q−1 ||
||Λ(Q)||l ||Λ(Q) || ≤ 4C(α(Q)δl−r + αl+N (Q)) ≤ 4Caδl−r + 4C
1 − ax−M
−1
≤ 4Caδl−r + 8C||Q||l+N ||Q−1 ||.
(11)
Finally, using (8), (11) and the choice of M and N , we obtain
−1
|||Λ(Q)|||M,x ||Λ(Q) || = α(Λ(Q)) +
∞
X
||Λ(Q)||l ||Λ(Q)−1 ||xl
l=M
∞
∞
X
X
a
+ 4Ca
δl−r xl + 8C
≤
||Q||l+N ||Q−1 ||xl
2
l=M
l=M
a a
≤
+ + 8Cax−N ≤ a.
2 4
2
Corollary III.9 For any x > 1, there exist Cx , qx > 0 such that
|||LN +n (Q) − ν(Q)h|||1,x ≤ Cx e−qx n ||Q||
∀Q ∈ A[0,N ] ∀n, N.
Proof. It is enough to prove that there exists a constant Cx independent of N such that
|||LN (Q)|||1,x ≤ Cx ||Q|| ∀Q ∈ A[0,N ] . As in the proof of Proposition III.8, Step 2, it suffices
to consider Q ∈ A[0,N ] , 1 ≤ Q ≤ 3. By the same arguments as given there, we have
α(LN (Q)) ≤ 3C and, for l ≥ r,
3C(3C + 1)
(3δl−r + αl+N (Q)).
2
But αl+N (Q) = 0 for any Q ∈ A[0,N ] , Q > 0, and we obtain
||LN (Q)||l ≤
|||LN (Q)|||1,x ≤ rxr−1 ||LN (Q)|| +
∞
X
||LN (Q)||l xl
l=r
≤ 3rxr−1 ||LN || +
∞
9C(3C + 1) X
δl−r xl .
2
l=r
From Corollary III.7 we conclude that the latter expression is bounded by a constant C̃x
independent of Q and N .
2
12
Lemma III.10 The sequence {νγ n }∞
n=1 converges to a state φ on A[0,∞) in norm. Moreover,
there exist C, q > 0 such that ||φ − νγ n || ≤ Ce−qn .
Proof. For n and N , N + n0 < n, let
1
1
Kn−N,n = E − γ n−N −1 (Ψ); − [H(0, n − N − n0 − 1) + H(n − N, n − n0 − 1)] .
2
2
According to Lemma III.4, Corollary III.7 and Proposition III.8, there exist C, q > 0 such
that
−1
||Kn−N,n Kn−N
− 1|| ≤ Ce−qN ,
||Ln || ≤ C,
||Ln (1) − h|| ≤ Ce−qn ,
where Kn−N ’s are defined in Lemma III.5.
Since γ n (Q) and Kn−N,n commute for any Q ∈ A[0,∞) , we have
−1
−1
−1
−1
Ln−N ((Kn−N,n Kn−N
)∗ γ n (Q)Kn−N,n Kn−N
) = γ N (Q)Ln−N ((Kn−N,n Kn−N
)∗ Kn−N,n Kn−N
).
So
−1
−1
||Ln−N (γ n (Q)) − γ N (Q)Ln−N (1)|| ≤ C||γ n (Q) − (Kn−N,n Kn−N
)∗ γ n (Q)Kn−N,n Kn−N
||
−1
−1
+C||(Kn−N,n Kn−N
)∗ Kn−N,n Kn−N
− 1|| ||Q|| ≤ C 0 e−qN ||Q||,
−1
where C 0 = 2C 2 (2 + C) (note that ||Kn−N,n Kn−N
|| ≤ 1 + C). Then
||Ln−N (γ n (Q))−γ N (Q)h|| ≤ C 0 e−qN ||Q||+||Ln−N (1)−h|| ||Q|| ≤ (C 0 e−qN +Ce−q(n−N ) )||Q||,
hence |ν(γ n (Q)) − ν(γ N (Q)h)| ≤ (C 0 e−qN + Ce−q(n−N ) )||Q||, and
||νγ n − νγ m || ≤ 2C 0 e−qN + Ce−q(n−N ) + Ce−q(m−N )
for n, m > N + n0 . We see that the sequence {νγ n }n converges. Denoting its limit by φ and
letting m → ∞ in the latter inequality we obtain
||νγ n − φ|| ≤ 2C 0 e−qN + Ce−q(n−N ) .
2
Taking N = [ 21 n] we obtain the desired.
The state φ of A[0,∞) is obviously γ-invariant. Hence there exists a unique γ-invariant
state of A that, being restricted to A[0,∞) , coincides with φ. We denote this state by the
same letter φ, and our aim is to prove that φ is the required Gibbs state.
First, we shall prove that φ satisfies the conclusion of Theorem III.2.
13
Proposition III.11 There exist C, q > 0 such that
|ν(Q1 Q2 ) − ν(Q1 )ν(Q2 )| ≤ Ce−qn ||Q1 || ||Q2 ||
for Q1 ∈ A[0,N ] and Q2 ∈ A[N +n,∞) for any N, n.
Proof. Replacing Q1 by Q1 − ν(Q1 )1 we can suppose that ν(Q1 ) = 0.
By the same arguments as in Lemma III.10, we can find C, q such that
||LN +m (Q1 Q2 ) − γ −N −m (Q2 )LN +m (Q1 )|| ≤ Ce−q(n−m) ||Q1 || ||Q2 ||
for m < n − n0 . On the other hand, ||LN +m (Q1 )|| ≤ Ce−qm ||Q1 || by Corollary III.9. Hence
||LN +m (Q1 Q2 )|| ≤ C(e−qm + e−q(n−m) )||Q1 || ||Q2 ||,
so that
|ν(Q1 Q2 )| ≤ C(e−qm + e−q(n−m) )||Q1 || ||Q2 ||.
2
Taking m = [ 21 n] we obtain the required.
Now we can prove that φ is uniformly exponentially clustering in the sense of Theorem
III.2:
If C, q are chosen according to Proposition III.11 then, for m > N > n, Q1 ∈ A[−N,−n] ,
Q2 ∈ A[n,N ] , we have
|ν(γ m (Q1 Q2 ) − ν(γ m (Q1 ))ν(γ m (Q2 ))| ≤ Ce−2qn ||Q1 || ||Q2 ||.
Letting m → ∞ we obtain
|φ(Q1 Q2 ) − φ(Q1 )φ(Q2 )| ≤ Ce−2qn ||Q1 || ||Q2 ||.
Lemma III.12 Let Z̄n = τ (e−H(0,n)−γ
A[0,∞) . Then
1
(i) n→∞
lim log Z̄n = log λ ;
n
(ii) there exist C, q > 0 such that
n (∆+ )
) and φ̄n (Q) = Z̄n−1 τ (Qe−H(0,n)−γ
n (∆+ )
) for Q ∈
|ν(Q) − φ̄n+N (Q)| ≤ Ce−qn ||Q|| ∀Q ∈ A[0,N ] ∀N, n.
Proof. See Lemma 8.1 in [1].
Let Q ∈ A[0,N ] . Proving (ii) we can suppose that Q > 0 and α(Q) ≤ 2 as in Corollary
III.7.
For n > m + r, let
KN +m,N
=
+n
1 N +m−1
1
=E − γ
(Ψ); − [H(0, N + m − n0 − 1) + H(N + m, N + n) + γ N +n (∆+ )] .
2
2
14
Then
1
1
e− 2 H(0,N +n)− 2 γ
N +n (∆+ )
1
1
1
= KN +m,N +n e− 2 H(0,N +m−n0 −1)− 2 H(N +m,N +n)− 2 γ
N +n (∆+ )
,
so that
φ̄N +n (Q) = Z̄N−1+n τ KN∗ +m,N +n QKN +m,N +n e−H(0,N +m−n0 −1)−H(N +m,N +n)−γ
N +n (∆+ )
= Z̄N−1+n τ γ −N −m EN +m (KN∗ +m,N +n QKN +m,N +n e−H(0,N +m−n0 −1) ) ×
×e−H(0,n−m)−γ
n−m (∆+ )
= Z̄N−1+n Z̄n−m φ̄n−m γ −N −m EN +m (KN∗ +m,N +n QKN +m,N +n e−H(0,N +m−n0 −1) )
By Corollary III.9 and a version of Lemma III.4, there exist C, q > 0 such that
||KN +m,N +n ||, ||KN +m ||, ||KN−1+m || ≤ C,
||LN +m (Q) − ν(Q)h|| ≤ Ce−qm ||Q||,
||KN +m − KN +m,N +n || ≤ Ce−q(n−m) .
−H(0,N +m−n0 −1)
), being a
Let pN +m = τ (e−H(0,N +m−n0 −1) ). Then x 7→ p−1
N +m EN +m (xe
unital completely positive mapping, has norm one. From this observation, the choice of C,
and Lemma III.5, we infer
||γ −N −m EN +m (KN∗ +m,N +n QKN +m,N +n e−H(0,N +m−n0 −1) ) − LN +m (Q)||
≤ pN +m ||KN∗ +m,N +n QKN +m,N +n − KN∗ +m QKN +m || ≤ pN +m 2C 2 e−q(n−m) ||Q||
2
−1
and LN +m (Q) ≥ pN +m ||KN−1+m ||−2 ||Q−1 ||−1 , so that ||LN +m (Q)−1 || ≤ p−1
N +m C ||Q ||.
Hence
||λ−N −m γ −N −m EN +m (KN∗ +m,N +n QKN +m,N +n e−H(0,N +m−n0 −1) ) − LN +m (Q)|| ||LN +m (Q)−1 ||
≤ 2C 4 e−q(n−m) α(Q) ≤ 4C 4 e−q(n−m) .
Applying φ̄n−m we obtain
−1
|λ−N −m Z̄N +n Z̄n−m
φ̄N +n (Q) − φ̄n−m (LN +m (Q))| ≤ 4C 4 e−q(n−m) ||LN +m (Q)−1 ||−1
≤ 4C 4 e−q(n−m) ν(LN +m (Q)) ≤ 4C 4 e−q(n−m) ||Q||.
Hence
−1
|λ−N −m Z̄N +n Z̄n−m
φ̄N +n (Q) − ν(Q)φ̄n−m (h)| ≤ (4C 4 e−q(n−m) + Ce−qm )||Q||.
(12)
In particular,
−1
|λ−N −m Z̄N +n Z̄n−m
− φ̄n−m (h)| ≤ 4C 4 e−q(n−m) + Ce−qm .
15
(13)
Having fixed n − m sufficiently large we see from ||h−1 ||−1 ≤ φ̄n−m (h) ≤ ||h|| and the
latter inequality that the sequence {−(N + n) log λ + log Z̄N +n }n is bounded. In particular,
lim n−1 log Z̄n = log λ. So we have proved (i).
n→∞
Multiplying (13) by φ̄N +n (Q) we also deduce from (12) and (13) that
|φ̄N +n (Q) − ν(Q)| ≤ 2(4C 4 e−q(n−m) + Ce−qm )||Q|| ||h−1 ||.
2
Taking m = [ 21 n], we obtain (ii).
Proof of Theorem III.1. Let Q ∈ A[a,b] .
τ (Qe−U (−n+a,b+n) )
τ (e−U (−n+a,b+n) )
τ (γ n−a (Q)e−U (0,b−a+2n) )
=
τ (e−U (0,b−a+2n) )
φ̄b−a+2n (∆∗b−a+2n γ n−a (Q)∆b−a+2n )
=
,
φ̄b−a+2n (∆∗b−a+2n ∆b−a+2n )
φ[−n+a,b+n] (Q) =
where φ̄b−a+2n is introduced in Lemma III.12,
1
1
= E − ∆− ; − [H(0, b − a + 2n) + γ b−a+2n (∆+ )] .
2
2
∆b−a+2n
Let C, q be chosen as in Lemma III.10, Proposition III.11 and Lemma III.12. By a version
of Lemma III.4, we can also suppose that
−1
||∆m ||, ||∆m
|| ≤ C and ||∆m − ∆k || ≤ Ce−q min{m,k} .
We have the following inequalities (m < n − n0 ):
|φ̄b−a+2n (∆∗b−a+2n γ n−a (Q)∆b−a+2n ) − φ̄b−a+2n (γ n−a (Q)∆∗m ∆m )| ≤ 2C 2 e−qm ||Q||,
(14)
since γ n−a (Q) and ∆m commute.
|φ̄b−a+2n (γ n−a (Q)∆∗m ∆m ) − ν(γ n−a (Q)∆∗m ∆m )| ≤ C 3 e−qn ||Q||
(15)
by Lemma III.12.
|ν(γ n−a (Q)∆∗m ∆m ) − ν(γ n−a (Q))ν(∆∗m ∆m )| ≤ C 3 e−q(n−m) ||Q||
(16)
by Proposition III.11.
|ν(γ n−a (Q))ν(∆∗m ∆m ) − φ(Q)ν(∆∗m ∆m )| ≤ C 3 e−qn ||Q||
(17)
by Lemma III.10.
|φ(Q)ν(∆∗m ∆m ) − φ(Q)φ̄b−a+2n (∆∗m ∆m )| ≤ C 3 e−q(b−a+2n−m) ||Q|| ≤ C 3 e−qn ||Q||
16
(18)
by Lemma III.12. Finally,
|φ(Q)φ̄b−a+2n (∆∗m ∆m ) − φ(Q)φ̄b−a+2n (∆∗b−a+2n ∆b−a+2n )| ≤ 2C 2 e−qm ||Q||,
(19)
by the property of ∆m ’s. Summing the inequalities (14)-(19) we obtain
|φ̄b−a+2n (∆∗b−a+2n γ n−a (Q)∆b−a+2n ) − φ(Q)φ̄b−a+2n (∆∗b−a+2n ∆b−a+2n )|
≤ C̃(e−qn + e−qm + e−q(n−m) )
for a constant C̃ depending only on the potential.
Since
1
|φ̄b−a+2n (∆∗b−a+2n ∆b−a+2n )| ≥ ||(∆∗b−a+2n ∆b−a+2n )−1 ||−1 ≥ 2 ,
C
we have
|φ[−n+a,b+n] (Q) − φ(Q)| ≤ C̃C 2 (e−qn + e−qm + e−q(n−m) ).
Taking m = [ 12 n] we obtain (ii).
τ (e−U (a,b) ) = Z̄b−a φ̄b−a (∆∗b−a ∆b−a ).
Since
1
C2
≤ φ̄b−a (∆∗b−a ∆b−a ) ≤ C 2 and
1
n
log Z̄n → log λ, we have (i).
2
Remark III.13 The Gibbs state φ is a σt -KMS-state (β = 1). It is known that for a onedimensional quantum lattice this is a unique σt -KMS-state (see [5], or [4]). The same result
holds for binary shifts (Example II.4). Indeed, by [16] and [17], there exist p, k ∈ N such
that A[0,k+pn] is a full matrix algebra for any n ∈ N. Then one can apply [5].
IV
Entropic properties of Gibbs states
In the sequel φ denotes a Gibbs state on (A, {A[n,m] }n≤m , τ, γ) corresponding to an interaction
Φ(X).
Theorem IV.1 Let M = πφ (A)00 . Then (M, φ, γ) is an entropic K-system.
Proof. Let M0 be the W ∗ -subalgebra of M generated by πφ (A(−∞,0] ). Using Theorem III.2
one concludes that ∩n γ n (M0 ) = C1. Hence (M, φ, γ) is an entropic K-system by [7, Theorem
3.1].
2
Corollary IV.2 If πφ (A) 6= C1, then hφ (γ) > 0.
2
17
Corollary IV.3 For all the examples of Section II we have 0 < hφ (γ) < ∞.
Proof. Let dn be the tracial dimension of A[1,n] , i. e. the dimension of a maximal abelian
subalgebra in A[1,n] . Then
hφ (γ) ≤ lim
inf
n→∞
S(φ|A[1,n] )
log dn
≤ sup
.
n
n
n
So to prove the finiteness it suffices to show that supn n1 log dn < ∞.
It is obvious for binary shifts (Example II.4), since dim A[1,n] = 2n . For the automorphism
θλ (Example II.2) and for the canonical shift on the tower of relative commutants (Example
II.3) we note that the tracial dimension of M 0 ∩ Mn does not exceed [Mn : M ] = [M : N ]n
[18].
To prove that the entropy is non-zero we have to show that each of the algebras under
consideration has no ideals of codimension one. The algebra of a binary shift is simple as well
as the algebra of relative commutants corresponding to a finite depth inclusion. In general,
an algebra of higher relative commutants can be non-simple. Nevertheless it always contains
a Temperley-Lieb algebra that has no ideals of codimension one.
2
Next we shall discuss mean entropy. First we need a form of the Gibbs condition (called
the Gibbs condition in the strong sense in [9]).
Proposition IV.4 For a finite interval Λ = [k, l] ⊂ Z, let W (Λ) denotes the surface energy,
i. e.
X
X
W (Λ) =
Φ(X) +
Φ(X).
X:
X∩Λ6=∅
X∩Λc 6=∅
X⊂Λc :
X∩([k−n0 ,k−1]∪[l+1,l+n0 ])6=∅
Then for the perturbed state φW (Λ) (see, for example, [4], Section 5.4.1) we have
φW (Λ) (ab) = φΛ (a)φW (Λ) (b) ∀a ∈ AΛ ∀b ∈ A(−∞,k−n0 )∪(l+n0 ,∞) .
Proof. Let K = E( 21 W (Λ); − 12 U (Z)),
1
1
1
1
1
Kn = E
W (Λ); − U (−n, n) = e 2 W (Λ)− 2 U (−n,n) e 2 U (−n,n) .
2
2
Then K and Kn are invertible, Kn converges to K in norm, and
φW (Λ) =
φ[−n,n] (Kn∗ · Kn )
φ(K ∗ · K) W (Λ)
,
φ
=
.
[−n,n]
φ(K ∗ K)
φ[−n,n] (Kn∗ Kn )
W (Λ)
Since φ[−n,n] pointwise converges to φ, we see that φ[−n,n] pointwise converges to φW (Λ) . For
n sufficiently large, we have
W (Λ)
φ[−n,n] (ab)
τ (abe−U (−n,n)+W (Λ) )
=
τ (e−U (−n,n)+W (Λ) )
18
τ (abe−U (Λ) e−U ([−n,k−n0 −1]∪[l+n0 +1,n]) )
τ (e−U (−n,n)+W (Λ) )
τ (ae−U (Λ) )τ (be−U ([−n,k−n0 −1]∪[l+n0 +1,n]) )
=
τ (e−U (−n,n)+W (Λ) )
τ (ae−U (Λ) ) τ (be−U (Λ) e−U ([−n,k−n0 −1]∪[l+n0 +1,n]) )
=
·
τ (e−U (Λ) )
τ (e−U (−n,n)+W (Λ) )
=
W (Λ)
= φΛ (a)φ[−n,n] (b).
2
With the help of the Gibbs condition the mean entropy is computed in a standard way
(see [4],[9]). Namely, we have the following.
Corollary IV.5 Let Pn be the density operator for τ |A[1,n] . Then
lim
n→∞
1
(φ(log Pn ) + S(φ|A[1,n] )) = φ(Φ) + P (Φ) = φ(EΦ ) + P (Φ),
n
where EΦ = X30 Φ(X)
. In particular, the mean entropy s(φ) = limn n1 S(φ|A[1,n] ) exists iff
|X|
the limit limn n1 φ(log Pn ) exists.
P
Proof. Let Qn be the density operator for φ|A[1,n] . The operator
operator for φ[1,n] |A[1,n] = φW (1,n) |A[1,n] . Then by [4], p. 278,
0 ≤ TrA[1,n] Qn
e−U (1,n)
log Qn − log
Pn
τ (e−U (1,n) )
!!
e−U (1,n)
P
τ (e−U (1,n) ) n
is the density
≤ S(φW (1,n) , φ) ≤ 2||W (1, n)||,
so that
0 ≤ −S(φ|A[1,n] ) − φ(log Pn ) + φ(U (1, n)) + log τ (e−U (1,n) ) ≤ 2||W (1, n)||.
Since n1 φ(U (1, n)) → φ(Φ) = φ(EΦ ),
is bounded, we obtain the desired.
1
n
log τ (e−U (1,n) ) → P (Φ), and the sequence {W (1, n)}n
2
Corollary IV.6 The mean entropy exists for each of the following cases:
1) the automorphism θλ , λ ≥ 41 ,
s(φ) = φ(EΦ ) + P (Φ) −
1
log λ ;
2
2) the canonical shift on the tower of relative commutants corresponding to a subfactor
of finite depth,
s(φ) = φ(EΦ ) + P (Φ) + log[M : N ] ;
3) a binary shift,
s(φ) = φ(EΦ ) + P (Φ) +
19
1
log 2.
2
Proof. For all these systems the limit limn n1 log Pn exists in norm. The result is well-known
for the systems in 1) and 2) (see e.g. [9], Examples 1.3, 1.4). Concerning binary shifts the
sequence {A[1,n] }n is periodic [17]. Hence the limit limn − n1 log Pn exists, is scalar and equals
to the entropy hτ (γ) of γ with respect to τ [19]. This entropy is equal to 12 log 2 [19].
2
Remark IV.7 It is not known whether the mean entropy equals to the dynamical entropy
even for a quantum spin lattice system. But such an equality is possible. This is the
case when an interaction potential lies in the diagonal (then, by the general theory [8], the
dynamical entropy coincides with entropy of the restriction of the automorphism to the
diagonal). A bit less trivial example can be obtained as follows.
Suppose an interaction potential on a one-dimensional quantum lattice has the property
σt (A0 ) ⊂ A[−n,n] for some n ∈ N. Then it is easy to see that hφ (γ) = s(φ) (in fact, the state
φ is n-Markov, i. e. it would be Markov if we consider A as a quantum lattice system with
the one-site algebra A[0,n−1] , [20], and we can refer to [21], see also [7]). One can construct
potentials with such a property, but that don’t lie in the diagonal. For example, we can
choose selfadjoint a0 , a1 , b ∈ M atd (C), d ≥ 3, such that a0 commutes with a1 and b, but a1
and b don’t commute. Define Φ({0}) = b, Φ({0, 1}) = a0 ⊗ a1 , Φ(X) = 0 for |X| ≥ 3. Then
σt (A0 ) ⊂ A[−1,1] , but the centralizer Aσ does not contain any diagonal of A0 .
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21