Dynamical entropy for Bogoliubov actions of
torsion-free abelian groups on the CAR-algebra
Valentin Ya. Golodets
Sergey V. Neshveyev
Abstract
We compute dynamical entropy in Connes, Narnhofer and Thirring sense for a
Bogoliubov action of a torsion-free abelian group G on the CAR-algebra. A formula
analogous to that found by Størmer and Voiculescu in the case G = Z is obtained.
The singular part of a unitary representation of G is shown to give zero contribution
to the entropy. A proof of these results requires new arguments since a torsion-free
group may have no finite index proper subgroups. Our approach allows to overcome
these difficulties, it differs from that of Størmer-Voiculescu.
Introduction.
Entropy is an important notion in classical statistical mechanics and information theory.
Initially the notion of entropy for automorphisms of a measure space was introduced by
Kolmogorov and Sinai in 1958. This invariant proved to be extremely useful, it generated
an entire field in the theory of classical dynamical systems and topological dynamics. The
extension of the notion of entropy onto quantum systems was treated as a difficult mathematical problem. It was solved by Connes and Størmer [CS] only in 1975 for dynamical
systems of type II1 . Then Connes, Narnhofer and Thirring [CNT] extended this theory to
general C ∗ - and W ∗ -dynamical systems.
The computation of dynamical entropy for specific models is one of the principal trends
in the theory (see [BG],[GN] for a bibliography). One of the main results in this sphere
belongs to Størmer and Voiculescu [StV]. They showed that the CNT-entropy of a Bogoliubov automorphism of the CAR-algebra is computed by a simple formula (predicted by A.
Connes for the tracial state), and only the absolutely continuous part of the unitary operator defining the Bogoliubov automorphism gives a contribution to the entropy. Bezuglyi
and Golodets [BG] obtained the same results for Bogoliubov actions of free abelian groups.
It is quite natural to extend these results to Bogoliubov actions of arbitrary countable
torsion-free abelian groups.
Note that in Størmer-Voiculescu’s approach it is very important that the group Z
has a lot of finite index subgroups (see Theorem 2.1, condition (iv), in [StV]). But, for
example, the group Q of rational numbers contains no finite index (proper) subgroups. So
the methods of [StV] and [BG] cannot be immediately applied.
It is interesting to note that the problem of studying entropic properties of actions of
the group Q is well-known in the commutative entropic theory. As far as we know there
are no methods to describe the Pinsker algebra and asymptotic properties of systems with
completely positive entropy for such actions. In particular, the Conze approach [Co] does
not allow to solve these problems. It strengthens our interest to Bogoliubov actions of
torsion-free abelian groups.
1
In this paper we prove that again there is a simple formula for the CNT-entropy of a
Bogoliubov action of a torsion-free abelian group, and only the absolutely continuous part
of a unitary representation gives a contribution to the entropy (see Theorems 3.1, 4.5 and
Corollary 4.2 below). To prove these results we apply new arguments. In particular, our
Lemma 2.4 allows to manage without finite index subgroups. We apply also results and
methods of [StV] and [BG].
1
Definition of entropy.
Throughout the paper G denotes a discrete countable torsion-free abelian group.
By a C ∗ -dynamical system we mean a triple (A, φ, αG ), where A is a C ∗ -algebra, αG
is an action of G on A by *-automorphisms, and φ is a G-invariant state on A.
For given channels γi : Bi → A, 1 ≤ i ≤ n, i.e. completely positive unital mappings
of finite-dimensional C ∗ -algebras Bi , Hφ (γ1 , . . . , γn ) denotes their mutual entropy with
respect to φ (see [CNT]).
If γ is a channel and A is a finite subset of G, we denote by Hφ (γ A ) the mutual entropy
of the channels αg ◦ γ, g ∈ A.
Definition 1.1. A parallelepiped in G is a finite subset A of G such that there exist
n ∈ N, a monomorphism I: Zn → G, and m1 , . . . , mn ∈ N such that
A = I ({z ∈ Zn | 0 ≤ zk ≤ mk , 1 ≤ k ≤ n}) .
Definition 1.2. The entropy of the system (A, φ, αG ) with respect to a channel γ is the
quantity
Hφ (γ A )
hφ (γ; αG ) = inf
,
|A|
where A runs over the set of all parallelepipeds in G. The dynamical entropy of the system
is
hφ (αG ) = sup hφ (γ; αG ).
γ
Remark 1.3. If A is commutative and γ is the inclusion of a finite-dimensional subalgebra
H (P A )
P of A, hφ (P; αG ) may be defined as the infimum of φ|A| over all finite subsets A of
G. Then one proves that this infimum is equal to the limit along a net of Følner sets, and
this holds for any amenable group G [M]. The proof relies on the strong subadditivity of
the function A 7→ Hφ (P A ), i.e.
Hφ (P A∪B ) + Hφ (P A∩B ) ≤ Hφ (P A ) + Hφ (P B ).
Apparently the function A 7→ Hφ (γ A ) is not strongly subadditive in the non-commutative
case. But it is at least subadditive [CNT], i.e.
Hφ (γ A∪B ) ≤ Hφ (γ A ) + Hφ (γ B ).
The following result is an immediate consequence of the subadditivity.
2
Proposition 1.4. Let G = Zn , n ∈ N. For N ∈ N, let AN denotes the cube {z ∈ Zn | 0 ≤
zk ≤ N }. Then, for any channel γ,
Hφ (γ AN )
.
N →∞
|AN |
hφ (γ; αG ) = lim
In particular, for G = Zn our definition of entropy coincides with the usual one
[CNT],[BG].
Remark 1.5. A statement analogous to Proposition 1.4 may be formulated for any G. Let
{gi }N
i=1 , N ≤ ∞, be a maximal linear independent system in G, G1 the subgroup of G
generated by this system. Since G is torsion-free, we can consider G as a subgroup of
G = Q ⊗Z G. Then {gi }i is a basis of the vector space G over Q. For an element x ∈ G,
let xi ∈ Q denotes the i-th coordinate of x in this basis.
Choose a set {skn }1≤k≤N,n∈N of non-negative numbers such that
(i) limn→∞ skn = ∞ for any k;
(ii) for any n, only finitely many of skn ’s are non-zero. Set
1
An = x ∈
G1 ∩ G | 0 ≤ xk < skn (xk = 0 if skn = 0)
n!
(note that An is not a parallelepiped in the sense of Definition 1.1). Then
hφ (γ; αG ) = inf
n
Hφ (γ An )
Hφ (γ An )
= lim
.
n
|An |
|An |
This result will not be used in the sequel.
The next lemma follows from the definitions.
Lemma 1.6. Let {Gn }∞
n=1 be an increasing sequence of subgroups of G such that ∪n Gn =
G. Then hφ (γ; αGn ) & hφ (γ; αG ).
Proposition 1.7. Let (A, φ, αG ) be a C ∗ -dynamical system, {Gn }∞
n=1 a sequence of subgroups of G, An a Gn -invariant C ∗ -subalgebra of A, Fn : A → An a completely positive
unital mapping, Fn → id pointwise-norm (we don’t require An ⊂ An+1 ). Then
(i) hφ (αG ) ≤ lim inf hφ (αGn |An );
n→∞
(ii) if An ’s are G-invariant and Fn ’s are φ-preserving conditional expectations then
hφ (αG ) = lim hφ (αG |An ).
n→∞
Proof. See the proof of Lemma 3.3 in [StV].
(i) For a fixed channel γ: B → A, let εn = ||Fn ◦ γ − γ||. By [CNT], Proposition IV.3,
|Hφ (γ A ) − Hφ ((Fn ◦ γ)A )| ≤ |A|δ(εn , d)
for any finite A ⊂ G, where δ(εn , d) depends only on εn and d = dim B, and δ(ε, d) → 0
as ε → 0. Thus, for any parallelepiped A in Gn , we have
Hφ|An ((Fn ◦ γ)A )
Hφ ((Fn ◦ γ)A )
Hφ (γ A )
≥
≥
− δ(εn , d) ≥ hφ (γ; αG ) − δ(εn , d),
|A|
|A|
|A|
3
so that
hφ (αGn |An ) ≥ hφ (Fn ◦ γ; αGn |An ) ≥ hφ (γ; αG ) − δ(εn , d),
whence lim inf hφ (αGn |An ) ≥ hφ (γ; αG ).
(ii) follows from (i) (Gn = G ∀n ∈ N) and the fact that if there exists a φ-preserving
conditional expectation onto a G-invariant subalgebra D of A then Hφ|D (γ1 , . . . , γn ) =
Hφ (γ1 , . . . , γn ) for any channels γ1 , . . . , γn in D, hence hφ (αG |D ) ≤ hφ (αG ).
If H is a subgroup of G, then hφ (γ; αH ) ≥ hφ (γ; αG ), whence hφ (αH ) ≥ hφ (αG ). The
following proposition makes this relation more precise.
Proposition 1.8. Let (A, φ, αG ) be a C ∗ -dynamical system, A nuclear, H a subgroup of
G. Then
(i) if [G : H] < ∞, then hφ (αH ) = [G : H] hφ (αG ) ;
(ii) if [G : H] = ∞ and hφ (αG ) > 0, then hφ (αH ) = ∞.
Proof.
(i) Let p = [G : H] < ∞. First prove that hφ (γ; αH ) ≤ p hφ (γ; αG ).
Choose an increasing sequence {Gn }∞
n=1 of finitely generated subgroups of G such that
∪n Gn = G. By Lemma 1.6, for a given ε > 0, there exists n ∈ N such that hφ (γ; αGn ) <
hφ (γ; αG ) + ε. Since Gn is a finite rank, free abelian group and Hn = H ∩ Gn is a
subgroup of Gn of index≤ p, there exist a basis g1 , . . . , gm in Gn and k1 , . . . , km ∈ N such
that k1 g1 , . . . , km gm is a basis in Hn [L]. For N ∈ N, let AN be the cube
{l1 g1 + . . . + lm gm | 0 ≤ li ≤ N − 1}
in Gn . The set AN ∩ Hn is a parallelepiped in H, and if k1 , . . . , km divide N , then
|AN | = [Gn : Hn ]|AN ∩ Hn |, hence
hφ (γ; αH ) ≤
Hφ (γ AN )
Hφ (γ AN )
Hφ (γ AN )
Hφ (γ AN ∩Hn )
≤
= [Gn : Hn ]
≤p
,
|AN ∩ Hn |
|AN ∩ Hn |
|AN |
|AN |
and using Proposition 1.4 we obtain
hφ (γ; αH ) ≤ p hφ (γ; αGn ) ≤ p hφ (γ; αG ) + p ε.
Thus we have proved that hφ (αH ) ≤ p hφ (αG ), and the assumption of nuclearity has not
been used yet.
To prove the inequality hφ (αG ) ≤ p1 hφ (αH ) choose representatives ḡ1 , . . . , ḡp for cosets
G/H. Due to the nuclearity, for a fixed ε > 0 and a channel γ: B → A, d = dim B, there
exist a channel θ: D → A and channels θ1 , . . . , θp : B → D such that
||θ ◦ θi − αḡi ◦ γ|| < ε, 1 ≤ i ≤ p.
Let {Gn }∞
n=1 be an increasing sequence of finitely generated subgroups of G such that
∪n Gn = G and ḡ1 , . . . , ḡp ∈ Gn for any n. For a fixed n, there exist a basis g1 , . . . , gm in
Gn and numbers k1 , . . . , km ∈ N such that k1 g1 , . . . , km gm is a basis in Hn = H ∩ Gn . The
absolute values of coordinates of ḡ1 , . . . , ḡp in this basis don’t exceed a number N0 . For
N ∈ N, let
AN
= {l1 g1 + . . . + lm gm | 0 ≤ li ≤ ki N − 1},
ÃN
= {l1 g1 + . . . + lm gm | N0 ≤ li ≤ ki N − N0 − 1},
BN
= AN ∩ H = {l1 k1 g1 + . . . + lm km gm | 0 ≤ li ≤ N − 1}.
4
The sets ḡi + BN , 1 ≤ i ≤ p, are mutually disjoint and ÃN ⊂ ∪i (ḡi + BN ). Thus
!
Hφ (γ AN )
Hφ (γ ÃN )
|ÃN |
hφ (γ; αG ) ≤
≤
+ 1−
Hφ (γ),
|AN |
|AN |
|AN |
Hφ (γ ÃN ) ≤ Hφ (γ ∪i (ḡi +BN ) ) ≤ Hφ ({αg ◦ θ ◦ θi | g ∈ BN , 1 ≤ i ≤ p}) + p|BN |δ(ε, d)
≤ Hφ (θBN ) + p|BN |δ(ε, d),
where we have used [CNT], Proposition IV.3 and Proposition III.6(a),(c).
|AN |
Since |B
= [Gn : Hn ] = p, we obtain
N|
1 Hφ (θBN )
hφ (γ; αG ) ≤
+ δ(ε, d) +
p |BN |
|ÃN |
1−
|AN |
!
Hφ (γ).
Letting N → ∞ and using Proposition 1.4 we conclude that
1
hφ (γ; αG ) ≤ hφ (θ; αHn ) + δ(ε, d),
p
and by Lemma 1.6,
1
hφ (γ; αG ) ≤ hφ (θ; αH ) + δ(ε, d).
p
So, due to the arbitrariness of ε, hφ (γ; αG ) ≤ p1 hφ (αH ).
(ii) Suppose [G : H] = ∞ and hφ (αH ) < ∞, and prove that hφ (αG ) = 0. Consider two
cases.
a) The group G/H is periodic.
There exists an increasing sequence {Hn }∞
n=1 of subgroups of G such that H ⊂ Hn ,
[Hn : H] < ∞, [Hn : H] → ∞. Then
hφ (αG ) ≤ hφ (αHn ) =
1
hφ (αH ) → 0.
[Hn : H]
b) The rank of G/H is non-zero.
Let g ∈ G be an element, whose image in G/H has infinite order. Let Hn = H + nZg.
Then
1
1
hφ (αG ) ≤ hφ (αH1 ) = hφ (αHn ) ≤ hφ (αH ) → 0.
n
n
2
Bogoliubov actions on the CAR-algebra.
Let H be a complex Hilbert space. Recall (see [StV],[BR2]) that the CAR-algebra A(H)
over H is a C ∗ -algebra generated by elements a(f ), f ∈ H, such that f 7→ a(f ) is a linear
map and
a(f )a(g)∗ + a(g)∗ a(f ) = (f, g)1, a(f )a(g) + a(g)a(f ) = 0.
If K is a closed subspace of H, we consider A(K) as a subalgebra of A(H).
The even part of the CAR-algebra is the C ∗ -subalgebra A(H)e generated by even
products of a(f )’s and a(g)∗ ’s.
5
If H1 and H2 are mutually orthogonal subspaces of H, then A(H1 ) and A(H2 )e commute and the C ∗ -algebra they generate is identified with A(H1 ) ⊗ A(H2 )e .
If 0 ≤ A ≤ 1 is an operator on H, then the quasi-free state ωA on A(H) is given by
ωA (a(fn )∗ . . . a(f1 )∗ a(g1 ) . . . a(gm )) = δnm det((Agi , fj )).
We will write ωλ instead of ωλI . The state ω 1 is the unique tracial state on A(H).
2
If H = H1 ⊕ H2 , Ai ∈ B(Hi ), 0 ≤ Ai ≤ 1, A = A1 ⊕ A2 , then
ωA |A(H1 )⊗A(H2 )e = ωA1 |A(H1 ) ⊗ ωA2 |A(H2 )e .
Suppose there exists an orthonormal basis {fn }N
n=1 , N ≤ ∞, such that A fn = λn fn .
Then
N
(A(H), ωA ) ∼
(2.1)
= ⊗ (Mat2 (C)n , ρλn )
n=1
via the homomorphism sending a(fn )
n−1
Y
(n)
(n)
(1 − 2a(fi )∗ a(fi )) to e12 (so a(fn )a(fn )∗ 7→ e11 ),
i=1
where ρλ is the state on Mat2 (C) given by
a b
ρλ
= (1 − λ)a + λd.
c d
Each unitary operator U on H defines a Bogoliubov automorphism α of A(H) by
α(a(f )) = a(U f ). So, for any unitary representation of G on H, we obtain an action of G
on A(H) called Bogoliubov. If U and A commute, then α preserves ωA .
If K is an invariant subspace for A and σ is the Bogoliubov automorphism correspondis an ωA -preserving conditional
ing to the operator 1 ⊕ −1 on H = K ⊕ K ⊥ , then id+σ
2
expectation of A(H) onto A(K) ⊗ A(K ⊥ )e , and composing it with id ⊗ ωA (·) we obtain
an ωA -preserving conditional expectation A(H) → A(K).
Lemma 2.1. Let {Pn }∞
n=1 be a sequence of projections in B(H), Pn → 1 strongly. Let
En be a conditional expectation of A(H) onto A(Pn H). Then En → id pointwise-norm.
Proof. The result easily follows from the facts that En is a projection of norm one and
||a(f )|| = ||f || for any f ∈ H.
Using the existence of conditional expectations and (2.1) we obtain also the following.
Lemma 2.2. Let H1 , . . . , Hn be mutually orthogonal finite-dimensional subspaces of H
invariant for A. Then
HωA (A(H1 ), . . . , A(Hn )) = S(ωA |A(H1 ⊕...⊕Hn ) ) =
n
X
TrHi (η(A) + η(1 − A)),
i=1
where η(x) = −x log x.
Proof. Cf. [CNT], Corollary VIII.8.
Lemma 2.3. Let U : G → B(H) be a unitary representation, αG the corresponding Bo∞
goliubov action on A(H), {Gn }∞
n=1 a sequence of subgroups of G, {Pn }n=1 a sequence of
projections in B(H) such that Hn = Pn H is Gn -invariant and Pn → 1 strongly. Then
6
(i) for any G-invariant state φ on A(H), we have
hφ (αG ) ≤ lim inf hφ (αGn |A(Hn ) ) ;
n→∞
(ii) if Hn ’s are G-invariant then, for A ∈ B(H), 0 ≤ A ≤ 1, Ug A = A Ug , A Hn ⊂ Hn ,
we have
hωA (αG ) = lim hωA (αG |A(Hn ) ).
n→∞
Proof. This is a consequence of Lemma 2.1 and Proposition 1.7.
The next simple observation plays the central role in the subsequent computations.
Lemma 2.4. Let U (n) : G → B(Hn ) be a unitary representation (n ∈ N), {χn }∞
n=1 ⊂ Ĝ a
sequence of characters of G. Consider two unitary representations of G on H = ⊕∞
n=1 Hn ,
∞
∞
n=1
n=1
Ug0 = ⊕ Ug(n) , Ug00 = ⊕ χn (g)Ug(n) ,
0 and α00 be the corresponding Bogoliubov actions. Then, for any α0 - and α00 and let αG
G
invariant state φ on A(H), we have
0
00
hφ (αG
) = hφ (αG
).
∗
Proof. For n ∈ N, let {fkn }∞
k=1 be an orthonormal basis in Hn . Let Am be the C subalgebra of A(H) generated by a(fkn ), 1 ≤ k, n ≤ m. Then
0
0
00
00
hφ (αG
) = lim hφ (Am ; αG
), hφ (αG
) = lim hφ (Am ; αG
)
m→∞
m→∞
by the proof of [CNT], Theorem V.2. On the other hand, since αg00 (a(fkn )) =
00 ) =
χn (g)αg0 (a(fkn )), we have αg00 (Am ) = αg0 (Am ) for any g ∈ G, hence hφ (Am ; αG
0
hφ (Am ; αG ).
3
Entropy formula: the case of absolutely continuous spectrum.
Recall some notions of the theory of representations that will be used below (see [K]).
Let U : G → B(H) be a unitary representation. Considering elements of G as characters
of the dual group Ĝ we can extend it to a *-representation f 7→ Uf of the algebra of
bounded Borel functions on Ĝ. Then the spectral projection for a Borel subset X ⊂ Ĝ is
the projection U´IX .
For a vector ξ ∈ H, the spectral measure µξ is a positive Borel measure on Ĝ such
that
Z
(Ug ξ, ξ) =
χ(g)dµξ (χ), g ∈ G.
Ĝ
The representation U is decomposed into a direct sum U = U a ⊕ U s , H = H a ⊕ H s , of
two representations such that, for any ξ ∈ H a (resp. ξ ∈ H s ), the spectral measure µξ is
7
absolutely continuous (resp. singular) with respect to the Haar measure λ on Ĝ. We say
that U a has absolutely continuous spectrum and U s has singular spectrum.
The representation U a is decomposed in a direct integral
Z ⊕
Z ⊕
a
H=
Hχ dλ(χ), Ug =
χ(g)dλ(χ).
Ĝ
Ĝ
The function m(χ) = dim Hχ is called the multiplicity function of the representation U a .
Our main result in this section is as follows.
Theorem 3.1. Let U : G → B(H) be a unitary representation with absolutely continuous
spectrum and the multiplicity function m. Then, for the corresponding Bogoliubov action
αG and β ∈ [0, 1],
Z
hωβ (αG ) = (η(β) + η(1 − β))
m(χ) dλ(χ),
Ĝ
where η(x) = −x log x.
The proof of Theorem is divided onto several lemmas.
First note that if β = 0 or β = 1 then the state ωβ is pure, so that the entropy of any
channel is zero and there is nothing to prove. Thus we can suppose β ∈ (0, 1).
l−1 k k
Lemma 3.2. Let q, l ∈ N, X = ∪k=0
[l, l +
1
ql ].
Then
{q 1/2 e2πi(lqk+r)t | k ∈ Z, 0 ≤ r ≤ l − 1}
is an orthonormal basis in L2 (X, dt).
Proof. Note that
Z
2πint
e
dt =
X
l−1
X
2πi nk
l
!Z
e
1
ql
e2πint dt.
0
k=0
This expression is zero if either l does not divide n or n is divided by ql. This implies the
orthonormality.
The mapping exp(2πilkt) 7→ exp(2πikt) defines a unitary operator from
Lin{e2πilkt | k ∈ Z} ⊂ L2 (X)
onto L2 (0, 1q ). Hence, for any p ∈ Z, exp(2πilpt) belongs to the closed subspace of L2 (X)
spanned by exp(2πilqkt), k ∈ Z (since {exp(2πiqkt) | k ∈ Z} is an orthogonal basis in
L2 (0, 1q )). Then exp(2πi(lp + r)t), 0 ≤ r ≤ l − 1, lies in the closed subspace spanned by
exp(2πi(lqk + r)t), k ∈ Z.
Lemma 3.3. Let G = Zn . Suppose the multiplicity function m equals to p´IX , where
p ∈ N and
!!
l−1
[ k k
1
X = exp 2πi
, +
×T
. . × T} .
| × .{z
l l
ql
k=0
n−1
So the space H of the representation can be identified with the sum of p copies of L2 (X, dλ),
p
H = ⊕ L2 (X, dλ)i .
i=1
8
Let K be the subspace of H spanned by ´IX ∈ L2 (X)i , 1 ≤ i ≤ p. Then
p
hωβ (αG ) = hωβ (A(K); αG ) = (η(β) + η(1 − β)) .
q
Proof. The equality hωβ (αG ) = (η(β) + η(1 − β)) pq is known: see [StV], Lemma 4.5, or
[BG], Lemma 3.4. Thus we have only to prove that hωβ (A(K); αG ) ≥ (η(β) + η(1 − β)) pq .
For N ∈ N, let
AN
= {z ∈ Zn | 0 ≤ zk ≤ N },
ÃN
= AN ∩ {z ∈ Zn | z1 = lqk + r, k ∈ Z, 0 ≤ r ≤ l − 1}.
By Lemma 3.2, the subspaces Ug K, g ∈ ÃN , are mutually orthogonal. So by Lemma 2.2,
Hωβ (A(K)ÃN ) = |ÃN |(η(β) + η(1 − β))p,
and using Proposition 1.4 we obtain
|ÃN |
p
(η(β) + η(1 − β))p = (η(β) + η(1 − β)) .
N →∞ |AN |
q
hωβ (A(K); αG ) ≥ lim
It is worth to note that the inequality hωβ (αG ) ≤ (η(β) + η(1 − β)) pq may also be
deduced from the completeness assertion of Lemma 3.2.
Lemma 3.4. Let g ∈ G\{0}, X = g −1 (exp(2πi[0, 1q ])) ⊂ Ĝ, where we consider g as a
character of Ĝ, m = p´IX . Then
p
hωβ (αG ) = (η(β) + η(1 − β)) .
q
Proof. The space H of the representation is identified with the sum of p copies of L2 (X, dλ),
H = ⊕pi=1 L2 (X, dλ)i . Let K be the subspace of H spanned by ´IX ∈ L2 (X)i , 1 ≤ i ≤ p.
There exists an increasing sequence {Gn }∞
n=1 of finitely generated subgroups of G such
that ∪n Gn = G and g ∈ Gn for any n. Let Hn be the minimal Gn -invariant subspace of
H containing K.
For a fixed k ∈ N, consider g as a character of Ĝk , and set Yk = g −1 (exp(2πi[0, 1q ])) ⊂
Ĝk . Then the representation of Gk on Hk has the multiplicity function p´IYk . There exist
a basis g1 , . . . , gn of Gk and l ∈ N such that g = lg1 . Having fixed a basis we can identify
Ĝk with Tn . Then g maps (t1 , . . . , tn ) ∈ Tn = Ĝk to tl1 , so that



l−1
[ j j
1
 × T × . . . × T,
Yk = exp 2πi 
, +
|
{z
}
l l
ql
j=0
n−1
By virtue of Lemma 3.3,
p
hωβ (αGk |A(Hk ) ) = hωβ (A(K); αGk |A(Hk ) ) = (η(β) + η(1 − β)) ,
q
9
and using Lemma 1.6 and Lemma 2.3(i) we obtain
p
hωβ (αG ) = hωβ (A(K); αG ) = (η(β) + η(1 − β)) .
q
Lemma 3.5. Let H be a locally compact group, λ its left Haar measure, X1 and X2
measurable subsets of H, 0 < λ(X1 ), λ(X2 ) < ∞. Then there exist a measurable subset Y
of X1 , λ(Y ) > 0, and h ∈ H such that hY ⊂ X2 .
Proof. The mapping h 7→ ´IX2 h ∈ L1 (H, dλ) is continuous. Hence there exists a neighbourhood U of the unit such that
1
||´IX2 − ´IX2 h−1 ||1 ≤ λ(X2 ) for any h ∈ U.
2
We can find h0 ∈ H with λ(h0 U ∩ X1 ) > 0. Let Y1 = U ∩ h−1
0 X1 . Then λ(Y1 ) > 0 and
Z
1
|1 − ´IX2 (xy)|dλ(x) ≤ λ(X2 )
2
X2
for any y ∈ Y1 , hence
Z
Z
dλ(y)
Y1
1
dλ(x)|1 − ´IX2 (xy)| ≤ λ(X2 )λ(Y1 ),
2
X2
and changing the order of integration,
Z
Z
1
1
dλ(x)
dλ(y)|1 − ´IX2 (xy)| ≤ λ(Y1 ).
λ(X2 ) X2
2
Y1
Hence there exists x0 ∈ X2 such that
Z
1
|1 − ´IX2 (x0 y)|dλ(y) ≤ λ(Y1 ).
2
Y1
In other words, if we set Ỹ = {y ∈ Y1 | x0 y ∈
/ X2 }, then λ(Ỹ ) ≤ 12 λ(Y1 ). Thus, for
Y = h0 (Y1 \Ỹ ), we have λ(Y ) ≥ 21 λ(Y1 ) > 0, Y ⊂ X1 and x0 h−1
0 Y ⊂ X2 .
For a multiplicity function m and the corresponding Bogoliubov action αG , set
µβ (m) =
hωβ (αG )
.
η(β) + η(1 − β)
R
Lemma 3.6. For integrable multiplicity functions, µβ (m) depends only on Ĝ m dλ.
R
R
Proof. Suppose m0 dλ = m00 dλ < ∞. Since m0 and m00 are at most countable sums of
indicator functions, Lemma 3.5 and a simple maximality argument ensure the existence
of measurable subsets Yn ⊂ Ĝ, n ∈ N, and a sequence {χn }∞
n=1 ⊂ Ĝ such that
m0 =
∞
X
´IYn and m00 =
n=1
∞
X
n=1
Then the result follows from Lemma 2.4.
10
´Iχn Yn a.e.
Lemma 3.7. If mn % m a.e. then µβ (mn ) % µβ (m).
Proof. If H (resp. Hn ) is the space of the representation U (resp. U (n) ) with the multiplicity function m (resp. mn ), then we may assume H1 ⊂ H2 ⊂ . . . ⊂ H and U (n) = U |Hn .
It remains to apply Lemma 2.3(ii).
R
Proof of Theorem.RWe have to prove that µβ (m) = m dλ.
First suppose m dλ < ∞. Choose g ∈ G\{0} and set X(r) = g −1 (exp(2πi[0, r])),
r ∈ [0, 1]. Then, for p, q, n ∈ N,
µβ (p´IX( nq ) ) = µβ (pn´IX( 1 ) ) (Lemma 3.6)
q
pn
=
(Lemma 3.4).
q
By
R Lemma 3.7, µβ (p´IX(r) ) = prR ∀p ∈ N ∀r ∈ [0, 1]. Finding p ∈ N and r ∈ [0, 1] with
m dλ = pr Rwe obtain µβ (m) = m dλ by Lemma 3.6.
Suppose m dλ = ∞. Letting mn = m ∧ (n´I) and applying Lemma 3.7 we obtain
Z
µβ (m) = lim µβ (mn ) = lim
mn dλ = ∞.
n→∞
n→∞
Remark 3.8. An inspection of the proof shows that the same entropy formula is valid for
the restriction of a Bogoliubov action to the even part of the CAR-algebra.
4
Entropy formula.
Theorem 4.1. Let U (i) : G → B(Hi ) be a unitary representation, i = 1, 2, A1 ∈ B(H1 ),
(1)
(1)
0 ≤ A1 ≤ 1, A1 Ug = Ug A1 , m(2) the multiplicity function of the absolutely continuous
part of U (2) . Suppose the representationR U (1) has absolutely continuous
spectrum and
R⊕
⊕
A1 has pure point spectrum. Let H1 = Ĝ Hχ dλ(χ) and A1 = Ĝ A(χ)dλ(χ) be direct
integral decompositions (λ is the Haar measure on Ĝ). Then, for the Bogoliubov action
αG corresponding to the representation U (1) ⊕ U (2) and for any G-invariant state φ on
A(H1 ⊕ H2 ) such that φ|A(H1 ) = ωA1 , we have
Z
Z
hφ (αG ) ≤
Tr (η(A(χ) + η(1 − A(χ))) dλ(χ) + (log 2)
m(2) (χ)dλ(χ).
Ĝ
Ĝ
Corollary 4.2. If the spectrum of a unitary representation of G is singular then the
entropy of the corresponding Bogoliubov action is zero with respect to any invariant state.
The proof of Theorem is a slight modification of the method used in [BG] to handle
the case of singular spectrum. First, we need two lemmas.
Lemma 4.3. Under the assumptions of Theorem 4.1 suppose that we are given a one
(3)
(3)
more representation U (3) : G → B(H3 ) and A3 ∈ B(H3 ), 0 ≤ A3 ≤ 1, A3 Ug = Ug A3 .
Let α̃G be the Bogoliubov action corresponding to U (1) ⊕ U (2) ⊕ U (3) . Then there exists a
11
G-invariant state ψ on A(H1 ⊕ H2 ⊕ H3 ) such that ψ|A(H1 ⊕H3 ) = ωA1 ⊕A3 and hψ (α̃G ) ≥
hφ (αG ).
Proof. Let σ be the Bogoliubov automorphism corresponding to the operator 1 ⊕ 1 ⊕
−1. Then E = id+σ
is a G-invariant conditional expectation of A(H1 ⊕ H2 ⊕ H3 ) onto
2
A(H1 ⊕ H2 ) ⊗ A(H3 )e (see Section 2). Set ψ = (φ ⊗ ωA3 ) ◦ E. Then ψ is G-invariant,
ψ|A(H1 ⊕H3 ) = ωA1 ⊕A3 , and since there exists a ψ-preserving conditional expectation onto
A(H1 ⊕ H2 ) (namely (idA(H1 ⊕H2 ) ⊗ ωA3 (·)) ◦ E), we have hψ (α̃G ) ≥ hφ (αG ).
Lemma 4.4. Let U be a unitary representation of G on H, m the multiplicity function of
the absolutely continuous part of U , {Gn }∞
n=1 an increasing sequence of subgroups of G with
∪n Gn = G. Then there exist a sequence kn % ∞ and, for each n ∈ N, a Gkn -invariant
subspace Hn of H such that
(i) if Pn is the projection onto Hn , then Pn → id strongly ;
(ii) if mn is the multiplicity function of the absolutely continuous part of the representation UGkn |Hn , then
Z
Z
mn dλkn ≤
lim sup
n→∞
Ĝkn
m dλ.
Ĝ
Proof. It suffices to prove Lemma for a cyclic representation. So let ξ ∈ H be a cyclic
vector, µ its spectral measure, µ = µa +µs the decomposition into the sum of the absolutely
continuous and the singular parts,
dµa
X = χ ∈ Ĝ |
(χ) > 0 .
dλ
Then the representation U is equivalent to the canonical representation on
L2 (Ĝ, dµ) = L2 (Ĝ, dµs ) ⊕ L2 (Ĝ, dµa ) ∼
= L2 (Ĝ, dµs ) ⊕ L2 (X, dλ).
In particular, m = ´IX .
For each n ∈ N, there exists a compact subset Xn of Ĝ such that
λ(X 4 Xn ) <
1
1
and µ(Ĝ\Xn ) < .
n
n
Then we can find an open Yn ⊂ Ĝ such that
Xn ⊂ Yn and λ(Yn \Xn ) <
1
.
n
Denote by In the inclusion Gn ,→ G. Due to the compactness of Xn and the equality
Ĝ = lim Ĝn , there exist kn ≥ n and a compact Z̃n ⊂ Ĝkn such that, for Zn = Iˆk−1
(Z̃n ), we
n
←
have Xn ⊂ Zn ⊂ Yn . Then
λ(Zn 4 X) ≤ λ(Zn 4 Xn ) + λ(Xn 4 X) <
1
2
and µ(Ĝ\Zn ) < .
n
n
Let E(Zn ) be the spectral projection corresponding to Zn . Set ξn = E(Zn )ξ, and let
Hn be the minimal Gkn -invariant subspace containing ξn . Then the spectral measure of
ξn (with respect to Gkn ) is supported by Z̃n , so if mn is the multiplicity function of the
absolutely continuous part of the representation of Gkn on Hn , we have
Z
Z
2
2
mn dλkn ≤ λkn (Z̃n ) = λ(Zn ) ≤ + λ(X) = +
m dλ.
n
n
Ĝkn
Ĝ
12
Thus the condition (ii) is satisfied. (i) follows from the estimate
||ξ − ξn ||2 = µ(Ĝ\Zn ) <
1
,
n
since ξ is cyclic and any g ∈ G is eventually contained in Gkn .
Proof of Theorem. Let {λn }N
n=1 , N ≤ ∞, be the point spectrum of A1 , and en the
spectral projection corresponding to λn . Note that if mn is the multiplicity function of
the representation U (1) |en H1 then
Z
Tr (η(A(χ) + η(1 − A(χ))) dλ(χ) =
Ĝ
N
X
Z
(η(λn ) + η(1 − λn ))
mn (χ)dλ(χ).
Ĝ
n=1
Choosing an increasing sequence of finitely generated subgroups of G and applying
Lemma 4.4 to each subspace H2 , en H1 , 1 ≤ n ≤ N , we infer from Lemma 2.3(i) that
it suffices to consider the case where G ∼
= Zn and N < ∞. By Lemma 2.3(i), we can also
suppose that the multiplicity functions m(2) , mn , 1 ≤ n ≤ N , are finite sums of indicator
functions of compact sets.
Let G = Z. Suppose the assertion is proved under the additional assumption that
m(2) and mn , 1 ≤ n ≤ N , are finite sums of indicator functions of (closed) arcs of rational
length. Consider the general case. For a fixed ε > 0, since for any compact set X ⊂ T
there exists a set Y such that X ⊂ Y , Y is a finite union of disjoint arcs of rational length
and λ(Y \X) is arbitrary small, we can find multiplicity functions m̃(2) , m̃n , 1 ≤ n ≤ N ,
such that the functions m(2) + m̃(2) , mn + m̃n , 1 ≤ n ≤ N , are finite sums of indicator
functions of arcs of rational length and
N
X
Z
Z
(η(λn ) + η(1 − λn ))
m̃(2) dλ < ε.
m̃n dλ + (log 2)
n=1
Let H̃ (2) , H̃n , 1 ≤ n ≤ N , be the spaces of the corresponding representations. Set
1
H3 = H̃1 ⊕ . . . ⊕ H̃N ⊕ H̃ (2) and A3 = λ1 1H̃1 ⊕ . . . ⊕ λN 1H̃N ⊕ 1H̃ (2) .
2
Let ψ and α̃G be as in the formulation of Lemma 4.3. By assumption, Theorem is true
for α̃G , so that
hφ (αG ) ≤ hψ (α̃G ) ≤
N
X
Z
(η(λn ) + η(1 − λn ))
Z
(mn + m̃n )dλ + (log 2)
(m(2) + m̃(2) )dλ
n=1
<
N
X
Z
(η(λn ) + η(1 − λn ))
Z
mn dλ + (log 2)
m(2) dλ + ε.
n=1
Since this holds for all ε > 0, we conclude that
hφ (αG ) ≤
N
X
Z
(η(λn ) + η(1 − λn ))
Z
mn dλ + (log 2)
m(2) dλ.
n=1
It remains to consider the case where the multiplicity functions m(2) , mn , 1 ≤ n ≤ N ,
are finite sums of indicator functions of arcs of rational length. If q is a common
13
denominator of these rational numbers, we can pass to the subgroup qZ of Z (since
hφ (αZ ) = 1q hφ (αqZ ) ) thus supposing m(2) = p(2)´I, mn = pn´I, 1 ≤ n ≤ N , for certain
p(2) , p1 , . . . , pN ∈ N. Then the representations on en H and H2a (the absolutely continuous
part of U (2) ) are finite sums of bilateral shifts, so that there exist subspaces Kn ⊂ en H1 ,
K ⊂ H2a such that
(1)
dim Kn = pn , the spaces Uj Kn , j ∈ Z, are mutually orthogonal,
S
(1)
a
j∈Z Uj Kn = en H1 ; and analogously for K ⊂ H2 .
n
n
(1)
(2)
Set Zn = ⊕ Uj (K1 ⊕ . . . ⊕ KN ) and Xn = Zn ⊕ ⊕ Uj K . Then the proof of
j=1
j=1
Lemma 5.3 in [StV] shows that
hφ (αZ ) ≤ lim inf
n→∞
1
S(φ|A(Xn ) )
n
for any invariant state φ. Since A(Zn ) and A(Xn ) are full matrix algebras of dimensions
(2)
22n(p1 +...+pN ) and 22n(p +p1 +...+pN ) respectively, the subadditivity of the von Neumann
entropy implies
S(φ|A(Xn ) ) ≤ S(φ|A(Zn ) ) + np(2) log 2.
By (2.1), we have S(φ|A(Zn ) ) = n
N
X
(η(λk ) + η(1 − λk ))pk , so
k=1
hφ (αZ ) ≤
=
N
X
k=1
N
X
(η(λk ) + η(1 − λk ))pk + p(2) log 2
Z
(η(λk ) + η(1 − λk ))
mk dλ + (log 2)
T
k=1
Z
m(2) dλ,
T
and the proof for G = Z is complete.
The case G = Zn , n > 1, is analogous (see Lemma 3.7 and Theorem 3.8 in [BG]). We
leave the details to the reader.
a its absolutely
Theorem 4.5. Let U : G → B(H) be a unitary representation, U a |H
R⊕
continuous part, A ∈ B(H), 0 ≤ A ≤ 1, AUg = Ug A. Let H a = Ĝ Hχ dλ(χ) and
R⊕
A|H a = Ĝ A(χ)dλ(χ) be direct integral decompositions. If A|H a has pure point spectrum
then, for the Bogoliubov action αG corresponding to U , we have
Z
hωA (αG ) =
Tr (η(A(χ) + η(1 − A(χ))) dλ(χ).
Ĝ
Proof. The inequality ≤ is proved in Theorem 4.1. Let {λn }N
n=1 be the point spectrum
of A|H a , en the spectral projection corresponding to λn . Taking into account Remark 3.8
the same arguments as in [StV], Theorem 6.3, give us
hωA (αG ) ≥
N
X
n=1
Z
hωλn (αG |A(en H)e ) =
Tr (η(A(χ) + η(1 − A(χ))) dλ(χ).
Ĝ
14
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[BR2] O. Bratteli, D.W. Robinson. Operator Algebras and Quantum Statistical Mechanics, II. Springer, 1987.
[CNT] A. Connes, H. Narnhofer, W. Thirring, Dynamical entropy of C ∗ -algebras and von
Neumann algebras, Commun. Math. Phys., 112 (1987), 691–719.
[CS] A. Connes, E. Størmer, Entropy for automorphisms of II1 von Neumann algebras,
Acta Math., 134 (1975), 289–306.
[Co] J.P. Conze, Entropie d’un groupe abelien de transformations, Z. Wahrscheinlichkeitstheorie und verw. Geb., 25 (1972), 11–30.
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[K]
A. Kirillov. Elements of the Theory of Representations. Springer, 1976.
[L]
S. Lang. Algebra. Addison-Wesley, 1965.
[M]
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Institute for Low Temperature Physics & Engineering
Lenin Ave 47
Kharkov 310164, Ukraine
golodets@ilt.kharkov.ua
neshveyev@ilt.kharkov.ua
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