Dynamical entropy for Bogoliubov actions of torsion-free abelian groups on the CAR-algebra

Transcription

1 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 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 II 1. 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

2 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, α G, where A is a C -algebra, α G is an action of G on A by *-automorphisms, and φ is a G-invariant state on A. For given channels γ i : B i A, 1 i n, i.e. completely positive unital mappings of finite-dimensional C -algebras B i, H φ (γ 1,..., γ n 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 H φ (γ A the mutual entropy of the channels α g 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: Z n G, and m 1,..., m n N such that A = I ({z Z n 0 z k m k, 1 k n}. Definition 1.2. The entropy of the system (A, φ, α G with respect to a channel γ is the quantity h φ (γ; α G = inf H φ(γ A, A where A runs over the set of all parallelepipeds in G. The dynamical entropy of the system is h φ (α G = sup h φ (γ; α G. γ Remark 1.3. If A is commutative and γ is the inclusion of a finite-dimensional subalgebra P of A, h φ (P; α G may be defined as the infimum of H φ(p A 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 H φ (P A, i.e. H φ (P A B + H φ (P A B H φ (P A + H φ (P B. Apparently the function A H φ (γ A is not strongly subadditive in the non-commutative case. But it is at least subadditive [CNT], i.e. H φ (γ A B H φ (γ A + H φ (γ B. The following result is an immediate consequence of the subadditivity. 2

3 Proposition 1.4. Let G = Z n, n N. For N N, let A N denotes the cube {z Z n 0 z k N}. Then, for any channel γ, h φ (γ; α G = lim N H φ (γ A N. A N In particular, for G = Z n 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 {g i } N i=1, N, be a maximal linear independent system in G, G 1 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 {g i } i is a basis of the vector space G over Q. For an element x G, let x i Q denotes the i-th coordinate of x in this basis. Choose a set {s kn } 1 k N,n N of non-negative numbers such that (i lim n s kn = for any k; (ii for any n, only finitely many of s kn s are non-zero. Set A n = { x ( 1 n! G 1 } G 0 x k < s kn (x k = 0 if s kn = 0 (note that A n is not a parallelepiped in the sense of Definition 1.1. Then h φ (γ; α G = inf n H φ (γ An A n H φ (γ An = lim. n A n This result will not be used in the sequel. The next lemma follows from the definitions. Lemma 1.6. Let {G n } be an increasing sequence of subgroups of G such that ng n = G. Then h φ (γ; α Gn h φ (γ; α G. Proposition 1.7. Let (A, φ, α G be a C -dynamical system, {G n } a sequence of subgroups of G, A n a G n -invariant C -subalgebra of A, F n : A A n a completely positive unital mapping, F n id pointwise-norm (we don t require A n A n+1. Then (i h φ (α G lim inf h φ(α Gn An ; n (ii if A n s are G-invariant and F n s are φ-preserving conditional expectations then h φ (α G = lim n h φ(α G An. Proof. See the proof of Lemma 3.3 in [StV]. (i For a fixed channel γ: B A, let ε n = F n γ γ. By [CNT], Proposition IV.3, H φ (γ A H φ ((F n γ A A δ(ε n, d for any finite A G, where δ(ε n, d depends only on ε n and d = dim B, and δ(ε, d 0 as ε 0. Thus, for any parallelepiped A in G n, we have H φ An ((F n γ A A H φ((f n γ A A H φ(γ A A 3 δ(ε n, d h φ (γ; α G δ(ε n, d,

4 so that h φ (α Gn An h φ (F n γ; α Gn An h φ (γ; α G δ(ε n, d, whence lim inf h φ (α Gn An h φ (γ; α G. (ii follows from (i (G n = G n N and the fact that if there exists a φ-preserving conditional expectation onto a G-invariant subalgebra D of A then H φ D (γ 1,..., γ n = H φ (γ 1,..., γ n for any channels γ 1,..., γ n in D, hence h φ (α G D h φ (α G. If H is a subgroup of G, then h φ (γ; α H h φ (γ; α G, whence h φ (α H h φ (α G. The following proposition makes this relation more precise. Proposition 1.8. Let (A, φ, α G be a C -dynamical system, A nuclear, H a subgroup of G. Then (i if [G : H] <, then h φ (α H = [G : H] h φ (α G ; (ii if [G : H] = and h φ (α G > 0, then h φ (α H =. Proof. (i Let p = [G : H] <. First prove that h φ (γ; α H p h φ (γ; α G. Choose an increasing sequence {G n } of finitely generated subgroups of G such that n G n = G. By Lemma 1.6, for a given ε > 0, there exists n N such that h φ (γ; α Gn < h φ (γ; α G + ε. Since G n is a finite rank, free abelian group and H n = H G n is a subgroup of G n of index p, there exist a basis g 1,..., g m in G n and k 1,..., k m N such that k 1 g 1,..., k m g m is a basis in H n [L]. For N N, let A N be the cube {l 1 g l m g m 0 l i N 1} in G n. The set A N H n is a parallelepiped in H, and if k 1,..., k m divide N, then A N = [G n : H n ] A N H n, hence h φ (γ; α H H φ(γ A N H n A N H n and using Proposition 1.4 we obtain H φ(γ A N A N H n = [G n : H n ] H φ(γ AN A N h φ (γ; α H p h φ (γ; α Gn p h φ (γ; α G + p ε. p H φ(γ A N, A N Thus we have proved that h φ (α H p h φ (α G, and the assumption of nuclearity has not been used yet. To prove the inequality h φ (α G 1 p h φ(α H choose representatives ḡ 1,..., ḡ p 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 θ 1,..., θ p : B D such that θ θ i αḡi γ < ε, 1 i p. Let {G n } be an increasing sequence of finitely generated subgroups of G such that n G n = G and ḡ 1,..., ḡ p G n for any n. For a fixed n, there exist a basis g 1,..., g m in G n and numbers k 1,..., k m N such that k 1 g 1,..., k m g m is a basis in H n = H G n. The absolute values of coordinates of ḡ 1,..., ḡ p in this basis don t exceed a number N 0. For N N, let A N = {l 1 g l m g m 0 l i k i N 1}, Ã N = {l 1 g l m g m N 0 l i k i N N 0 1}, B N = A N H = {l 1 k 1 g l m k m g m 0 l i N 1}. 4

5 The sets ḡ i + B N, 1 i p, are mutually disjoint and ÃN i (ḡ i + B N. Thus ( h φ (γ; α G H φ(γ A N Hφ(γÃN + 1 ÃN H φ (γ, A N A N A N H φ (γãn H φ (γ i(ḡ i +B N H φ ({α g θ θ i g B N, 1 i p} + p B N δ(ε, d H φ (θ B N + p B N δ(ε, d, where we have used [CNT], Proposition IV.3 and Proposition III.6(a,(c. Since A N B N = [G n : H n ] = p, we obtain h φ (γ; α G 1 H φ (θ B N + δ(ε, d + p B N Letting N and using Proposition 1.4 we conclude that h φ (γ; α G 1 p h φ(θ; α Hn + δ(ε, d, ( 1 ÃN H φ (γ. A N and by Lemma 1.6, h φ (γ; α G 1 p h φ(θ; α H + δ(ε, d. So, due to the arbitrariness of ε, h φ (γ; α G 1 p h φ(α H. (ii Suppose [G : H] = and h φ (α H <, and prove that h φ (α G = 0. Consider two cases. a The group G/H is periodic. There exists an increasing sequence {H n } of subgroups of G such that H H n, [H n : H] <, [H n : H]. Then h φ (α G h φ (α Hn = 1 [H n : H] h φ(α H 0. b The rank of G/H is non-zero. Let g G be an element, whose image in G/H has infinite order. Let H n = H + nzg. Then h φ (α G h φ (α H1 = 1 n h φ(α Hn 1 n h φ(α H 0. 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 a(f is a linear map and a(fa(g + a(g a(f = (f, g1, a(fa(g + a(ga(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

6 If H 1 and H 2 are mutually orthogonal subspaces of H, then A(H 1 and A(H 2 e commute and the C -algebra they generate is identified with A(H 1 A(H 2 e. If 0 A 1 is an operator on H, then the quasi-free state ω A on A(H is given by ω A (a(f n... a(f 1 a(g 1... a(g m = δ nm det((ag i, f j. We will write ω λ instead of ω λi. The state ω 1 is the unique tracial state on A(H. 2 If H = H 1 H 2, A i B(H i, 0 A i 1, A = A 1 A 2, then ω A A(H1 A(H 2 e = ω A1 A(H1 ω A2 A(H2 e. Suppose there exists an orthonormal basis {f n } N, N, such that A f n = λ n f n. Then (A(H, ω A = N (Mat 2 (C n, ρ λn (2.1 n 1 via the homomorphism sending a(f n (1 2a(f i a(f i to e (n 12 (so a(f na(f n e (n 11, where ρ λ is the state on Mat 2 (C given by (( a b ρ λ = (1 λa + λd. c d i=1 Each unitary operator U on H defines a Bogoliubov automorphism α of A(H by α(a(f = a(uf. 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 ω A. If K is an invariant subspace for A and σ is the Bogoliubov automorphism corresponding to the operator 1 1 on H = K K, then id+σ 2 is an ω A -preserving conditional expectation of A(H onto A(K A(K e, and composing it with id ω A ( we obtain an ω A -preserving conditional expectation A(H A(K. Lemma 2.1. Let {P n } be a sequence of projections in B(H, P n 1 strongly. Let E n be a conditional expectation of A(H onto A(P n H. Then E n id pointwise-norm. Proof. The result easily follows from the facts that E n 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 H 1,..., H n be mutually orthogonal finite-dimensional subspaces of H invariant for A. Then H ωa (A(H 1,..., A(H n = S(ω A A(H1... H n = n Tr Hi (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, α G the corresponding Bogoliubov action on A(H, {G n } a sequence of subgroups of G, {P n} a sequence of projections in B(H such that H n = P n H is G n -invariant and P n 1 strongly. Then 6

7 (i for any G-invariant state φ on A(H, we have h φ (α G lim inf n h φ(α Gn A(Hn ; (ii if H n s are G-invariant then, for A B(H, 0 A 1, U g A = A U g, A H n H n, we have h ωa (α G = lim n h ω A (α G A(Hn. 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(H n be a unitary representation (n N, {χ n } a sequence of characters of G. Consider two unitary representations of G on H = H n, U g = U g (n, U g = χ n (gu g (n, and let α G and α G be the corresponding Bogoliubov actions. Then, for any α - and α - invariant state φ on A(H, we have h φ (α G = h φ (α G. Proof. For n N, let {f kn } k=1 be an orthonormal basis in H n. Let A m be the C - subalgebra of A(H generated by a(f kn, 1 k, n m. Then h φ (α G = lim m h φ(a m ; α G, h φ (α G = lim m h φ(a m ; α G by the proof of [CNT], Theorem V.2. On the other hand, since α g(a(f kn = χ n (gα g(a(f kn, we have α g(a m = α g(a m for any g G, hence h φ (A m ; α G = h φ (A m ; α G. 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 we can extend it to a *-representation f U f of the algebra of bounded Borel functions on. Then the spectral projection for a Borel subset X is the projection U IX. For a vector ξ H, the spectral measure µ ξ is a positive Borel measure on such that (U g ξ, ξ = χ(gdµ ξ (χ, g G. 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 µ ξ is 7

8 absolutely continuous (resp. singular with respect to the Haar measure λ on. 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 H = H χ dλ(χ, U a g = χ(gdλ(χ. The function m(χ = dim H χ 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 α G and β [0, 1], h ωβ (α G = (η(β + η(1 β m(χ dλ(χ, 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 ω β is pure, so that the entropy of any channel is zero and there is nothing to prove. Thus we can suppose β (0, 1. Lemma 3.2. Let q, l N, X = l 1 k=0 [ k l, k l + 1 ql ]. Then is an orthonormal basis in L 2 (X, dt. Proof. Note that {q 1/2 e 2πi(lqk+rt k Z, 0 r l 1} X e 2πint dt = ( l 1 k=0 nk 2πi e l 1 ql e 2πint dt. 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 exp(2πikt defines a unitary operator from Lin{e 2πilkt k Z} L 2 (X onto L 2 (0, 1 q. 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 L 2 (0, 1 q. Then exp(2πi(lp + rt, 0 r l 1, lies in the closed subspace spanned by exp(2πi(lqk + rt, k Z. Lemma 3.3. Let G = Z n. Suppose the multiplicity function m equals to p I X, where p N and X = exp ( 2πi ( l 1 k=0 0 [ k l, k l + 1 ] T... T ql } {{ }. n 1 So the space H of the representation can be identified with the sum of p copies of L 2 (X, dλ, H = p L 2 (X, dλ i. i=1 8

9 Let K be the subspace of H spanned by I X L 2 (X i, 1 i p. Then h ωβ (α G = h ωβ (A(K; α G = (η(β + η(1 β p q. Proof. The equality h ωβ (α G = (η(β + η(1 β p q is known: see [StV], Lemma 4.5, or [BG], Lemma 3.4. Thus we have only to prove that h ωβ (A(K; α G (η(β + η(1 β p q. For N N, let A N = {z Z n 0 z k N}, Ã N = A N {z Z n z 1 = lqk + r, k Z, 0 r l 1}. By Lemma 3.2, the subspaces U g K, g ÃN, are mutually orthogonal. So by Lemma 2.2, and using Proposition 1.4 we obtain H ωβ (A(KÃN = ÃN (η(β + η(1 βp, ÃN h ωβ (A(K; α G lim N A N (η(β + η(1 βp = (η(β + η(1 βp q. It is worth to note that the inequality h ωβ (α G (η(β + η(1 β p q deduced from the completeness assertion of Lemma 3.2. may also be Lemma 3.4. Let g G\{0}, X = g 1 (exp(2πi[0, 1 q ], where we consider g as a character of, m = p I X. Then h ωβ (α G = (η(β + η(1 β p q. Proof. The space H of the representation is identified with the sum of p copies of L 2 (X, dλ, H = p i=1 L2 (X, dλ i. Let K be the subspace of H spanned by I X L 2 (X i, 1 i p. There exists an increasing sequence {G n } of finitely generated subgroups of G such that n G n = G and g G n for any n. Let H n be the minimal G n -invariant subspace of H containing K. For a fixed k N, consider g as a character of k, and set Y k = g 1 (exp(2πi[0, 1 q ] k. Then the representation of G k on H k has the multiplicity function p I Yk. There exist a basis g 1,..., g n of G k and l N such that g = lg 1. Having fixed a basis we can identify k with T n. Then g maps (t 1,..., t n T n = k to t l 1, so that By virtue of Lemma 3.3, l 1 Y k = exp 2πi j=0 [ j l, j l + 1 ql ] T }. {{.. T }, n 1 h ωβ (α Gk A(Hk = h ωβ (A(K; α Gk A(Hk = (η(β + η(1 β p q, 9

10 and using Lemma 1.6 and Lemma 2.3(i we obtain h ωβ (α G = h ωβ (A(K; α G = (η(β + η(1 β p q. Lemma 3.5. Let H be a locally compact group, λ its left Haar measure, X 1 and X 2 measurable subsets of H, 0 < λ(x 1, λ(x 2 <. Then there exist a measurable subset Y of X 1, λ(y > 0, and h H such that hy X 2. Proof. The mapping h I X2 h L 1 (H, dλ is continuous. Hence there exists a neighbourhood U of the unit such that I X2 I X2 h λ(x 2 for any h U. We can find h 0 H with λ(h 0 U X 1 > 0. Let Y 1 = U h 1 0 X 1. Then λ(y 1 > 0 and 1 I X2 (xy dλ(x 1 X 2 2 λ(x 2 for any y Y 1, hence dλ(y dλ(x 1 I X2 (xy 1 Y 1 X 2 2 λ(x 2λ(Y 1, and changing the order of integration, 1 dλ(x dλ(y 1 I X2 (xy 1 λ(x 2 X 2 Y 1 2 λ(y 1. Hence there exists x 0 X 2 such that 1 I X2 (x 0 y dλ(y 1 Y 1 2 λ(y 1. In other words, if we set Ỹ = {y Y 1 x 0 y / X 2 }, then λ(ỹ 1 2 λ(y 1. Y = h 0 (Y 1 \Ỹ, we have λ(y 1 2 λ(y 1 > 0, Y X 1 and x 0 h 1 0 Y X 2. For a multiplicity function m and the corresponding Bogoliubov action α G, set µ β (m = h ωβ (α G η(β + η(1 β. Lemma 3.6. For integrable multiplicity functions, µ β (m depends only on m dλ. Thus, for Proof. Suppose m dλ = m dλ <. Since m and m are at most countable sums of indicator functions, Lemma 3.5 and a simple maximality argument ensure the existence of measurable subsets Y n, n N, and a sequence {χ n} such that m = I Yn and m = Then the result follows from Lemma 2.4. I χnyn a.e. 10

11 Lemma 3.7. If m n m a.e. then µ β (m n µ β (m. Proof. If H (resp. H n is the space of the representation U (resp. U (n with the multiplicity function m (resp. m n, then we may assume H 1 H 2... H and U (n = U Hn. It remains to apply Lemma 2.3(ii. Proof of Theorem. We have to prove that µ β (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, µ β (p I X( n q = µ β (pn I = pn q X( 1 q (Lemma 3.6 (Lemma 3.4. By Lemma 3.7, µ β (p I X(r = pr p N r [0, 1]. Finding p N and r [0, 1] with m dλ = pr we obtain µβ (m = m dλ by Lemma 3.6. Suppose m dλ =. Letting m n = m (n I and applying Lemma 3.7 we obtain µ β (m = lim µ β(m n = lim m n 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(H i be a unitary representation, i = 1, 2, A 1 B(H 1, 0 A 1 1, A 1 U g (1 = U g (1 A 1, m (2 the multiplicity function of the absolutely continuous part of U (2. Suppose the representation U (1 has absolutely continuous spectrum and A 1 has pure point spectrum. Let H 1 = H χdλ(χ and A 1 = A(χdλ(χ be direct integral decompositions (λ is the Haar measure on. Then, for the Bogoliubov action α G corresponding to the representation U (1 U (2 and for any G-invariant state φ on A(H 1 H 2 such that φ A(H1 = ω A1, we have h φ (α G Tr (η(a(χ + η(1 A(χ dλ(χ + (log 2 m (2 (χdλ(χ. 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 more representation U (3 : G B(H 3 and A 3 B(H 3, 0 A 3 1, A 3 U g (3 = U g (3 A 3. Let α G be the Bogoliubov action corresponding to U (1 U (2 U (3. Then there exists a 11

12 G-invariant state ψ on A(H 1 H 2 H 3 such that ψ A(H1 H 3 = ω A1 A 3 and h ψ ( α G h φ (α G. Proof. Let σ be the Bogoliubov automorphism corresponding to the operator Then E = id+σ 2 is a G-invariant conditional expectation of A(H 1 H 2 H 3 onto A(H 1 H 2 A(H 3 e (see Section 2. Set ψ = (φ ω A3 E. Then ψ is G-invariant, ψ A(H1 H 3 = ω A1 A 3, and since there exists a ψ-preserving conditional expectation onto A(H 1 H 2 (namely (id A(H1 H 2 ω A3 ( E, we have h ψ ( α G h φ (α G. Lemma 4.4. Let U be a unitary representation of G on H, m the multiplicity function of the absolutely continuous part of U, {G n } an increasing sequence of subgroups of G with n G n = G. Then there exist a sequence k n and, for each n N, a G kn -invariant subspace H n of H such that (i if P n is the projection onto H n, then P n id strongly ; (ii if m n is the multiplicity function of the absolutely continuous part of the representation U Gkn Hn, then lim sup m n dλ kn m dλ. n kn Proof. It suffices to prove Lemma for a cyclic representation. So let ξ H be a cyclic vector, µ its spectral measure, µ = µ a +µ s the decomposition into the sum of the absolutely continuous and the singular parts, { X = χ dµ } a dλ (χ > 0. Then the representation U is equivalent to the canonical representation on L 2 (, dµ = L2 (, dµ s L 2 (, dµ a = L 2 (, dµ s L 2 (X, dλ. In particular, m = I X. For each n N, there exists a compact subset X n of such that λ(x X n < 1 n and µ(\x n < 1 n. Then we can find an open Y n such that X n Y n and λ(y n \X n < 1 n. Denote by I n the inclusion G n G. Due to the compactness of X n and the equality = lim n, there exist k n n and a compact Z n k n such that, for Z n = Î 1 k n ( Z n, we have X n Z n Y n. Then λ(z n X λ(z n X n + λ(x n X < 2 n and µ(\z n < 1 n. Let E(Z n be the spectral projection corresponding to Z n. Set ξ n = E(Z n ξ, and let H n be the minimal G kn -invariant subspace containing ξ n. Then the spectral measure of ξ n (with respect to G kn is supported by Z n, so if m n is the multiplicity function of the absolutely continuous part of the representation of G kn on H n, we have m n dλ kn λ kn ( Z n = λ(z n 2 kn n + λ(x = 2 n + m dλ. 12

13 Thus the condition (ii is satisfied. (i follows from the estimate ξ ξ n 2 = µ(\z n < 1 n, since ξ is cyclic and any g G is eventually contained in G kn. Proof of Theorem. Let {λ n } N, N, be the point spectrum of A 1, and e n the spectral projection corresponding to λ n. Note that if m n is the multiplicity function of the representation U (1 enh1 then Tr (η(a(χ + η(1 A(χ dλ(χ = (η(λ n + η(1 λ n m n (χdλ(χ. Choosing an increasing sequence of finitely generated subgroups of G and applying Lemma 4.4 to each subspace H 2, e n H 1, 1 n N, we infer from Lemma 2.3(i that it suffices to consider the case where G = Z n and N <. By Lemma 2.3(i, we can also suppose that the multiplicity functions m (2, m n, 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 m n, 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 m (2, m n, 1 n N, such that the functions m (2 + m (2, m n + m n, 1 n N, are finite sums of indicator functions of arcs of rational length and (η(λ n + η(1 λ n m n dλ + (log 2 m (2 dλ < ε. Let H (2, Hn, 1 n N, be the spaces of the corresponding representations. Set H 3 = H 1... H N H (2 and A 3 = λ 1 1 H1... λ N 1 HN H(2. Let ψ and α G be as in the formulation of Lemma 4.3. By assumption, Theorem is true for α G, so that h φ (α G h ψ ( α G < (η(λ n + η(1 λ n (η(λ n + η(1 λ n Since this holds for all ε > 0, we conclude that h φ (α G (η(λ n + η(1 λ n (m n + m n dλ + (log 2 m n dλ + (log 2 m n dλ + (log 2 m (2 dλ + ε. m (2 dλ. (m (2 + m (2 dλ It remains to consider the case where the multiplicity functions m (2, m n, 1 n N, are finite sums of indicator functions of arcs of rational length. If q is a common 13

14 denominator of these rational numbers, we can pass to the subgroup qz of Z (since h φ (α Z = 1 q h φ(α qz thus supposing m (2 = p (2 I, m n = p n I, 1 n N, for certain p (2, p 1,..., p N N. Then the representations on e n H and H2 a (the absolutely continuous part of U (2 are finite sums of bilateral shifts, so that there exist subspaces K n e n H 1, K H2 a such that dim K n = p n, the spaces U (1 j K n, j Z, are mutually orthogonal, j Z U (1 j K n = e n H 1 ; and analogously for K H2 a. Set Z n = ( n n U (1 j (K 1... K N and X n = Z n j=1 j=1 Lemma 5.3 in [StV] shows that h φ (α Z lim inf n 1 n S(φ A(X n U (2 j K. Then the proof of for any invariant state φ. Since A(Z n and A(X n are full matrix algebras of dimensions 2 2n(p p N and 2 2n(p(2 +p p N 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 (η(λ k + η(1 λ k p k, so k=1 h φ (α Z = (η(λ k + η(1 λ k p k + p (2 log 2 k=1 (η(λ k + η(1 λ k k=1 T m k dλ + (log 2 m (2 dλ, T and the proof for G = Z is complete. The case G = Z n, n > 1, is analogous (see Lemma 3.7 and Theorem 3.8 in [BG]. We leave the details to the reader. Theorem 4.5. Let U: G B(H be a unitary representation, U a H a its absolutely continuous part, A B(H, 0 A 1, AU g = U g A. Let H a = H χdλ(χ and A H a = A(χdλ(χ be direct integral decompositions. If A H a has pure point spectrum then, for the Bogoliubov action α G corresponding to U, we have h ωa (α G = Tr (η(a(χ + η(1 A(χ dλ(χ. Proof. The inequality is proved in Theorem 4.1. Let {λ n } N be the point spectrum of A H a, e n the spectral projection corresponding to λ n. Taking into account Remark 3.8 the same arguments as in [StV], Theorem 6.3, give us h ωa (α G h ωλn (α G A(enH e = Tr (η(a(χ + η(1 A(χ dλ(χ. 14

15 References [BG] S.I. Bezuglyi, V.Ya. Golodets, Dynamical entropy for Bogoliubov actions of free abelian groups on the CAR-algebra, Ergod. Th.&Dynam. Sys., 17 (1997, [BR2] O. Bratteli, D.W. Robinson. Operator Algebras and Quantum Statistical Mechanics, II. Springer, [CNT] A. Connes, H. Narnhofer, W. Thirring, Dynamical entropy of C -algebras and von Neumann algebras, Commun. Math. Phys., 112 (1987, [CS] [Co] A. Connes, E. Størmer, Entropy for automorphisms of II 1 von Neumann algebras, Acta Math., 134 (1975, J.P. Conze, Entropie d un groupe abelien de transformations, Z. Wahrscheinlichkeitstheorie und verw. Geb., 25 (1972, [GN] V.Ya. Golodets, S.V. Neshveyev, Non-Bernoullian quantum K-systems, Commun. Math. Phys., 195 (1998, [K] A. Kirillov. Elements of the Theory of Representations. Springer, [L] S. Lang. Algebra. Addison-Wesley, [M] J. Moulin Ollagnier. Ergodic theory and statistical mechanics. Lect. Notes in Math., 1115, Springer, [StV] E. Størmer, D. Voiculescu, Entropy of Bogoliubov automorphisms of the Canonical Anticommutation Relations, Commun. Math. Phys., 133 (1990, Institute for Low Temperature Physics & Engineering Lenin Ave 47 Kharkov , Ukraine golodets@ilt.kharkov.ua neshveyev@ilt.kharkov.ua 15