Entropy of the Shift on II₁-representations of the Group S(∞)
We have obtained the explicit formulae for the CNT-entropy of the shift on II₁-representations of the infinite symmetric group S(∞) in terms of Thoma-parameters.
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irk-123456789-1245792017-09-30T03:04:12Z Entropy of the Shift on II₁-representations of the Group S(∞) Boyko, M.S. Nessonov, N.I. We have obtained the explicit formulae for the CNT-entropy of the shift on II₁-representations of the infinite symmetric group S(∞) in terms of Thoma-parameters. 2005 Article Entropy of the Shift on II₁-representations of the Group S(∞) / M.S. Boyko, N.I. Nessonov // Український математичний вісник. — 2005. — Т. 2, № 1. — С. 15-37. — Бібліогр.: 20 назв. — англ. 1810-3200 2000 MSC. 37A35; 37B40; 20C32. http://dspace.nbuv.gov.ua/handle/123456789/124579 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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We have obtained the explicit formulae for the CNT-entropy of the shift on II₁-representations of the infinite symmetric group S(∞) in terms of Thoma-parameters. |
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Article |
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Boyko, M.S. Nessonov, N.I. |
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Boyko, M.S. Nessonov, N.I. Entropy of the Shift on II₁-representations of the Group S(∞) Український математичний вісник |
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Boyko, M.S. Nessonov, N.I. |
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Boyko, M.S. |
| title |
Entropy of the Shift on II₁-representations of the Group S(∞) |
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Entropy of the Shift on II₁-representations of the Group S(∞) |
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Entropy of the Shift on II₁-representations of the Group S(∞) |
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Entropy of the Shift on II₁-representations of the Group S(∞) |
| title_full_unstemmed |
Entropy of the Shift on II₁-representations of the Group S(∞) |
| title_sort |
entropy of the shift on ii₁-representations of the group s(∞) |
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Інститут прикладної математики і механіки НАН України |
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2005 |
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Entropy of the Shift on II₁-representations of the Group S(∞) / M.S. Boyko, N.I. Nessonov // Український математичний вісник. — 2005. — Т. 2, № 1. — С. 15-37. — Бібліогр.: 20 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT boykoms entropyoftheshiftonii1representationsofthegroups AT nessonovni entropyoftheshiftonii1representationsofthegroups |
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2025-07-09T01:39:18Z |
| last_indexed |
2025-07-09T01:39:18Z |
| _version_ |
1837131534639824896 |
| fulltext |
Український математичний вiсник
Том 2 (2005), № 1, 15 – 37
Entropy of the Shift on II1-representations of
the Group S(∞)
M. S. Boyko and N. I. Nessonov
(Presented by Yu. M. Beresanskii)
Abstract. We have obtained the explicit formulae for the CNT-entropy
of the shift on II1-representations of the infinite symmetric group S(∞)
in terms of Thoma-parameters.
2000 MSC. 37A35; 37B40; 20C32.
Key words and phrases. CNT-entropy, factor representation, infinite
symmetric group.
1. Introduction
Entropy is one of the most important notion in the information the-
ory and the ergodic theory. Initially entropy has appeared in the Claude
Shannon’s applied works. Next Kolmogorov and Sinai developed the im-
portant invariant, namely the entropy for an automorphism of an Abelian
W ⋆-algebra (see [9], [15], [16]). In 1975 the entropy for an automorphism
of a non-abelianW ⋆-algebra with a central state was defined by A. Connes
and E. Størmer (see [5]). The final definition was given in the paper of
Connes, Narnhofer and Tirring in 1987 (see [4]). This one is usually called
by the quantum dynamical entropy or the CNT-entropy.
The CNT-entropy is calculated for many non-commutative dynami-
cal systems of the topological, algebraic or physical origin. We consider
in our work the dynamical systems generated by the shift automorphism
on the II1-representations of the infinite symmetric group S(∞). The
group S(∞) has been often quoted as a typical example of ICC-groups
and hence of groups of non-type I. For that reason S(∞) involves a
number of interesting features which not observed in groups of type I.
Dynamical systems generated by the non-commutative shift have been
Received 25.02.2004
Supported in part by CRDF grant UM1-2546
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
16 Entropy of the Shift...
investigated beginning from the introduction of the notion of the CNT-
entropy. Connes and Størmer obtained the explicit formulae for the
non-commutative Bernoulli shift (see [5]). In the work of Størmer and
Golodets the similar results was obtained for the binary shift on a CAR-
algebra (see [7]). The main examples for which the C∗-algebra entropy
have been computed, are those of quasifree states of the CAR and CCR-
algebras and invariant Bogoliubov (or quasifree) automorphisms (see [2],
[6], [11], [12], [14], [17]). In our work [3] the Bogoliubov automorphisms
on the II1-representations of U(∞) are defined and the explicit formulae
for the CNT-entropy are obtained in the case of elementary characters.
Using the results of the present work for a low estimation of the CNT-
entropy of the shift on the II1-representations of U(∞) we obtain the
formulae for the Bogoliubov automorphism in the case of a general char-
acter (this results will be published in the separate paper).
Denote by S(2n+1)=S (Bn) the group of permutations of the set Bn =
{−n, . . . , 0, . . . , n}. If A and B are two sets and B ⊂ A, then we identify
S (B) with the subgroup {g ∈ S (A) : ga = a ∀ a ∈ A\B} of S (A). Let
S(∞) =
⋃
n
S(2n+ 1). Thoma has obtained the full description of II1-
factor-representations of group S(∞). Corresponding normalized charac-
ters χ
(S)
α,β are labelled by a pair of sequences of real numbers {αi} = α,
{βi} = β, i = 1, 2, . . ., such that αi ≥ αi+1 ≥ 0, βj ≥ βj+1 ≥ 0 ∀ i, j ∈ N,∑
αi +
∑
βj ≤ 1. The value of a character χ
(S)
α,β on a permutation with a
single cycle of length k is equal to
∑
j
αk
j + (−1)k−1
∑
j
βk
j (1.1)
Its value on a permutation with several disjoint cycles equals to the prod-
uct of its values on each cycle. As usual, it is assumed that an empty
product equals to 1. In particular, the character of the regular represen-
tation of the group S(∞) corresponds to the sequences αj ≡ 0, βj ≡ 0.
The bijection i ∈ Z → i + 1 ∈ Z defines naturally an automorphism
ϑS of the group S(∞), which extends up to the automorphism ϑχ
S of the
II1-factor built by the representation that corresponds to the character
χ. We denote by Hχ(θ) the CNT-entropy of an automorphism θ of the
II1-factor.
The main result of our work is following
Theorem 1.1. Let χ=χ
(S)
α,β, let η(t)=−t ln t and γ=1− (
∑
αi+
∑
βj).
(i) If γ > 0 then Hχ
(
ϑχ
S
)
= ∞.
(ii) If γ = 0 then Hχ
(
ϑχ
S
)
=
∑
j
η (αj) +
∑
j
η (βj).
M. S. Boyko, N. I. Nessonov 17
2. The Case γ > 0
In this section we will consider the case γ > 0.
Theorem 2.1. Let χ = χ
(S)
α,β and γ = 1 − (
∑
αi +
∑
βj) > 0, then
Hχ
(
ϑχ
S
)
= ∞.
We will prove several subsidiary statements.
Consider the complex type II1 factor-representation Πχ of the group
S(∞) which corresponds to the normalize character χ (see (1.1)). We
assume that Πχ is realized in Hilbert space Hχ which is the closure of the
linear span of vectors u ∈ S(∞) with the scalar product 〈u, v〉χ = χ(uv∗).
In Hχ we define the unitary representations lχ and rχ of the group S(∞):
lχ(u)v = uv, rχ(u)v = vu∗. (2.1)
Let us denote by Lχ (Rχ) the W ⋆-algebra generated by lχ (S(∞))
(rχ (S(∞))) and denote by Hχ (N1, N2, . . . , Nk) a CNT-entropy of a sys-
tem of finite-dimensional subalgebras N1, N2, . . . , Nk ⊂ Lχ (see [5]).
If A is an operator family and A′ is the commutant of A then L′
χ = Rχ.
Definition 2.1. A normalize character χ on G is called an indecompos-
able one if algebra Lχ (Rχ) is a factor.
Lemma 2.1. Let A ⊂ Bn and let W ⋆-algebra Lχ (S (A)) be generated by
operators lχ(g) (g ∈ S (A)). If χ is an indecomposable normalize char-
acter on S(∞) then
Hχ(ϑχ
S) ≥ Hχ (Lχ (S (A)))
2n+ 1
(2.2)
Proof. Let trχ be a trace on Lχ that corresponds to character χ. If
α = (ϑχ
S)2n+1, Nk = αk (Lχ (S (A))), then the following properties hold
true:
i) Nk are pairwise commute for any k ∈ Z, where Z is the set of
integers;
ii) if n1, n2 ∈ Z and n1 < n2, then ∃ a masa 1 A ⊂
n2∨
n1
Nk for which
Ak = A
⋂
Nk is the masa in Nk;
iii) A =
n2∨
n1
Ak and trχ
(
n2∏
k=n1
ak
)
=
n2∏
k=n1
trχ (ak) ∀ ak ∈ Ak.
1maximal abelian subalgebra
18 Entropy of the Shift...
From these statements and properties (D), (E) [5] it follows that
Hχ (Nn1 ,Nn1+1, . . . ,Nn2)=Hχ
(n2∨
n1
Ak
)
=(n2 − n1 + 1)Hχ(Lχ(S (A))).
Thus (2n+ 1)Hχ
(
ϑχ
S
)
= Hχ(α) ≥ Hχ (Lχ (S (A))).
Next statement allows a lower boundary for the entropyHχ(Lχ(S(A)))
in a case of the regular representation.
Lemma 2.2. Let A be the same one as in Lemma 2.1. If χ is a character
of the regular representation, then there exists a number C which does not
depend on A and Hχ (Lχ (S (A))) ≥ C · |A| · ln |A|.
Proof. Let |A| = m, and let χ(λ) be a character of an irreducible repre-
sentation πλ of the group S(m) = S (A) which corresponds to the Young
diagram λ, dimλ = dimπλ, χ
(λ)
norm = χ(λ)
dim λ . If χm is a restriction of χ on
S(m) and |λ| is the number of boxes in λ, then
χm =
∑
λ:|λ|=m
(dimλ)2
m!
χ(λ)
norm. (2.3)
We denote by eλ the minimal projection in W ∗-algebra (πλ (S(m)))′′
which is generated by operators πλ (g) (g ∈ S(m)). h(p, q) will denote
the corresponding hook length for a box (p, q) ∈ λ. Recall the well-known
hooks-formula
dimλ = m! ·
∏
(p,q)∈λ
1
h(p, q)
. (2.4)
Using (2.3) and (2.4), we obtain
χm (eλ) =
∏
(p,q)∈λ
1
h(p, q)
. (2.5)
It implies that
Hχ (Lχ (S (A))) =
∑
λ:|λ|=m
−dimλ · χm (eλ) · ln (χm (eλ))
=
∑
λ:|λ|=m
(dimλ)2
m!
· ln
( ∏
(p,q)∈λ
h(p, q)
)
. (2.6)
M. S. Boyko, N. I. Nessonov 19
Using the following inequality belonged to Vershik and Kerov (see [18])
and (2.3)
exp
[c0
2
√
m
]
·
√
m! ≤ min
λ:|λ|=m
∏
(p,q)∈λ
h(p, q) ≤ exp
[c1
2
√
m
]
·
√
m!,
where c0 and c1 are positive integers , from (2.6) we obtain
Hχ (Lχ (S (A))) ≥ c0
2
√
m+
1
2
· ln (m!) .
So the statement of our lemma follows from Stirling’s formula.
Now let us take for χ an arbitrary indecomposable normalize character
on S(∞). If M is an injective finite factor with normalize trace tr, then
there is a representation
πχ : S(∞) → U(M)
with the property
χ(g) = tr (πχ(g)) .
Here U(M) denotes a group of unitary operators in M.
Consider the following operator limit in the weak operator topology
lim
n→∞
πχ ((i, n)) = Ai, (2.7)
where (i, n) ∈ S(∞) is a transposition. It is obviously, that Ai = A∗
i . Let
µ be a spectral measure of operator Ai:
∫
xkµ(dt) = tr
(
Ak
i
)
∀ k ∈ N.
We denote by N/g a set of orbits of a permutation g on the set N. Denote
by |p| the cardinality of an orbit p ∈ N/g. The following statement
belongs to A. Okounkov (see [13]).
Lemma 2.3. The following properties are true:
a) AiAj = AjAi ∀ i, j ∈ Z and tr
(∏
l
Akl
jl
)
=
∏
l
tr
(
Akl
jl
)
∀ kl ∈ Z+ =
N
⋃{0};
b) πχ (g)Aiπχ
(
g−1
)
= Ag(i);
c) suppµ ⊂ [−1, 1], the measure µ is discrete and ∀ ε > 0 a set
[−1,−ε]⋃[ε, 1] contains at the most 2
ε its atoms;
20 Entropy of the Shift...
d) Let fi, gi (i ∈ Z) are functions on [−1, 1] which are pointwise limits
of uniformly bounded sequences of continuous functions. If all of
fi, gi (i ∈ Z) but finitely many identically equal to 1, then
tr
(∏
i∈Z
ḡi (Ai)πχ (g)
∏
i∈Z
fi (Ai)
)
=
∏
p∈N/g
∫
x|p|−1
∏
i∈p
fi(t)gi(t) dµ;
e) ∀x 6= 0 ν(x) = µ(x)
|x| ∈ Z+;
f) if χ = χ
(S)
α,β (see (1.1)), x 6= 0 and x ∈ suppµ, then ∃ i ∈ N, for
which
{
αi = x, . . . , αi+ν(x)−1 = x if x > 0,
βi = |x|, . . . , βi+ν(x)−1 = |x| if x < 0.
Denote by δx the function that equals to 1 at the point x, and that
equals to 0 at all the rest points. Let Ei = δ0 (Ai).
The next statement easily follows from the previous lemma.
Corollary 2.1. Let χ = χ
(S)
α,β, γ = tr (En) = 1 −∑
i
(αi + βi), and let
Ak = {i1, i2, . . . , ik} be a set of different numbers from Z. If EAk
=
k∏
j=1
Eij , γ > 0, then for g ∈ S (A)
ϕγ,k (g) = γ−k · tr (EAk
πχ(g)EAk
) =
{
1 if g = e,
0 otherwise.
From here and from lemma 2.2 it follows the next
Lemma 2.4. If γ ∈]0, 1[, and if Ei (|i| ≤ n) and πχ (S (Bn)) generates
a W ∗-algebra Mn, then there is some constant C1 which doesn’t depends
on n and such that
Hχ (Mn) ≥ C1 · n lnn.
Proof. Let us use the notations of Corollary 2.1. By Lemma 2.2 there
exists a constant C which does not depend on k and C is such that
Hϕγ,k
(EAk
πχ (S (Ak))EAk
) ≥ C · k ln k.
M. S. Boyko, N. I. Nessonov 21
Taking into consideration this result, we obtain
Hχ (Mn) ≥
n∑
k=0
∑
λk:|λk|=k
(
n
n
)
(1 − γ)n−k γk (dimλk)
2
k!
×
[
ln
( ∏
(p,q)∈λk
h(p, q)
)
− k ln γ − (n− k) ln(1 − γ)
]
≥ −n (γ ln γ + (1 − γ) ln(1 − γ)) + C
n∑
k=0
(
n
k
)
(1 − γ)n−k γkk ln k.
(2.8)
Now we take a constant d > 0 for which
[nγ+d
√
n]∑
k=[nγ−d
√
n]
(
n
k
)
(1 − γ)n−k γk >
1
2
∀ n ∈ N.
Taking into account this and (2.8), we have
Hχ (Mn)≥−n (γ ln γ + (1 − γ) ln(1 − γ))+
C
2
[
nγ − d
√
n
]
ln
[
nγ − d
√
n
]
.
Thus, the statement of Lemma 2.4 is proved.
Proof of Theorem 2.1. If γ = 1, then the statement of Theorem 2.1
follows from Lemmas 2.1 and 2.2. Let γ < 1. Using a method we have
proved Lemma 2.1, we receive the following estimation
Hχ
(
ϑχ
S
)
≥ Hχ (Mn)
2n+ 1
(see Lemma 2.4).
Thus, the statement of Theorem 2.1 follows from Lemma 2.4.
3. The Case of γ = 0
In this section we will present two different entropy estimation meth-
ods developed for the case of finite cardinality of set I = {i : αi > 0}∪{i :
βi > 0} and for the case of infinite one correspondingly. First method
is based on the important formulaes from the theory of symmetric func-
tions. The second one uses the structural properties of von Neumann
factors constructed by the representations of S(∞). It will be clear, that
the case of finite cardinality can be included in the second one, but we
would like to show special technic in the Subsection 3.1.
22 Entropy of the Shift...
3.1. The Subcase |I| <∞
In this Section we will prove the following theorem.
Theorem 3.1. Let η(t) = −t ln t, ∑αi +
∑
βj = 1, χ = χ
(S)
α,β and let
N ∈ N exist for which αj = βj = 0 ∀ j > N . Then
Hχ
(
ϑχ
S
)
=
∑
j
η (αj) +
∑
j
η (βj) .
Consider the restriction of χ onto a finite symmetric group S (A).
The characters of the finite symmetric group S (A) are labeled by the
Young diagrams with |A| boxes. Let χ(λ) be a (non normalized) character
corresponding to an irreducible representation λ. The restriction χS(A)
to the group S (A) is a non-negative linear combination of the functions
χ(λ)
χ
∣∣
S(A)
=
∑
λ:|λ|=|A|
s̃λ(α, β) · χ(λ). (3.1)
The Fourier coefficient s̃λ(α, β) is given by the extended Schur function
(see [8]), which can be formally defined by Jacoby-Trudi determinant
s̃λ(α, β) =
∣∣∣∣∣∣∣∣∣∣
hλ1 hλ1+1 hλ1+2 . . . hλ1+m−1
hλ2−1 hλ2 hλ2+1 . . . hλ2+m−2
hλ3−2 hλ3−1 hλ3 . . . hλ3+m−3
. . . . . . . . . . . . . . .
hλm−m+1 hλm−m+2 hλm−m+3 . . . hλm
∣∣∣∣∣∣∣∣∣∣
, (3.2)
where the extended complete homogeneous symmetric functions hl =
hl (α, β) arise as the coefficients of the generating series
ezγ
∞∏
j=1
1 + zβj
1 − zαj
= 1 +
∞∑
l=1
hl(α, β)zl.
We denote by d = d(λ) the number of diagonal boxes in the Young
diagram λ and we will use the Frobenius notation [10]
λ = (p1, . . . , pd|q1, . . . , qd) .
Here pi = λi − i is a number of boxes in the i−th row of λ on the right
of the i−th diagonal box; likewise, qi = λ′i − i is the number of boxes
in the i−th column of λ below the i−th diagonal box (λ′ stands for the
transposed diagram).
M. S. Boyko, N. I. Nessonov 23
Lemma 3.1. Let α = {αi}∞i=1, β = {βi}∞i=1 be Thoma-parameters,
∞∑
i=1
(αi + βi) = 1, Nα =max {i ∈ N : αk > 0}, Nβ =max {i ∈ N : βk > 0}.
If max {Nα,Nβ} <∞, then sλ(α, β) = 0 in each of the following cases
i) d(λ) > d = max {Nα,Nβ};
ii) λi > d ∀ i = Nβ + 1, . . . , d;
iii) λ′i > d ∀ i = Nα + 1, . . . , d.
Proof. We consider a sequence of the Young diagrams
λ(2n+1) =
(
p
(2n+1)
1 , . . . , p
(2n+1)
d |q(2n+1)
1 , . . . , q
(2n+1)
d
)
with properties:
i)
∣∣λ(2n+1)
∣∣ = 2n + 1 and d = d
(
λ(2n+1)
)
= max {Nα,Nβ} for n
sufficiently great;
ii) αi = lim
n→∞
p
(2n+1)
i
2n+1 , βi = lim
n→∞
q
(2n+1)
i
2n+1 ∀ i = 1, . . . , d.
It follows from the approximation Theorem [19] that
χ(g) = χ
(S)
α,β(g) = lim
n→∞
χ(λ(2n+1))(g)
dimλ(2n+1)
∀ g ∈ S(∞).
Using this claim, property i) and the Young branching rule
χ(Λ)
∣∣
S(|Λ|) =
∑
λ:Λցλ
χ(λ),
where the notation Λ ց λ means that diagram λ ⊂ Λ is obtained from
the diagram Λ by removing a box, we obtain the statement of the lemma.
Further we will need the Berele-Regev formula (see [1]) for the super-
symmetric Schur functions sλ
sλ (x1, . . . , xd; y1, . . . , yd) =
det
[
x
pj
i
]d
i,j=1
V (x1, . . . , xd)
·
det
[
y
qj
i
]d
i,j=1
V (y1, . . . , yd)
d∏
i,j=1
(xi + yj) .
(3.3)
Here λ = (p1, . . . , pd|q1, . . . , qd), V (. . .) is the Vandermonde determinant
and the parameters x1, . . . , xd, as well as y1, . . . , yd, are assumed to be
pairwise distinct.
24 Entropy of the Shift...
If
∞∑
i=1
(αi + βi) = 1, then the extended Schur (3.1) function coincides
with the supersymmetric Schur function
s̃λ(α, β) = sλ(α, β).
Now we will obtain the lower boundary for entropy Hχ (Lχ (S (n))) (see
Lemma 2.1).
Lemma 3.2. Let parameters α = {αi}∞i=1 and β = {βi}∞i=1 satisfy the
conditions of Lemma 3.1, χ = χ
(S)
α,β. Then ∀ ε > 0 ∃ N (ε) ∈ N for which
Hχ (Lχ (S (n))) ≥ −n(1 − ε)
{Nα∑
j=1
[αj − ε] · lnαj
+
Nβ∑
j=1
[βj − ε] · lnβj
}
+ N lnn ∀n > N (ε),
where N is a constant, which does not depend on n.
Proof. Let Yn(d) be a set of Young diagrams λ such, that |λ| = n and
d(λ) ≤ d. For k < d we define two sets
Yn(d, k) =
{
λ ∈ Yn(d) : λ′i ≤ d ∀ i = k + 1, k + 2, . . .
}
,
Y ′
n(d, k) = {λ ∈ Yn(d) : λi ≤ d ∀ i = k + 1, k + 2, . . .} .
We assume, that Nα ≥ Nβ. By Lemma 3.1, we have
χ
∣∣
S(n)
=
∑
λ∈Yn(Nα,Nβ)
sλ(α, β) · χ(λ). (3.4)
Let
Yn(d, k, ε) =
{
λ ∈ Yn(d, k) : λ′i = Nα ∀ i = Nβ + 1, . . . ,Nα,
|pi(λ) − nαi| < nε and |qj(λ) − nβj | < nε
∀ i = 1, . . . , d; j = 1, . . . , k
}
. (3.5)
Using (3.4) and (3.5), by the approximation Theorem [19] we obtain,
that there exists N (ε) ∈ N for which
1 ≥
∑
λ∈Yn(Nα,Nβ ,ε)
dimλ · sλ(α, β) > 1 − ε ∀n > N (ε). (3.6)
M. S. Boyko, N. I. Nessonov 25
Formula (3.3) can be extended by a continuity to the case, when the
number of parameters x1, . . . , xn is not equal to the number of parame-
ters y1, . . . , ym. We assume that, the parameters {α1 ≥ . . . ≥ αNα > 0}
are pairwise distinct as well as the parameters
{
β1 ≥ . . . ≥ βNβ
> 0
}
.
The next statement is obtained for the diagram λ = (p1, . . . , pNα |
q1, . . . , qNα) ∈ Yn (Nα,Nβ, ε) from relation (3.3) by passing to the limit
( βNβ+1 → 0, . . ., βNα → 0)
sλ
(
α1, . . . , αNα ;β1, . . . , βNβ
)
=
det
[
α
pj
i
]Nα
i,j=1
V (α1, . . . , αNα)
×
det
[
β
qj
i
]Nβ
i,j=1
V
(
β1, . . . , βNβ
)
Nα∏
i=1
[
α
Nα−Nβ
i
Nβ∏
j=1
(αi + βj)
]
. (3.7)
Now we consider the case, when there are the coincident parameters. Let
{ni(α)}kα
i=1 and {ni(β)}kβ
i=1 be subsets in N with the properties:
kα∑
i=1
ni(α) = Nα,
kβ∑
i=1
ni(β) = Nβ,
αn1(α)+...+nj(α)+1 = . . . = αn1(α)+...+nj(α)+nj+1(α) = tj ,
βn1(β)+...+nj(β)+1 = . . . = βn1(β)+...+nj(β)+nj+1(β) = sj ,
the parameters t1, . . . , tkα , are pairwise distinct as well as
s1, . . . , skβ
.
(3.8)
If
Tjk =
tpk
r if j =
r+1∑
i=1
ni(α),
m−1∏
i=1
(pk − i+ 1) tpk−m+1
r if j = −m+
r+1∑
i=1
ni(α),
where m = 1, . . . , nr+1(α) − 1;
Sjk =
sqk
r if j = 1 +
r+1∑
i=1
ni(β),
m−1∏
i=1
(qk − i+ 1) sqk−m+1
r if j = −m+
r+1∑
i=1
ni(β),
where m = 1, . . . , nr+1(β) − 1;
then we can rewrite (3.7) as follows:
sλ (α, β) =
detT
∏
1≤l<j≤kα
(tl − tj)
nl(α)·nj(α) · (nj(α) − 1)! (nl(α) − 1)!
26 Entropy of the Shift...
× detS
∏
1≤l<j≤kβ
(sl − sj)
nl(β)·nj(β) · (nj(β) − 1)! (nl(β) − 1)!
×
kα∏
j=1
kβ∏
i=1
(tj + si)
nj(α)·ni(β) . (3.9)
Using inequality
xn1
1 xn2
2 . . . xnk
k ≥ xn1
π(1)x
n2
π(2) . . . x
nk
π(k),
where π is a permutation, 0 < xk ≤ . . . ≤ x2 ≤ x1, ni ∈ N (1 ≤ i ≤ k)
and 0 < nk ≤ . . . ≤ n2 ≤ n1, from (3.9) we have
sλ (α, β) ≥
PT (p1, . . . , pNα)
kα∏
i=1
t
ni(α)
(
2pi−ni(α)+1
2
)
i
∏
1≤l<j≤kα
(tl − tj)
nl(α)·nj(α)
Nα∏
i=1
[
α
Nα−Nβ
i
]
×
PS
(
q1, . . . , qNβ
) kβ∏
i=1
s
ni(β)
(
2pi−ni(β)+1
2
)
i
∏
1≤l<j≤kβ
(sl − sj)
nl(β)·nj(β)
·
kα∏
j=1
kβ∏
i=1
(tj + si)
nj(α)·ni(β) .
(3.10)
Here PT (PS) is a polynomial of Nα − kα (Nβ − kβ) degree with coeffi-
cients which does not depend on n. Thus, we have
Hχ (Lχ (S (n))) = −
∑
λ:|λ|=n
dimλ · sλ (α, β) · ln sλ (α, β)
≥ −
∑
λ∈Yn(Nα,Nβ)
dimλ · sλ(α, β) · ln sλ (α, β)
see (3.10), (3.8)
≥ −
∑
λ∈Yn(Nα,Nβ)
dimλ · sλ(α, β)
×
[( Nα∑
j=1
pi · lnαi +
Nβ∑
j=1
qi · lnβi
)
+(Nα + Nβ − kα − kβ) lnn+C(α, β)
]
.
Here C(α, β) is a constant that does not depend on n. From here, taking
into account (3.5) and (3.6), we obtain the statement of the lemma.
The case when Nα < Nβ can be considered analogous by taking Yn(·, ·)
instead of Y ′
n(·, ·).
M. S. Boyko, N. I. Nessonov 27
3.2. The Subcase of Infinite Cardinality
Next we consider the case of the infinite number of nonzero param-
eters {αi} = α, {βi} = β and obtain a lower boundary for the entropy
Hχ (Mn), where Mn is generated by A0 and πχ (S(n)) as a W ∗-algebra.
Lemma 3.3. If
∑
αi +
∑
βj = 1, then
Hχ
(
ϑχ
S
)
≥
∑
i
(η (ν(αi) · αi)
ν(αi)
+
η (ν(βi) · βi)
ν(βi)
)
,
where ν is the multiplicity function (see Lemma 2.3).
Proof. Let χ = χ
(S)
α,β and let πχ be the representation that corresponds
to χ. We denote by A the W ∗-algebra which is generated by operators
{Ai}i∈Z
(see Lemma 2.3). Since ϑχ
S (Ai) = Ai+1 (2.7), ϑχ
S restricts to
an automorphism of A. So we get, using properties a), c), d), e), f) of
Lemma 2.3, that the Abelian dynamical system
(
A, ϑχ
S , tr
)
is the classical
Bernoulli shift with the entropy
∑
i
(η (ν(αi) · αi)
ν(αi)
+
η (ν(βi) · βi)
ν(βi)
)
.
Let us consider the following union {αi} =
⋃
j Uj , where ∀αk, αl ∈
Uj , αk = αl, ∀αk ∈ Uj , αl ∈ Um, αk > αl if j < m. Next we define
α′ = {α′
i} such that ∀ i α′
i ∈ Ui, α
′
i 6= 0 and ∀ i, j, i 6= j α′
i 6= α′
j . In the
same way we define the sequence β′ = {β′i}.
Let ai = ν (α′
i)α
′
i, bi = ν (β′i)β
′
i and let
Nα,β (m, kα, kβ , D)
=
{(
mα
1 , . . . ,m
α
kα
,mβ
1 , . . . ,m
β
kβ
,mkα+kβ+1
)
∈
kα+kβ+1
×
j=1
N :
(
aim−D
√
m ≤ mα
i ≤ aim+D
√
m
)
∧(
bjm−D
√
m ≤ mβ
j ≤ bjm+D
√
m
)
∧( kα∑
j=1
mα
j +
kβ∑
j=1
mβ
j +mkα+kβ+1 = m
)
∀ i = 1, 2 . . . , kα; j = 1, 2 . . . , kβ
}
.
(3.11)
The next statements follows from the central limit theorem.
28 Entropy of the Shift...
Lemma 3.4. Let
∑
αi +
∑
βj = 1, aj = ν
(
α′
j
)
α′
j, bj = ν
(
β′j
)
β′j,
γkl = 1 −
k∑
j=1
aj −
l∑
j=1
bj and let δ1, δ2 be given. Then there are N (δ1) ,
N (δ1, δ2) ∈ N and D = D (δ1, δ2) > 0 with properties:
i) γkl < δ1 ∀ k, l ≥ N (δ1);
ii) if kα = min {N (δ1) , |α′|}, kβ = min {N (δ1) , |β′|}, m ≥ N (δ1, δ2),
then
∑
m̃∈Nα,β(m,kα,kβ ,D)
m! ·
kα∏
j=1
a
mα
j
j
kβ∏
j=1
b
mβ
j
j
mkα+kβ+1! ·
kα∏
j=1
mα
j !
kβ∏
j=1
mβ
j !
· γmkα+kβ+1
kα kβ
> 1 − δ2,
where m̃=
(
mα
1 , . . . ,m
α
kα
,mβ
1 , . . . ,m
β
kβ
,mkα+kβ+1
)
and D, N(δ1, δ2)
are constants which do not depend on m.
Let Ei (x) = δx (Ai) (see Lemma 2.3) and let Ak = {i1, i2, . . . , ik}
be a set of different numbers from Z. We denote by EAk
(x) projection
k∏
j=1
Eij (x). If g ∈ S (Ak) then by Lemma 2.3 b)
[EAk
(x), πχ(g)] = 0. (3.12)
Therefore, the positive definite function τAk
on S (Ak), which is defined
by formula
τAk,x(g) =
tr (EAk
(x)πχ(g))
tr (EAk
(x))
, (3.13)
where x ∈ {α′}⋃ {β′}, is the normalize character.
The next Lemma is an auxiliary one.
Lemma 3.5. The next “dual” formula for extended Schur functions
s̃λ(α, β) = s̃λ′(β, α), (3.14)
where λ′ is the transposed diagram for λ, is valid.
Proof. The formula (3.14) is the generalization of the formula (2.9′) from
[10]. We will repeat the main ideas of that proof as applied to our case.
Let us denote
H(α,β)(z) = ezγ
∞∏
j=1
1 + zβj
1 − zαj
= 1 +
∞∑
l=1
hl(α, β)zl. (3.15)
M. S. Boyko, N. I. Nessonov 29
Then
H(α,β)(z)H(β,α)(−z) ≡ 1. (3.16)
Let us consider two matrices
H = (hi−j(α, β))0≤i,j≤N (3.17)
and
H̃ =
(
(−1)i−jhi−j(β, α)
)
0≤i,j≤N
, (3.18)
where N is some positive integer. We remind that hk(α, β) = 0 for k < 0
and hence the both matrices are upper-triangular with
det H̃ = detH = 1. (3.19)
Moreover, in view of (3.16)
H̃H = HH̃ = I (3.20)
holds. Hence H̃ = H−1. Let H̃′ be the transposed matrix for H̃, M
is an arbitrary minor of the matrix H and A is the algebraic adjunct
corresponding to the minor M ′ of the matrix H̃′ with the same numbers
of columns and rows as the numbers of ones in M . By the Laplace
theorem and by the (3.19)-(3.20) we obtain the equation M = A.
Let λ = (λ1, λ2, . . . , λn) be a Young diagram, λ′ = (λ′1, λ
′
2, . . . , λ
′
m)
be the transposed diagram. Then by the (3.2) s̃λ(α, β) can be consider
as the minor of the matrix H with the raw numbers λi − i+n, 1 ≤ i ≤ n
and the column numbers n−j, 1 ≤ j ≤ n. It is well-known that the m+n
numbers λi−i+n, 1 ≤ i ≤ n and (m+n−1)−(λ′j−j+m) = n−1−λ′j+j,
1 ≤ j ≤ m are the permutation of the {0, 1, 2, . . . ,m+ n− 1} (see [10]).
Below we assume that the dimension of the matrixesN = m+n−1. Then
the corresponded algebraic adjunct has the raw numbers n−1−λ′i+i, 1 ≤
i ≤ m and the column numbers n−1+j, 1 ≤ j ≤ m. Since the elements of
the matrix H̃′ look like (−1)j−ihj−i(β, α) the algebraic adjunct consists
of such elements (−1)λ′
i+j−ihλ′
i+j−i(β, α). Besides
∑n
i=1(λi − i + n) −∑n
j=1(n− j) = |λ|. Thus
s̃λ(α, β) = det (hλ+j−i(α, β))1≤i,j≤n
= (−1)|λ| det
(
(−1)λ′
i+j−ihλ′+j−i(β, α)
)
1≤i,j≤m
= det
(
hλ′+j−i(β, α)
)
1≤i,j≤m
= s̃λ′(β, α),
30 Entropy of the Shift...
Lemma 3.6. Let l(g) (g ∈ S (Ak)) is the number of cycles of a permu-
tation g. Then
τAk,x(g) =
(signx)k−l(g)
ν(x)k−l(g)
. (3.21)
Therefore, τAk,x is the restriction of characters χ
(S)
αν(x),0
for x ∈ α′
(
χ
(S)
0,βν(x)
for x ∈ β′
)
to the group S (Ak). Here αν , βν =
{
ν−1, . . . , ν−1
︸ ︷︷ ︸
ν
}
.
Proof. We denote by Ak/g a set of orbits of the permutation g on the
set Ak. If µ is a spectral measure of operator Ai, then µ(x) = ν(x) · |x|
(Lemma 2.3 e)) and by (Lemma 2.3 d)) we obtain
τAk,x(g) =
∏
p∈Ak/g
[
x|p|−1µ(x)
]
|xk|νk(x)
=
xk−l(g)|x|l(g)νl(g)(x)
|x|k · νk(x)
=
(sign x)k−l(g)
νk−l(g)(x)
.
Let parameters α = {αi}∞i=1 and β = {βi}∞i=1 satisfy the conditions
of Lemma 3.2, χ = χ
(S)
α,β , the W ∗-algebra Lχ (S (Ak)) be generated by
operators πχ (S (Ak)). We denote by Cx
k (Ak) the center of theW ∗-algebra
Mx
k (Ak) = EAk
(x)Lχ (S (Ak)).
At first we assume that x > 0. Then from (3.1) and Lemma 3.4 we
obtain
τAk,x = χ
(S)
α(ν(x)),0
∣∣
S(Ak)
=
∑
λ:|λ|=k
s̃λ
(
αν(x), 0
)
· χ(λ). (3.22)
The coefficients s̃λ
(
αν(x), 0
)
in the expansion can be easily evaluated by
using (3.1)
s̃λ
(
αν(x), 0
)
= |ν(x)|−k
∣∣∣∣∣∣∣∣∣
(
ν+λ1−1
ν−1
) (
ν+λ1
ν−1
)
. . .
(
2ν+λ1−2
ν−1
)
(
ν+λ2−2
ν−1
) (
ν+λ2−1
ν−1
)
. . .
(
2ν+λ2−3
ν−1
)
. . . . . . . . . . . .(
λν
ν−1
) (
λν+1
ν−1
)
. . .
(
ν+λν−1
ν−1
)
∣∣∣∣∣∣∣∣∣
. (3.23)
If x < 0, then
τAk,x = χ
(S)
0,βν(x)
∣∣
S(Ak)
=
∑
λ:|λ|=k
s̃λ
(
0, βν(x)
)
· χ(λ) and by the Lemma 3.5
s̃λ
(
0, βν(x)
)
= s̃λ′
(
βν(x), 0
)
= |ν(x)|−k
∣∣∣∣∣∣∣∣∣
(ν+λ′
1−1
ν−1
) (ν+λ′
1
ν−1
)
. . .
(2ν+λ′
1−2
ν−1
)
(ν+λ′
2−2
ν−1
) (ν+λ′
2−1
ν−1
)
. . .
(2ν+λ′
2−3
ν−1
)
. . . . . . . . . . . .(
λ′
ν
ν−1
) (
λ′
ν+1
ν−1
)
. . .
(
ν+λ′
ν−1
ν−1
)
∣∣∣∣∣∣∣∣∣
.
(3.24)
M. S. Boyko, N. I. Nessonov 31
Here λ′ stands for the transposed diagram and ν = ν(x).
Let Yk be a set of a Young diagrams λ such that |λ| = k. We in-
troduce further the set Yk(x) =
{{
λ ∈ Yk : s̃λ
(
αν(x), 0
)
6= 0
}
if x > 0,{
λ ∈ Yk : s̃λ
(
0, βν(x)
)
6= 0
}
if x < 0
and denote by Sx
k (Ak) the set of all minimal projections in Cx
k (Ak). By
virtue of (3.22) and(3.24), the mapping
λ ∈ Yk(x) →
dimλ
k!
EAk
(x) ·
∑
g∈S(Ak)
χ(λ)(g)πχ(g) = ex
k(λ) ∈ Sx
k (Ak)
is one-to-one correspondence and true
Lemma 3.7. Let Sx
k λ (Ak) be the set of all minimal projections in
ex
k(λ)Mx
k (Ak). If e ∈ Sx
k λ (Ak) then
χ
(S)
α,β (e) = [ν(x)|x|]k ·
{
s̃λ
(
αν(x), 0
)
if x > 0,
s̃λ
(
0, βν(x)
)
if x < 0
= oλ
x(k)|x|k,
where oλ
x(k) ∈ N ∀ k ∈ N and lim
k→∞
oλ
x(k)
kν2(x)
= 0 uniformly on the set Yk.
Proof. Using (3.22), (3.23) and (3.24), we obtain the statement of Lemma
3.7 from the next chain of equalities
χ
(S)
α,β (e) = χ
(S)
α,β (EAk
(x)) · τAk,x (e) = [ν(x)|x|]k ·
{
s̃λ
(
αν(x), 0
)
if x > 0,
s̃λ
(
0, βν(x)
)
if x < 0.
Let us denote for x > 0
oλ
x(k) =
∣∣∣∣∣∣∣∣∣
(
ν+λ1−1
ν−1
) (
ν+λ1
ν−1
)
. . .
(
2ν+λ1−2
ν−1
)
(
ν+λ2−2
ν−1
) (
ν+λ2−1
ν−1
)
. . .
(
2ν+λ2−3
ν−1
)
. . . . . . . . . . . .(
λν
ν−1
) (
λν+1
ν−1
)
. . .
(
ν+λν−1
ν−1
)
∣∣∣∣∣∣∣∣∣
and for x < 0
oλ
x(k) =
∣∣∣∣∣∣∣∣∣
(ν+λ′
1−1
ν−1
) (ν+λ′
1
ν−1
)
. . .
(2ν+λ′
1−2
ν−1
)
(ν+λ′
2−2
ν−1
) (ν+λ′
2−1
ν−1
)
. . .
(2ν+λ′
2−3
ν−1
)
. . . . . . . . . . . .(
λ′
ν
ν−1
) (
λ′
ν+1
ν−1
)
. . .
(
ν+λ′
ν−1
ν−1
)
∣∣∣∣∣∣∣∣∣
.
The function oλ
x(k) can be considered as a polynomial from ν(x) variables
((λ1, λ2, . . . , λν) or (λ′1, λ
′
2, . . . , λ
′
ν)) with the degree equals ν(x)(ν(x)−1).
Thus the lemma is proved.
32 Entropy of the Shift...
By means of m̃ ∈ Nα,β (m, kα, kβ , D) (see (3.11), p. 27) we introduce
pairwise disjoint subsets Q
(
α′
j
)
(1 ≤ j ≤ kα) and Q
(
β′j
)
(1 ≤ j ≤ kβ)
in I(m) = {1, 2, . . . ,m} with properties
∣∣Q
(
α′
j
)∣∣ = mα
j ,
∣∣Q
(
β′j
)∣∣ = mβ
j .
Let Q (γ) = I(m) \
(
kα⋃
j=1
Q
(
α′
j
) kβ⋃
j=1
Q
(
β′j
))
and
EQ̃(m̃)
=
[ kα∏
j=1
kβ∏
l=1
∏
s∈Q(α′
j)
Es
(
α′
j
) ∏
s∈Q(β′
l)
Es
(
β′l
)] ∏
s∈Q(γ)
Fs, (3.25)
where Fs = I −
kα∑
j=1
Es
(
α′
j
)
−
kβ∑
j=1
Es
(
β′j
)
, Q̃ (m̃) is an ordered set
(
Q (α′
1) , . . . ,Q
(
α′
kα
)
;Q (β′1) , . . . ,Q
(
β′kβ
)
;Q (γ)
)
. If
G
(
Q̃ (m̃)
)
=
kα×
j=1
S
(
Q
(
α′
j
)) kβ
×
i=1
S
(
Q
(
β′i
))
and Lχ
(
G
(
Q̃ (m̃)
))
is generated by operators πχ
(
G
(
Q̃ (m̃)
))
as a W ∗-
algebra, then M
(
Q̃ (m̃)
)
= EQ̃(m̃)
Lχ
(
G
(
Q̃ (m̃)
))
is isomorphic to
kα⊗
j=1
M
α′
j
mα
j
(
Q
(
α′
j
)) kβ⊗
i=1
M
β′
i
mβ
i
(
Q
(
β′i
))
(see p.30). (3.26)
Lemma 3.8. Let trχ is the central normalize state on Lχ (S (∞)) which
corresponds to χ = χ
(S)
α,β. If ϕm̃ is the restriction trχ to the algebra
M
(
Q̃ (m̃)
)
and Hϕm̃
(
M
(
Q̃ (m̃)
))
is the CNT-entropy of M
(
Q̃ (m̃)
)
cor-
responding to ϕm̃, then
Hϕm̃
(
M
(
Q̃ (m̃)
))
= −γmkα+kβ+1
kα kβ
kα∏
j=1
a
mα
j
j
kβ∏
j=1
b
mβ
j
j
×
{(
mkα+kβ+1
)
ln
(
γkα kβ
)
+
kα∑
j=1
mα
j lnα′
j +
kβ∑
j=1
mβ
j lnβ′j +O (lnm)
}
,
where 0 ≤ lim sup
m→∞
O(ln m)
ln m <∞, γkα kβ
= χ (Fs) = 1 −
kα∑
j=1
aj −
kβ∑
j=1
bj.
M. S. Boyko, N. I. Nessonov 33
Proof. We denote by ϕx
Ak
the restriction trχ to the algebra Mx
k (Ak). In
view of (3.25) and (3.26), we have
ϕm̃ = γ
mkα+kβ+1
kα kβ
kα⊗
j=1
ϕ
α′
j
Q(α′
j)
kβ
⊗
i=1
ϕ
β′
i
Q(β′
i)
. (3.27)
Let
fλ
x =
{
s̃λ
(
αν(x), 0
)
if x > 0,
s̃λ
(
0, βν(x)
)
if x < 0.
Further, using (3.22) and (3.24), we obtain
Hϕx
Ak
(Mx
k (Ak))
= − (ν(x)|x|)k
{ ∑
λ:|λ|=k
dimλ · fλ
x
[
ln fλ
x + ln
(
(ν(x)|x|)k
)]}
= − (ν(x)|x|)k
{ ∑
λ:|λ|=k
dimλ · fλ
x
[
k ln |x| + ln
(
(ν(x))k fx
)]}
(see Lemma 3.7)
= − (ν(x)|x|)k
{ ∑
λ:|λ|=k
dimλ · fλ
x
[
k ln |x| + ln
(
oλ
x(k)
)]}
.
Since
∑
λ:|λ|=k
dimλ · fλ
x = 1, we may rewrite Hϕx
Ak
(Mx
k (Ak)) as follows:
Hϕx
Ak
(Mx
k (Ak)) = −k (ν(x)|x|)k ln |x| − (ν(x)|x|)k O (x, ln k) , (3.28)
where O (x, ln k) =
∑
λ:|λ|=k
dimλ · fλ
x ln
(
oλ
x(k)
)
and by Lemma 3.7
0 ≤ lim sup
k→∞
O (x, ln k)
ln k
<∞. (3.29)
It follows from (3.26), (3.27) and (3.28) that
Hϕm̃
(
M
(
Q̃ (m̃)
)) (by definition ϕm̃)
= −γmkα+kβ+1
kα kβ
kα∏
j=1
a
mα
j
j
kβ∏
j=1
b
mβ
j
j
×
{(
mkα+kβ+1
)
ln
(
γkα kβ
)
+
kα∑
j=1
mα
j lnα′
j +
kβ∑
j=1
mβ
j lnβ′j
+
kα∑
j=1
O
(
α′
j , lnm
α
j
)
+
kβ∑
j=1
O
(
β′j , lnm
β
j
)}
.
34 Entropy of the Shift...
Using (3.29) and the equality
kα∑
j=1
mα
j +
kβ∑
j=1
mβ
j +mkα+kβ+1 = m in latter
assertion, we have
0 ≤ lim sup
m→∞
kα∑
j=1
O
(
α′
j , lnm
α
j
)
+
kβ∑
j=1
O
(
β′j , lnm
β
j
)
lnm
<∞
and
Hϕm̃
(
M
(
Q̃ (m̃)
))
= −γmkα+kβ+1
kα kβ
kα∏
j=1
a
mα
j
j
kβ∏
j=1
b
mβ
j
j
×
{(
mkα+kβ+1
)
ln
(
γkα kβ
)
+
kα∑
j=1
mα
j lnα′
j +
kβ∑
j=1
mβ
j lnβ′j +O (lnm)
}
.
In the next statements we will use the notations from Lemma 3.4.
Proposition 3.1. Let Mm be a W ∗-algebra generated by A0 and
πχ (S(m)) and let δ1, δ2 be given. Then there exists a natural numbers:
kα (δ1), kβ (δ1), C(δ1, δ2), and M (δ1, δ2) such that
i) γkαkβ
= 1 −
kα(δ1)∑
j=1
aj −
kβ(δ1)∑
j=1
bj < δ1;
ii) ∀ m > M (δ1, δ2)
Hχ (Mm) ≥ −m (1 − δ2)
{ kα(δ1)∑
j=1
ν
(
α′
j
)
α′
j lnα′
j
+
kβ(δ1)∑
j=1
ν
(
β′j
)
β′j lnβ′j
}
+ C(δ1, δ2)
√
m (see Lemma 3.8).
Proof. It is clear that M
(
Q̃ (m̃)
)
⊂ Mm. Therefore,
Hχ (Mm) ≥
∑
m̃
m! ·Hϕm̃
(
M
(
Q̃ (m̃)
))
mkα(δ1)+kβ(δ1)+1! ·
kα(δ1)∏
j=1
mα
j !
kβ(δ1)∏
j=1
mβ
j !
M. S. Boyko, N. I. Nessonov 35
( Lemma 3.8)
≥ −
{
∑
m̃
m! · γ
mkα(δ1)+kβ(δ1)+1
kα(δ1)kβ(δ1)
kα(δ1)∏
j=1
a
mα
j
j
kβ(δ1)∏
j=1
b
mβ
j
j
mkα(δ1)+kβ(δ1)+1! ·
kα(δ1)∏
j=1
mα
j !
kβ(δ1)∏
j=1
mβ
j !
×
[ kα(δ1)∑
j=1
mα
j lnα′
j +
kβ(δ1)∑
j=1
mβ
j lnβ′j
]
+ O (lnm)
}
( Lemma 3.4, (3.11))
≥ −m(1−δ2)
[ kα(δ1)∑
j=1
aj lnα′
j+
kβ(δ1)∑
j=1
bj lnβ′j
]
+C(δ1, δ2)
√
m.
The latter inequality is true for all sufficiently large m.
Proposition 3.2. (An upper bound for the entropy) η(t) = −t ln t,∑
αi +
∑
βi = 1, χ = χ
(S)
α,β. Then
Hχ
(
ϑχ
S
)
≤
∑
j
η (αj) +
∑
j
η (αj) .
Proof. First we recall the well-known construction (see [20]) of the
embedding of the group S(∞) in the Powers factor. Let n(α) =
min {i : αi > 0}, n(β) = min {i : βi > 0}, n = n(α) + n(β), Nα
n =
{1, 2, . . . , n(α)}, N
β
n = {−1,−2, . . . ,−n(β)} and Nn = Nα
n
⊔
N
β
n. We
consider the algebra Mn (C) of all complex n×n-matrices with system of
matrix units {ei,j}i,j∈Nn
. Let hαβ = diag
(
α1, . . . , αn(α);β1, . . . , βn(β)
)
∈
Mn (C) and let ϕ(·) = Tr (·hαβ) is the state on Mn (C), where Tr is
ordinary trace. For j ∈ Z let Mj = Mn (C) and ϕj = ϕ. Let (M, ϕ̃) =
⊗j∈Z (Mj , ϕj), where ϕ̃ = ⊗j∈Zϕj . For sequence ik = (ij ∈ Nn)j=k
j=−k let
j (ik) =
({
j1 < j2 < . . . < jl(ik)
}
: ijl
∈ Nβ
n ∀ l = 1, 2, . . . , l (ik)
)
.
If g is any permutation of the set Bk = {−k, . . . , 0, . . . , k}, then there is
permutation s (g, ik) ∈ S (g (j (ik))) such that
s (g, ik) (g (j1)) < s (g, ik) (g (j2)) < . . . < s (g, ik)
(
g
(
jl(ik)
))
.
Let ψ (g, ik) = sgn (s (g, ik) ) and let Ik be the set of all sequences
(ij ∈ Nn)j=k
j=−k. Now by a direct checking we can make sure, that opera-
tors Ug =
∑
ik∈Ik
ψ (g, ik) e
ik g(ik)
, where
e
ik g(ik)
= e
i−k ig(−k)
⊗ e
i−k+1 ig(−k+1)
⊗ . . . e
ik ig(k)
∈ Mϕ̃,
36 Entropy of the Shift...
Mϕ̃ is the centralizer of ϕ̃, define a unitary representation of the group
S(2k + 1) = S (Bk) and ϕ̃ (Ug) = χ
(S)
α,β(g). Further, we notice that au-
tomorphism ϑχ
S of the W ∗-algebra generated by Ug
(
g ∈ ⋃
k
S (Bk)
)
ex-
tends to the automorphism θ of the W ∗-algebra M. But it is well-known
(see [5]), that θ is a noncommutative Bernouli shift with the entropy∑
j
η (αj) +
∑
j
η (βj).
Proof of Theorem 3.1. The statement of the theorem follows from Lemma
3.2 and Proposition 3.2 when sets {αi 6= 0} and {βi 6= 0} are finite ones.
In the general case we consider the algebra Mm that is generated by
A0 and πχ (S(m)) as a W ∗-algebra. Using a method by which we have
proved Lemma 2.1, we receive the following estimation
Hχ
(
ϑχ
S
)
≥ Hχ (Mn)
m
.
Hence, tacking into account Propositions 3.2 and 3.1, we complete the
proof.
References
[1] A. Berele, A. Regev, Hook Young diagrams with applications to combinatories
and to representations of Lie superalgebras // Adv. Math. 64 (1987), 118–175.
[2] S. Bezuglyi, V. Ya. Golodets, Dynamical entropy for Bogoliubov actions of free
abelian groups on the CAR-algebra // Ergod. Th. and Dynam. Sys., 17 (1997),
757–782.
[3] M. Boyko, N. Nessonov, Entropy of Bogoliubov automorphisms on II1-
representations of the group U(∞) // Metods Funct. Anal. Topology, 9 (2003),
No. 4, 317–332.
[4] A. Connes, H. Narnhofer, W. Thirring, Dynamical entropy of C∗-algebras and
von Neumann algebras // Commun. Math. Phys., 112 (1987), 691–719.
[5] A. Connes, E. Størmer, Entropy for automorphisms of II1 von Neumann alge-
bras // Acta Math. 134 (1975), 289–306.
[6] V. Ya. Golodets, S. Neshveyev, Entropy for Bogoliubov actions of torsion-free
abelian groups on the CAR-algebra // Ergod. Th. and Dynam. Sys., 20 (2000),
No. 4, 1111–1125.
[7] V. Ya. Golodets, E. Størmer, Entropy of C∗ dynamical systems defined by bit-
streams // Ergod. Th. and Dynam. Sys., 18 (1998), 859–874.
[8] S. Kerov, A. Okounkov, G. Olshanski, The boundary of Young graph with Jack
edge multiplicities // Intern. Math. Res. Notices (1998), No. 4, 173–199.
[9] A. N. Kolmogorov, The new metric invariant of transitive dynamical system and
automorphisms of Lebesgue spaces // Dokl. Acad. Nauk SSSR, (1958), No. 5,
861–864. (in Russian)
[10] I. G. Macdonald, Symmetric Functions and Hall Polinomials. Clarendon
Press·Oxford 1979.
M. S. Boyko, N. I. Nessonov 37
[11] H. Narnhofer, W. Thirring, Dynamical entropy of quantum systems and
their abelian counterpart. On Klauder Path: A field trip, Emch, G. G.,
Hegerfeldt, G. C., Streit, L. World Scientific; Singapore, 1994.
[12] S. Neshveyev, Entropy of Bogoliubov automorphismes of CAR and CCR algebras
with respect to quasi-free states // Rev. Math. Phys., 13 (2001), No. 1, 29–50.
[13] A. Okounkov, The Thoma theorem and representations of the infinite bisymmetric
group // Funct. Anal. Appl. 28 (1994), No. 2, 100–107.
[14] Y. M. Park, H. H. Shin, Dynamical entropy of space translations of CAR and
CCR algebras with respect to quasi-free states // Commun. Math. Phys., 152
(1993), 497–537.
[15] V. A. Rokhlin, Ya. G. Sinai, Construction and properties of invariant measurable
partitions // Dokl. Acad. Nauk SSSR, (1961), No. 5, 1038–1041. (in Russian)
[16] Ya. G. Sinai, On the concept of the entropy for a dynamical system // Dokl. Acad.
Nauk SSSR, (1959), No. 4, 768–771. (in Russian)
[17] E. Størmer, D. Voiculescu, Entropy of Bogoliubov automorphisms of the Canonical
Anticommutation Relations // Commun. Math. Phys., 133 (1990), 521–542.
[18] A. M. Vershik, S. V. Kerov, Asymptotic form of minimal and typical dimensions of
irreducible representations for the symmetric group // Funk. Anal. i ego Prilozh.
19 (1985), No. 1, 25–36. (in Russian)
[19] A. M. Vershik, S. V. Kerov, Asymptotic theory of characters of the symmetric
group // Funct. Anal. Appl. 15 (1981), No. 1, 246–255.
[20] A. M. Vershik, S. V. Kerov, Characters and factor representations of the infinite
symmetric group // Soviet Math. Dokl., 23 (1981), No. 2, 389–392.
Contact information
M. S. Boyko,
N. I. Nessonov
Institute For Low Temperature
Physics and Engineering,
Department of Mathematics,
47 Lenin Avenue, Kharkiv,
Ukraine
E-Mail: mboyko@land.ru,
nessonov@ilt.kharkov.ua
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