Entropy of subshifts and the Macaev norm

ENTROPY OF SUBSHIFTS AND THE MACAEV NORM

RUI OKAYASU

Abstract. We obtain the exact value of Voiculescu’s invariant k−∞(τ), which is anobstruction of the existence of quasicentral approximate units relative to the Macaevideal in perturbation theory, for a tuple τ of operators in the following two classes:(1) creation operators associated with a subshift, which are used to define Matsumotoalgebras, (2) unitaries in the left regular representation of a finitely generated group.

1. Introduction

In the remarkable serial works [Voi1], [Voi2], [Voi3] and [DV] on perturbation of Hilbertspace operators, Voiculescu investigated a numerical invariant kΦ(τ) for a family τ ofbounded linear operators on a separable Hilbert space, where kΦ(τ) is the obstruction of

the existence of quasicentral approximate units relative to the normed ideal S(0)Φ corre-

sponding to a symmetric norming function Φ, (see definitions in Section 2). The invariant

kΦ(τ) is considered to be a kind of dimension of τ with respect to the normed ideal S(0)Φ

(see [Voi1] and [DV]).In the present paper, we study the invariant kΦ(τ) for the Macaev ideal, which is

denoted by k−∞(τ). It is known that k−∞(τ) possesses several remarkable properties: for

instance, k−∞(τ) is always finite and kΦ(τ) = 0 if S(0)Φ is strictly larger than the Macaev

ideal. In [Voi3], Voiculescu investigated the invariant k−∞(τ) for several examples. Heproved that k−∞(τ) = log N for an N -tuple τ of isometries in extensions of the Cuntzalgebra ON . Here, log N can be interpreted as the value of the topological entropy of theN -full shift. Inspired by this result, we show that k−∞(τ) = htop(X) for a general subshiftX with a certain condition, where htop(X) is the topological entropy of X and τ is thefamily of creation operators on the Fock space associated with the subshift X, which isused to define the Matsumoto algebra associated with X (e.g. see [Mat]). In particular,we show that k−∞(τ) = htop(X) holds for every almost sofic shift X (cf. [Pet]).

Let Γ be a countable finitely generated group and S its generating set. We also studyk−∞((λa)a∈S), where λ is the left regular representation of Γ. For the related topic, see[Voi5], in which a relation between k−∞((λa)a∈S) and the entropy of random walks ongroups is discussed. By using a method introduced in [Oka], we can compute the exactvalue of k−∞((λa)a∈S) for certain amalgamated free product groups. Voiculescu provedthat log N ≤ k−∞((λa)a∈S) ≤ log(2N − 1) holds for the free group FN with the canonicalgenerating set S ([Voi3, Proposition 3.7. (a)]). As a particular case of our results, weshow that k−∞((λa)a∈S) = log(2N − 1) actually holds.Acknowledgment. The author wishes to express his gratitude to Masaki Izumi for hisconstant encouragement and important suggestions. He is grateful to Kengo Matsumoto

2000 Mathematics Subject Classification. Primary 47B10; Secondary 47B06, 37B10, 37B40.Key Word and Phases. perturbation theory, Macaev ideal, symbolic dynamics.

1

2 RUI OKAYASU

for useful comments about symbolic dynamics. He also thanks the referee for carefulreading.

2. Preliminary

Let H be a separable infinite dimensional Hilbert space. By B(H), K(H), F(H) andF(H)+

1 , we denote the bounded linear operators, the compact operators, the finite rankoperators and the finite rank positive contractions on H, respectively.

We begin by recalling some facts concerning normed ideal in [GK]. Let c0 be theset of real valued sequences ξ = (ξj)j∈N with limj→∞ ξj = 0, and c0,0 the subspace ofc0 consisting of the sequences with finite support. A function Φ on c0,0 is said to be asymmetric norming function if Φ satisfies:

(1) Φ is a norm on c0,0;(2) Φ((1, 0, 0, . . . )) = 1;(3) Φ((ξj)j∈N) = Φ((|ξπ(j)|)j∈N) for any bijection π : N→ N.

For ξ = (ξj)j∈N ∈ c0, we define

Φ(ξ) = limn→∞

Φ(ξ∗(n)) ∈ [0,∞],

where ξ∗(n) = (ξ∗1 , . . . , ξ∗n, 0, 0, . . . ) ∈ c0,0 and ξ∗1 ≥ ξ∗2 ≥ · · · is the decreasing rearrange-

ment of the absolute value (|ξj|)j∈N. If T ∈ K(H) and Φ is a symmetric norming function,then let us denote

||T ||Φ = Φ((sj(T ))j∈N),

where (sj(T ))j∈N is the singular numbers of T . We define two symmetrically normedideals

SΦ = T ∈ K(H) | ||T ||Φ < ∞,and S

(0)Φ by the closure of F(H) with respect to the norm || · ||Φ. Note that S

(0)Φ does not

coincide with SΦ in general. If S is a symmetrically normed ideal, i.e. S is a ideal ofB(H) and a Banach space with respect to the norm || · ||S satisfying:

(1) ||XTY ||S ≤ ||X|| · ||T ||S · ||Y || for T ∈ S and X, Y ∈ B(H),(2) ||T ||S = ||T || if T is of rank one,

where || · || is the operator norm in B(H), then there exists a unique symmetric norming

function Φ such that ||T ||S = ||T ||Φ for T ∈ F(H) and S(0)Φ ⊆ S ⊆ SΦ.

We introduce some symmetrically normed ideals. For 1 < p ≤ ∞, the symmetricallynormed ideal C−p (H) is given by the symmetric norming function

Φ−p (ξ) =

∞∑j=1

ξ∗jj1−1/p

.

We define C−p (H) = S(0)

Φ−p. We remark that it coincides with SΦ−p . For 1 ≤ p < ∞, the

symmetrically normed ideal C+p (H) is given by the symmetric norming function

Φ+p (ξ) = sup

n∈N

∑nj=1 ξ∗j∑n

j=1 j−1/p.

ENTROPY OF SUBSHIFTS AND THE MACAEV NORM 3

We define C+p (H) = SΦ+

p. However S

(0)

Φ+p

is strictly smaller than C+p (H). For 1 ≤ p < q <

r ≤ ∞, we haveCp(H) $ C−q (H) $ Cq(H) $ C+

q (H) $ Cr(H),

where Cp(H) is the Schatten p class.For a given symmetric norming function Φ, which is not equivalent to the l1-norm,

there is a symmetric norming function Φ∗ such that SΦ∗ is the dual of S(0)Φ , where the

dual pairing is given by the bilinear form (T, S) 7→ Tr(TS). If 1/p + 1/q = 1, thenCp(H)∗ ' Cq(H) and C−p (H)∗ ' C+

q (H). In particular, C−∞(H) and C+1 (H) are called the

Macaev ideal and the dual Macaev ideal, respectively.

Let S(0)Φ be a symmetrically normed ideal with a symmetric norming function Φ. If

τ = (T1, . . . , TN) is an N -tuple of bounded linear operators, then the number kΦ(τ) isdefined by

kΦ(τ) = lim infu∈F(H)+1

max1≤a≤N

||[u, Ta]||Φ,

where the inferior limit is taken with respect to the natural order on F(H)+1 and [A,B] =

AB−BA. Throughout this paper, we denote || · ||Φ−p by || · ||−p and kΦ−p by k−p . A relation

between the invariant kΦ and the existence of quasicentral approximate units relative to

the symmetrically normed ideal S(0)Φ is discussed in [Voi1]. A quasicentral approximate

unit for τ = (T1, . . . , TN) relative to S(0)Φ is a sequence un∞n=1 ⊆ F(H)+

1 such that un Iand limn→∞ ||[un, Ta]||Φ = 0 for 1 ≤ a ≤ N . Note that for an N -tuple τ = (T1, . . . , TN),

there exists a quasicentral approximate unit for τ relative to S(0)Φ if and only if kΦ(τ) = 0

(e.g. see [Voi2, Lemma 1.1]).We use the following propositions to prove our theorem.

Proposition 2.1 ([Voi1, Proposition 1.1]). Let τ = (T1, . . . , TN) ∈ B(H)N and S(0)Φ be a

symmetrically normed ideal with a symmetric norming function Φ. If we take a sequenceun∞n=1 ⊆ F(H)+

1 with w-limn→∞ un = I, then

kΦ(τ) ≤ lim infn→∞

max1≤a≤N

||[un, Ta]||Φ.

Proposition 2.2 ([Voi3, Proposition 2.1]). Let τ = (T1, . . . , TN) ∈ B(H)N and Xa ∈C+

1 (H) for a = 1, . . . , N . If

N∑a=1

[Xa, Ta] ∈ C1(H) + B(H)+,

then we have ∣∣∣∣∣Tr

(N∑

a=1

[Xa, Ta]

)∣∣∣∣∣ ≤ k−∞(τ)N∑

a=1

||Xa||e+1 ,

where ||Xa||e+1 = infY ∈F(H) ||Xa − Y ||Φ+1.

The following proposition was shown in the proof of [GK, Theorem 14.1].

Proposition 2.3. For T ∈ C+1 (H), we have

||T ||e+1 = lim supn→∞

∑nj=1 sj(T )∑nj=1 1/j

.

4 RUI OKAYASU

3. Subshifts and Macaev Norm

Let A be a finite set with the discrete topology, which we call the alphabet, and AZ thetwo-sided infinite product space

∏∞i=−∞A endowed with the product topology. The shift

map σ on AZ is given by (σ(x))i = xi+1 for i ∈ Z. The pair (AZ, σ) is called the full shift.In particular, if the cardinality of the alphabet A is N , then we call it the N -full shift.

Let X be a shift invariant closed subset of AZ. The topological dynamical system(X, σX) is called a subshift of AZ, where σX is the restriction of the shift map σ. Wesometimes denote the subshift (X, σX) by X for short. A word over A is a finite sequencew = (a1, . . . , an) with ai ∈ A. For x ∈ AZ and a word w = (a1, . . . , an), we say that woccurs in x if there is an index i such that xi = a1, . . . , xi+n−1 = an. The empty wordoccurs in every x ∈ AZ by convention. Let F be a collection of words over AZ. We definethe subshift XF to be the subset of sequences in AZ in which no word in F occurs. It iswell-known that any subshift X of AZ is given by XF for some collection F of forbiddenwords over AZ. Note that for F = ∅, the subshift XF is the full shift AZ.

Let X be a subshift of AZ. We denote by Wn(X) the set of all words with length nthat occur in X and we set

W(X) =∞⋃

n=0

Wn(X).

Let ϕ : Wm+n+1(X) → A be a map, which we call a block map. The extension of ϕ fromX to AZ is defined by (xi)i∈Z 7→ (yi)i∈Z, where

yi = ϕ((xi−m, xi−m+1, . . . , xi+n)).

We also denote this extension by ϕ and call it a sliding block code. Let X, Y be twosubshifts and ϕ : X → Y a sliding block code. If ϕ is one-to-one, then ϕ is called anembedding of X into Y and we denote X ⊆ Y . If ϕ has an inverse, i.e. a sliding blockcode ψ : Y → X such that ψ ϕ = idX and ϕ ψ = idY , then two subshifts X and Y aretopologically conjugate.

The topological entropy of a subshift X is defined by

htop(X) = limn→∞

1

nlog |Wn(X)|,

where |Wn(X)| is the cardinality of Wn(X). The reader is referred to [LM] for an intro-duction to symbolic dynamics.

For a given subshift X, we next construct the creation operators on the Fock spaceassociated with X (cf. [Mat]). Let ξaa∈A be an orthonormal basis of N -dimensionalHilbert space CN , where N is the cardinality of A. For w = (a1, . . . , an) ∈ Wn(X), wedenote ξw = ξa1 ⊗ · · · ⊗ ξan . We define the Fock space FX for a subshift X by

FX = Cξ0 ⊕⊕

n∈Nspanξw | w ∈ Wn(X),

where ξ0 is the vacuum vector. The creation operator Ta on FX for a ∈ A is given by

Taξ0 = ξa,

Taξw =

ξa ⊗ ξw if aw ∈ W(X),

0 otherwise.


Note that Ta is a partial isometry such that

P0 +∑a∈A

TaT∗a = 1,

where P0 is the rank one projection onto Cξ0. We denote by Pn the projection onto thesubspace spanned by ξw for all w ∈ Wn(X). For w = (a1, . . . , an) ∈ Wn(X), we setTw = Ta1 · · ·Tan . The following proposition is essentially proved in [Voi3].

Proposition 3.1. If τ = (Ta)a∈A, then we have

k−∞(τ) ≤ htop(X).

Proof. We first assume that the topological entropy of X is non-zero. Let us denoteh = htop(X). By definition, for a given ε > 1, there exists K ∈ N such that for anyn ≥ K, we have

1

nlog |Wn(X)| < εh.

Thus|Wn(X)| < enεh,

for all n ≥ K. We set

Xn =n−1∑j=0

(1− j

n

)Pj.

One can show that

||[Xn, Ta]|| ≤ 1

n.

Since

rn = rank([Xn, Ta]) ≤n∑

j=1

|Wj(X)| ≤K−1∑j=1

|Wj(X)|+n∑

j=K

ejεh

for n ≥ K, we obtain

k−∞(τ) ≤ lim supn→∞

maxa∈A

||[Xn, Ta]||−∞ ≤ lim supn→∞

∑rn

j=1 1/j

n≤ εh.

In the case of h = 0, for any ε > 0, we have

|Wn(X)| < enε

for sufficiently large n. By the same argument, we can get

k−∞(τ) ≤ lim supn→∞

maxa∈A

||[Xn, Ta]||−∞ ≤ ε,

for arbitrary ε > 0. ¤Next we obtain the lower bound of k−∞(τ) by using Proposition 2.2. Before it, we

prepare some notations. For any m ∈ Z and w = (a1, . . . , an) ∈ Wn(X), let us denote

m[w] = (xi)i∈Z ∈ X | xm = a1, . . . , xm+n−1 = an.We sometimes denote the cylinder set 0[w] by [w] for short. Let µ be a shift invariantprobability measure on X. The following holds:

(1)∑

a∈A µ([a]) = 1;(2) µ([a1, . . . , an]) =

∑a0∈A µ([a0, a1, . . . , an]);

(3) µ([a1, . . . , an]) =∑

an+1∈A µ([a1, . . . , an, an+1]).

6 RUI OKAYASU

For any partition β = (B1, . . . , Bn) of X, we define a function on X by

Iµ(β) = −∑

B∈β

log µ(B)χB,

where χB is the characteristic function of B. Let β1, . . . , βk be partitions of X. Thepartition

∨ki=1 βi is defined by

k⋂

i=1

Bi | Bi ∈ βi, 1 ≤ i ≤ k

.

The value

Hµ(β) = −∑

B∈β

µ(B) log µ(B)

is called the entropy of the partition β. We define

hµ(β, σX) = limn→∞

1

nHµ(

n−1∨i=0

σ−iX (β)).

The entropy of (X, σX , µ) is defined by

hµ(σX) = suphµ(β, σX) | Hµ(β) < ∞.Note that hµ(σX) ≤ htop(X) in general. A shift invariant probability measure µ is said tobe a maximal measure if htop(X) = hµ(σX). The reader is referred to [DGS] for details.

Theorem 3.2. Let τ = (Ta)a∈A be the creation operators for a subshift X. If there existsa shift invariant probability measure µ on X such that for any ε > 0 we have

∞∑n=0

µ

(x ∈ X :

∣∣∣∣∣1

n + 1Iµ

(n∨

i=0

σ−iX β

)(x)− hµ(σX)

∣∣∣∣∣ > ε

)< ∞,

where β is the generating partition [a]a∈A of X, then

hµ(σX) ≤ k−∞(τ).

In particular, if we can take a maximal measure µ with the above condition, then we have

k−∞(τ) = htop(X).

Proof. Let µ be a shift invariant probability measure on X. For a ∈ A, we set

Xa =∑n≥0

∑

w∈Wn(X)

µ([aw])TwP0T∗aw.

Then∑a∈A

TaXa =∑n≥0

∑a∈A

∑

w∈Wn(X)

µ([aw])TawP0T∗aw

=∑n≥1

∑

w∈Wn(X)

µ([w])TwP0T∗w,


and

∑a∈A

XaTa =∑n≥0

∑

w∈Wn(X)

(∑a∈A

µ([aw])

)TwP0T

∗w

=∑n≥0

∑

w∈Wn(X)

µ([w])TwP0T∗w.

Hence we have ∑a∈A

[Xa, Ta] = P0.

We assume that hµ(σX) 6= 0 and denote it by h for short. To apply Proposition 2.2, we

need an estimate of ||Xa||e+1 . Fix ε > 0 and a ∈ A. We set

Dn =w ∈ Wn(X) | e−(n+1)(h+ε) ≤ µ([aw]) ≤ e−(n+1)(h−ε)

,

andεn =

∑

w∈Wn(X)\Dn

µ([aw]).

If µ satisfies the assumption, then we have∑n≥0

εn < ∞. (?)

Note that sj(Xa) = sj(XaTa) for all j ∈ N. Thus we have ||Xa||e+1 = ||XaTa||e+1 . We put

Xa =∑n≥0

∑w∈Dn

µ([aw])TwP0T∗w.

We remark that for each j ∈ N, there are n ∈ N, w ∈ Wn(X) such that sj(XaTa) =µ([aw]). By (?), we obtain

||Xa||e+1 = ||XaTa||e+1 = lim supn→∞

∑nj=1 sj(XaTa)∑n

j=1 1/j

≤ ||Xa||e+1 + lim supn→∞

∑∞k=0 εk∑n

j=1 1/j= ||Xa||e+1 .

Hence it suffices to give an estimate of ||Xa||e+1 . Let dn =∑n

j=0 |Dj|, where |Dj| is thecardinality of Dj. One can easily check that

||Xa||e+1 ≤ lim supn→∞

∑dn

j=1 sj(Xa)∑dn

j=1 1/j.

Note that if sj(Xa) = µ([aw]) for some w ∈ Dn, then we have

e−(n+1)(h+ε) ≤ sj(Xa) = µ([aw]) ≤ e−(n+1)(h−ε).

Assume that there are m > n such that sj(Xa) = µ([aw]) for some w ∈ Dm and j ≤ dn.Then it holds that

e−(m+1)(h−ε) ≥ e−(n+1)(h+ε). (??)

Indeed, if e−(m+1)(h−ε) < e−(n+1)(h+ε), then

sj(Xa) = µ([aw]) ≤ e−(m+1)(h−ε) < e−(n+1)(h+ε) ≤ µ([au]),

8 RUI OKAYASU

for all u ∈ Dk (1 ≤ k ≤ n). However, by our assumption, we have µ([av]) ≤ sj(Xa) =µ([aw]) for some v ∈ Dl and 1 ≤ l ≤ n. This is a contradiction.

Hence, by (??), we have

m + 1 ≤ (n + 1)h + ε

h− ε.

Let k ∈ N with

(n + 1)h + ε

h− ε− 1 < k + 1 ≤ (n + 1)

h + ε

h− ε.

Since∑dn

j=1 sj(Xa)∑dn

j=1 1/j≤

∑ki=0

∑w∈Di

µ([aw])

log dn

≤∑k

i=0 µ([a])

log dn

≤ n + 1

log dn

· h + ε

h− εµ([a]),

we obtain


n + 1

log dn

· h + ε

h− εµ([a]).

Moreover, because

µ([a]) =∑

w∈Dn

µ([aw]) +∑

w∈Wn(X)\Dn

µ([aw]) ≤ |Dn|e−(n+1)(h−ε) + εn,

we have

(µ([a])− εn) e(n+1)(h−ε) ≤ |Dn|.Note that εn → 0 (n →∞) by (?). Therefore


n + 1

log |Dn| ·h + ε

h− εµ([a])

≤ lim supn→∞

n + 1

log (µ([a])− εn) + (n + 1)(h− ε)· h + ε

h− εµ([a])

=h + ε

(h− ε)2µ([a]).

Since ε is arbitrary, we have

||Xa||e+1 ≤1

hµ([a]).

By Proposition 2.2, the proof is complete. ¤

We now give some examples of subshifts with a maximal measure satisfying the condi-tion in Theorem 3.2.

Corollary 3.3. Let A be a 0-1 N × N matrix. We denote by ΣA the Markov shiftassociated with A, i.e.

ΣA = (ai)i∈Z ∈ SZ | A(ai, ai+1) = 1,


where S = 1, . . . , N is an alphabet. If τ = (Ta)a∈S is the creation operators for theMarkov shift ΣA, then we have

k−∞(τ) = htop(ΣA).

Proof. It suffices to show that the unique maximal measure of ΣA satisfies the conditionin Theorem 3.2. For simplicity, we may assume that A is irreducible with the Perronvalue α. Note that the topological entropy htop(ΣA) is equal to log α. If l and r are the

left and right Perron vectors with∑N

a=1 lara = 1, then the unique maximal measure µ isgiven by

µ([a0, a1, . . . , an]) =la0ran

αn,

where (a0, a1, . . . , an) ∈ Wn+1(ΣA) (e.g. see [Kit]). For any ε > 0, there exists K ∈ Nsuch that for any n ≥ K, we have

∣∣∣∣log larbα

n + 1

∣∣∣∣ < ε,

for all 1 ≤ a, b ≤ N . Therefore for any w ∈ Wn+1(ΣA), we have∣∣∣∣−

1

n + 1log µ([w])− log α

∣∣∣∣ < ε,

for all n ≥ K, i.e. the maximal measure µ satisfies the condition in Theorem 3.2. ¤

More generally, there is a class of subshifts, which is called almost sofic (see [Pet]).A subshift X is said to be almost sofic if for any ε > 0, there is an SFT Σ ⊆ X suchthat htop(X) − ε < htop(Σ), where a shift of finite type or SFT is a subshift that can bedescribed by a finite set of forbidden words, i.e. a subshift having the form XF for somefinite set F of words.

Corollary 3.4. If τ = (Ta)a∈A is the creation operators for an SFT Σ, then we have

k−∞(τ) = htop(Σ).

Proof. We recall that every SFT Σ is topologically conjugate to a Markov shift ΣA asso-ciated with a 0-1 matrix A. Now we give a short proof of this result. Let Σ be an SFTthat can be described by a finite set F of forbidden words. We may assume that all words

in F have length N + 1. We set A[N ]Σ = WN(Σ) and the block map ϕ : WN(Σ) → A[N ]

Σ ,

w 7→ w. We define the N -th higher block code βN : Σ → (A[N ]Σ )Z by

(βN(x))i = (xi, . . . , xi+N−1) ∈ A[N ]Σ ,

for x = (xi)i∈N ∈ Σ. Note that βN is the sliding block code with respect to ϕ. Thesubshift βN(Σ) is given by a Markov shift, i.e. there is a 0-1 matrix A with βN(Σ) = ΣA.

Let µ be the maximal measure of ΣA. The maximal measure of Σ is given by ν = µβN .We recall that µ is the Markov measure given by the left and right eigenvectors l, r andthe eigenvalue α. For w ∈ Wn(Σ) with n ≥ N , we have

ν([w]) = µ([ϕ(w[1,N ]), . . . , ϕ(w[n−N+1,n])])

=larb

αn−N,

10 RUI OKAYASU

where a = ϕ(w[1,N ]), b = ϕ(w[n−N+1,n]) and w[k,l] = (wk, . . . , wl) for k ≤ l. Hence one canshow that the maximal measure ν of Σ satisfies the condition in Theorem 3.2 by the sameargument as in the proof of Corollary 3.3. ¤

Corollary 3.5. Let X be an almost sofic shift. If τ = (Ta)a∈A is the creation operatorsfor X, then we have

k−∞(τ) = htop(X).

Proof. Let ε > 0. Since X is almost sofic, there is an SFT Σ ⊆ X such that htop(X)−ε <htop(Σ). Let ϕ : Σ → X be an embedding. Note that the subshift ϕ(Σ) is also an SFT.Thus we may identify ϕ(Σ) with Σ. Let µ be the unique maximal measure of Σ. Fora ∈ A, we set

Xa =∑n≥0

∑w

µ([aw])TwP0T∗aw,

where w runs over all elements in Wn(Σ) with aw ∈ W(Σ). We have shown that themaximal measure µ of Σ satisfies the condition of Theorem 3.2 in the proof of Corollary3.4. Hence by the same argument as in the proof of Theorem 3.2, we have

htop(Σ) ≤ k−∞(τ).

Thus for arbitrary ε > 0, the following holds:

htop(X)− ε < htop(Σ) ≤ k−∞(τ).

It therefore follows from Proposition 3.1 that htop(X) = k−∞(τ) if X is an almost soficshift. ¤

For β > 1, the β-transformation Tβ on the interval [0, 1] is defined by the multiplicationwith β (mod 1), i.e. Tβ(x) = βx− [βx], where [t] is the integer part of t. Let N ∈ N withN − 1 < β ≤ N and A = 0, 1, . . . , N − 1. The β-expansion of x ∈ [0, 1] is a sequenced(x, β) = di(x, β)i∈N of A determined by

di(x, β) = [βT i−1β (x)].

We set

ζβ = supx∈[0,1)

(di(x, β))i∈N,

where the above supremum is taken in the lexicographical order, and we define the shiftinvariant closed subset Σ+

β of the full one-sided shift AN by

Σ+β =

x ∈ AN | σi(x) ≤ ζβ, i = 0, 1, . . .

,

where ≤ is the lexicographical order on AN = 0, 1, . . . , N − 1N. The β-shift Σβ is thenatural extension given by

Σβ =(xi)i∈Z ∈ AZ | (xi)i≥k ∈ Σ+

β , k ∈ Z.

It is known that htop(Σβ) = log β, (see [Hof]).The following result might be known among specialists. However, we give a proof here

as we can not find it in the literature.

Proposition 3.6. For β > 1, the β-shift Σβ is an almost sofic shift.


Proof. In [Par], it is shown that Σβ is an SFT if and only if d(1, β) is finite, i.e. there isK ∈ N such that dk(1, β) = 0 for all k ≥ K. Thus we may assume that d(1, β) is notfinite. Let ζβ = (ξi)i∈N. For n ∈ N, there is β(n) < β such that

1 =ξ1

β(n)+

ξ2

β(n)2+ · · ·+ ξn

β(n)n.

In [Par, Theorem 5], it is proved that

limn→∞

β(n) = β.

Hence we may assume that N − 1 ≤ β(n) < β for sufficiently large n. Since the maximalelement ζβ(n) has the form

(ξ1, ξ2, . . . , (ξn − 1), ξ1, ξ2, . . . , (ξn − 1), ξ1, . . . ),

we have ζβ(n) < ζ, where < is the lexicographical order. Therefore we obtain

Σ+β(n) ⊆ Σ+

β ⊆ 0, 1, . . . , N − 1N.It follows that Σβ(n) is the shift invariant closed subset of Σβ with topological entropylog β(n). Since d(1, β(n)) is finite, the subshift Σβ(n) is an SFT. It therefore follows form[Par, Theorem 5] that Σβ is an almost sofic. ¤

Hence it holds that k−∞(τ) = htop(Σβ) for every β-shift by Corollary 3.5.

Corollary 3.7. Let Σβ be the β-shift for β > 1. If τ = (Ta)a∈A is the creation operatorsfor Σβ, then we have

k−∞(τ) = htop(Σβ) = log β.

4. Groups and Macaev Norm

We discuss a relation between groups and the Macaev norm. Let Γ be a countablefinitely generated group, S a symmetric set of generators of Γ. We denote by | · |S theword length and by Wn(Γ, S) the set of elements in Γ with length n, with respect tothe system of generators S. The logarithmic volume of a group Γ in a given system ofgenerators S is the number

vS = limn→∞

log |Wn(Γ, S)|n

,

(cf. [Ver]). The following proposition can be proved in the same way as in the free groupcase [Voi3, Proposition 3.7. (a)].

Proposition 4.1. Let Γ be a finitely generated group with a finite generating set S andλ the left regular representation of Γ. If we set λS = (λa)a∈S, then

k−∞(λS) ≤ vS.

Proof. Let us denote by Pn the projection onto the subspace spanδg ∈ l2(Γ) | |g|S = n.If we set

Xn =n−1∑j=0

(1− j

n

)Pj,

then we have

||Xnλa − λaXn|| = ||λ∗aXnλa −Xn|| ≤ 1

n.

12 RUI OKAYASU

for a ∈ S. Hence

k−∞(λS) ≤ lim supn→∞

maxa∈S

||[Xn, λa]||−∞ ≤ limn→∞

log∑n

j=0 |Wn(Γ, S)|n

= vS.

¤

Now we compute the exact value of k−∞(λS) for certain amalgamated free productgroups.

Proposition 4.2. Let A be a finite group, G1, · · · , GM nontrivial finite groups contain-ing A as a subgroup and H1, . . . , HN the product group of the infinite cyclic group Zand the finite group A, (N + M > 1). Let Γ be the amalgamated free product group ofG1, · · · , GM , H1, · · · , HN with amalgamation over A. Set S = G1 ∪ · · · ∪GM ∪ (S1×A)∪· · · ∪ (SN × A) \ e, where Sj is the canonical generating set xj, x

−1j of the infinite

cyclic group Z and e is the group unit. Let λ be the left regular representation of Γ andλS = (λa)a∈S. Then we have

k−∞(λS) = vS.

In particular, for the free group FN (N ≥ 2), we have

k−∞(λS) = log(2N − 1).

Proof. By Proposition 4.1, it suffices to show that vS ≤ k−∞(λS). Let Ωi be the set of therepresentatives of Gi/A with e ∈ Ωi for i = 1, . . . , M . We identify xj with (xj, e) ∈ Hj

for j = 1, . . . , N , and set ΩM+j = xj, x−1j , e. Let

S =M+N⋃i=1

Ωi \ e.

We define the 0-1 matrix A with index S by

A(a, b) =

1 if |ab|S = 2;0 otherwise.

One can easily check that the above matrix A is irreducible and the topological entropyhtop(ΣA) of the Markov shift ΣA coincides with the logarithmic volume vS of Γ withrespect to the generating set S.

We denote by Γ0 the subset of Γ consisting of the group unit e and elements a1 · · · an ∈Γ, (n ∈ N) of the form

ak ∈ Ωik \ e for k = 1, . . . , n,ik 6= ik+1 if 1 ≤ ik ≤ M,ak = ak+1 if M + 1 ≤ ik ≤ M + N, ik = ik+1.

Note that the subspace l2(Γ0) can be identified with the Fock space FA of the Markovshift ΣA by the following correspondence:

δe ←→ ξ0,δa1···an ←→ ξa1 ⊗ · · · ⊗ ξan .

Let us denote by Pn the projection onto the subspace

spanδg ∈ l2(Γ) | |g|S = n.


For a ∈ S, we define the partial isometry Ta ∈ B(l2(Γ)) by

Ta =∑n≥0

Pn+1λaPn.

Under the identification with FA, the partial isometry Ta|l2(Γ0) for a ∈ S is the creationoperator on FA, (cf. [Oka]). We also identify Γ0 and W(ΣA). For w = a1 · · · an ∈ Γ0, we

set Tw = Ta1 · · ·Tan . Let µ be the maximal measure of ΣA. For a ∈ S, we put

Xa =∑n≥0

∑w

µ([aw])TwP0T∗aw,

where w runs over all w ∈ Γ0 with |w|S = n and |aw|S = |w|S + 1. For a ∈ S \ S, we

set Xa = 0. It can be easily checked that [λa, Xa] = [Ta, Xa] for a ∈ S. Therefore by thesame proof as in the sufshift case, we obtain

vS = htop(ΣA) = k−∞(λS).

¤

Remark 4.3. Let Γ be a finitely generated group with a finite generating set S. In [Voi5],Voiculescu proved that if the entropy h(Γ, µ) of a random walk µ on Γ with support Sis non-zero, then k−∞((λa)a∈S) is non-zero. However the above proposition suggests thatthe volume vS of Γ is more related to the invariant k−∞((λa)a∈S) rather than the entropyh(Γ, µ). It is an interesting problem to ask whether vS being non-zero implies k−∞((λa)a∈S)being non-zero. We also remark here that there is a relation between vS and h(Γ, µ): Ifh(Γ, µ) 6= 0, then vS 6= 0, (see [Ver, Theorem 1]). If the above mentioned problem wassolved affirmatively, then it would follow from Proposition 4.1 that k−∞((λa)a∈S) 6= 0 ifand only if vS 6= 0, i.e. Γ has exponential growth.

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[Hof] Hofbauer, F.: β-shifts have unique maximal measure. Monatsh. Math. 85 (1978), no. 3, 189–198.[Kit] Kitchens, B.P.: Symbolic dynamics. One-sided, two-sided and countable state Markov shifts, Uni-

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[Voi1] Voiculescu, D.: Some results on norm-ideal perturbations of Hilbert space operators. J. OperatorTheory 2 (1979), no. 1, 3–37.

[Voi2] Voiculescu, D.: Some results on norm-ideal perturbations of Hilbert space operators. II. J. OperatorTheory 5 (1981), no. 1, 77–100.

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Department of Mathematics, Kyoto University, Kyoto 606-8502, JAPANE-mail address: [email protected]

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Entropy of subshifts and the Macaev norm