Orbit Representations from Linear mod 1 Transformations
We show that every point x0∈[0,1] carries a representation of a C∗-algebra that encodes the orbit structure of the linear mod 1 interval map fβ,α(x)=βx+α. Such C∗-algebra is generated by partial isometries arising from the subintervals of monotonicity of the underlying map fβ,α. Then we prove that s...
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Correia Ramos, C. Martins, N. Pinto, P.R. 2019-02-18T13:25:02Z 2019-02-18T13:25:02Z 2012 Orbit Representations from Linear mod 1 Transformations / C. Correia Ramos, N. Martins, P.R. Pinto // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 17 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 46L55; 37B10; 46L05 DOI: http://dx.doi.org/10.3842/SIGMA.2012.029 https://nasplib.isofts.kiev.ua/handle/123456789/148466 We show that every point x0∈[0,1] carries a representation of a C∗-algebra that encodes the orbit structure of the linear mod 1 interval map fβ,α(x)=βx+α. Such C∗-algebra is generated by partial isometries arising from the subintervals of monotonicity of the underlying map fβ,α. Then we prove that such representation is irreducible. Moreover two such of representations are unitarily equivalent if and only if the points belong to the same generalized orbit, for every α∈[0,1[ and β≥1. First author acknowledges CIMA-UE for financial support. The other authors were partially supported by the Fundacao para a Ciencia e a Tecnologia through the Program POCI 2010/FEDER. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Orbit Representations from Linear mod 1 Transformations Article published earlier |
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Orbit Representations from Linear mod 1 Transformations |
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Orbit Representations from Linear mod 1 Transformations Correia Ramos, C. Martins, N. Pinto, P.R. |
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Orbit Representations from Linear mod 1 Transformations |
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Orbit Representations from Linear mod 1 Transformations |
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orbit representations from linear mod 1 transformations |
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We show that every point x0∈[0,1] carries a representation of a C∗-algebra that encodes the orbit structure of the linear mod 1 interval map fβ,α(x)=βx+α. Such C∗-algebra is generated by partial isometries arising from the subintervals of monotonicity of the underlying map fβ,α. Then we prove that such representation is irreducible. Moreover two such of representations are unitarily equivalent if and only if the points belong to the same generalized orbit, for every α∈[0,1[ and β≥1.
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Orbit Representations from Linear mod 1 Transformations / C. Correia Ramos, N. Martins, P.R. Pinto // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 17 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 029, 9 pages
Orbit Representations
from Linear mod 1 Transformations
Carlos CORREIA RAMOS †, Nuno MARTINS ‡ and Paulo R. PINTO ‡
† Centro de Investigação em Matemática e Aplicações,
R. Romão Ramalho, 59, 7000-671 Évora, Portugal
E-mail: ccr@uevora.pt
‡ Department of Mathematics, CAMGSD, Instituto Superior Técnico,
Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
E-mail: nmartins@math.ist.utl.pt, ppinto@math.ist.utl.pt
Received March 14, 2012, in final form May 09, 2012; Published online May 16, 2012
http://dx.doi.org/10.3842/SIGMA.2012.029
Abstract. We show that every point x0 ∈ [0, 1] carries a representation of a C∗-algebra
that encodes the orbit structure of the linear mod 1 interval map fβ,α(x) = βx + α. Such
C∗-algebra is generated by partial isometries arising from the subintervals of monotonicity
of the underlying map fβ,α. Then we prove that such representation is irreducible. Moreover
two such of representations are unitarily equivalent if and only if the points belong to the
same generalized orbit, for every α ∈ [0, 1[ and β ≥ 1.
Key words: interval maps; symbolic dynamics; C∗-algebras; representations of algebras
2010 Mathematics Subject Classification: 46L55; 37B10; 46L05
1 Introduction
A famous class of representations of the Cuntz algebra On called permutative representations
were studied and classified by Bratteli and Jorgensen in [2, 3]. From the applications viewpoint,
and besides its own right, applications of representation theory of Cuntz and Cuntz–Krieger
algebras to wavelets, fractals, dynamical systems, see e.g. [2, 3, 13], and quantum field theory
in [1] are particularly remarkable. For example, it is known that these representations of the
Cuntz algebra serve as a computational tool for wavelets analysts, see [12]. This is clear because
such a representation on a Hilbert space H induces a subdivision of H into orthogonal subspaces.
Then the problem in wavelet theory is to build orthonormal bases in L2(R) from these data.
Indeed this can be done [9] and these wavelet bases have the advantages over the earlier known
basis constructions (one advantage is the efficiency of computation). This method has also been
applied to the context of fractals that arise from affine iterated function systems [10]. Some of
these results have been extended to the more general class of Cuntz–Krieger algebras, see [7, 13]
and subshift C∗-algebras [4, 11, 15] (whose underlying subshift is not necessarily of finite type)
in [5].
Symbolic dynamics is one of the main tools that we have used in [5, 7] to construct representa-
tions of Cuntz, Cuntz–Krieger and subshift C∗-algebras. The C∗-algebra is naturally associated
to the given interval map and the Hilbert spaces naturally arise from the generalized orbits of
the interval map. For a particular family of interval maps, we were able to recover Bratteli and
Jorgensen permutative representations in [7] among the class of Markov maps (which underline
the Cuntz–Krieger algebras of the transition matrix). We remark that while these Cuntz and
Cuntz–Krieger algebras are naturally associated to the so-called Markov or periodic dynamical
systems, the subshift C∗-algebras are ready to incorporate bigger classes of interval maps.
mailto:ccr@uevora.pt
mailto:nmartins@math.ist.utl.pt
mailto:ppinto@math.ist.utl.pt
http://dx.doi.org/10.3842/SIGMA.2012.029
2 C. Correia Ramos, N. Martins and P.R. Pinto
The interval maps that we treat in [5] are unimodal maps (that have precisely two subintervals
of monotonicity). Then the representations of the subshift C∗-algebra constructed in [5] are
shown to coincide with the ones constructed in [7] from the Cuntz–Krieger algebra, provided the
underlying dynamical system is periodic and therefore has a finite transition Markov Matrix.
However, the proof of the irreducibility of the subshift C∗-algebras representations (for unimodal
maps without a finite transition Markov matrix) rely on the structure of these unimodal interval
maps where the C∗-algebra is generated by two partial isometries.
In this paper we construct representations (of a subshift C∗-algebra generated by n partial
isometries) from a family of interval maps and prove the irreducibility of these representations
(avoiding the unimodal maps techniques used in [5]).
Namely, we yield and study representations of a certain C∗-algebra on the generalized orbit⋃
i∈Z f
j
β,α(x0) of every point x0 ∈ [0, 1] from the interval map fβ,α : [0, 1]→ [0, 1] defined by
fβ,α(x) = βx+ α (mod 1) with β ≥ 1 and α ∈ [0, 1[, (1)
by fixing the parameters α and β. The underlying C∗-algebra OΛfβ,α
is generated by n partial
isometries where n is the number of monotonicity subintervals of fβ,α. See Fig. 1 for a graph of
one such map.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure 1. Graph of fβ,α with α =
√
2− 1 and β = 2 (n = 3).
We show that the representation is irreducible. Moreover the representations of the same
algebra on the orbits of the two points x0 and y0 are unitarily equivalent if and only if the orbits
coincide. If the parameters α and β are so that the dynamical system ([0, 1], fβ,α) is periodic, the
above results were obtained by [6, 7], where the relevant C∗-algebra is the Cuntz–Krieger OAfβ,α
and Afβ,α is the underlying Markov transition 0-1 (finite) matrix of fβ,α. A further particular
case is obtained when β = n is an integer and α = 0, in which case the n×n matrix Afn,0 = (ai,j)
is full, aij = 1 for all i, j, and thus recovering the Cuntz algebra On representations yielded
in [2, 3] using wavelet theory framework.
Of course we may fairly easily prove that we do get representations of such C∗-algebras in
the context of piecewise monotone maps, but the main concern is how to show irreducibility
(and unitarily equivalence) of such the representations, thus giving a rich family of representa-
tions attached to interval maps. The periodic or non-periodic cases for which we have n = 2
subintervals of monotonicity were carried out in [5]. We generalize here the construction of
the representations for generic piecewise monotone interval maps and prove irreducibility and
unitarily equivalence for the dynamical systems arising from equation (1) above – and obviously
we look for values of the parameters α and β for which we do not have a finite Markov transition
matrix for fβ,α and thus we really get representations that cannot be recovered from [5, 6, 7].
A more detailed description of the paper is as follows. In Section 2 we provide some back-
ground material first on the operator algebras setup and then in symbolic dynamics [16]. The
Orbit Representations from Linear mod 1 Transformations 3
main results are in Section 3. We consider a partition I of the interval I = [0, 1] into subin-
tervals so that the restriction of fβ,α to each of these subintervals is monotone. Then for every
x0 ∈ I we explicitly define in equation (6) a linear operator on the Hilbert space Hx0 that arises
from the generalized orbit of x0, for every such subinterval. The Hilbert space Hx0 encodes the
generalized orbit orbit(x0) of x0 and in fact every ξ ∈ orbit(x0) is regarded as a vector |ξ〉 in Hx0
using Dirac’s notation. Then we prove that these linear operators do satisfy the relation they
ought to satisfy, leading to a representation ρx0 : OAfβ,α → B(Hx0) of the C∗-algebra OAfβ,α ,
as in Proposition 2.
Then the main result of this paper is Theorem 1 where we show that representation ρx0 of the
underlying C∗-algebra OAfβ,α is irreducible and that two such representations ρx0 and ρy0 are
unitarily equivalent if and only if y0 belongs to the generalized orbit of x0. The new ingredient
involved in the proof of Theorem 1 is the computation of the commutant A′β,α of C∗-algebra Aβ,α
(generated by the operators defined in equations (9) and (10)) in B(Hx0). Indeed, as soon as
we prove that the commutant is trivial A′β,α = C1, we only have to show that A′β,α contains
ρx0(OAfβ,α )′ as in Proposition 3.
2 Preliminaries
In this section we provide some necessary background, starting with the operator algebras we
obtain from dynamical systems. A representation of a ∗-algebra A on a complex Hilbert space H
is a ∗-homomorphism π : A → B(H) into the ∗-algebra B(H) of bounded linear operators
on H. Usually representations are studied up to unitary equivalence. Two representations
π : A → B(H) and π̃ : A → B(H̃) are (unitarily) equivalent if there is a unitary operator
U : H → H̃ (i.e., U is a surjective isometry) such that
Uπ(a) = π̃(a)U for every a ∈ A,
and in this case we write π ∼ π̃. A representation π : A → B(H) of some ∗-algebra is said to be
irreducible if there is no non-trivial subspace of H invariant with respect to all operators π(a)
with a ∈ A. A well known result, see e.g. [17, Proposition 3.13], says that π is irreducible if and
only if
x ∈ B(H) : xπ(a) = π(a)x for all a ∈ A =⇒ x = λ1, (2)
for some complex number λ, where 1 denotes the identity of B(H). By the very definition of
commutant, (2) can be restated as follows: π(A)′ = C1. We will be interested in some classes
of C∗-algebras (= Banach ∗-algebras such that ||aa∗|| = ||a||2 holds for all a, see e.g. [17]).
Besides, if we have a representation π : A → B(H) of a C∗-algebra A, then π being a ∗-
homomorphism implies that ||π(a)|| ≤ ||a|| for all a ∈ A, thus π is automatically continuous, see
also e.g. [17, Secttion 1.5.7]).
Subshift C∗-algebras
Let Λ ⊆ ΣN be a subshift with a finite alphabet Σ = {1, . . . , n}. Exel [11] and Matsumoto [15]
constructed C∗-algebras associated to Λ. Carlsen and Silvestrov [4] unified the two constructions
that led them to a C∗-algebra OΛ, which is unital and generated partial isometries {ti}i∈Σ. Then
the partial isometries that generate OΛ obey the following relations:∑
tit
∗
i = 1, t∗αtαtβ = tβt
∗
αβtαβ, t∗αtαt
∗
βtβ = t∗βtβt
∗
αtα,
where tα = tα1 · · · tα|α| and tβ = tβ1 · · · tβ|β| with α, β admissible words (if α = (α1, . . . , αk) with
αi ∈ Σ we denote by |α| the length k of α). The algebra OΛ is called the C∗-algebra associated
4 C. Correia Ramos, N. Martins and P.R. Pinto
to the subshift Λ or subshift C∗-algebra. Important properties of the subshift C∗-algebra OΛ
(e.g. simplicity) are naturally inherited from properties of the subshift Λ. If Λ is a subshift
of finite type, then OΛ is nothing but the well known Cuntz–Krieger algebra [15], where the
Cuntz–Krieger algebra OA associated to a 0-1 matrix A = (aij) is the C∗-algebra [8] generated
by (non-zero) partial isometries s1, . . . , sn satisfying:
s∗i si =
n∑
j=1
aijsjs
∗
j (i = 1, . . . , n),
n∑
i=1
sis
∗
i = 1, (3)
and the Cuntz algebraOn is the Cuntz–Krieger algebraOA with A full aij = 1 for all 1 ≤ i, j ≤ n.
2.1 Symbolic dynamics on piecewise monotone interval maps
Let f : I → I be a piecewise monotone map of the interval I into itself, that is, there is a minimal
partition of open sub-intervals of I, I = {I1, . . . , In} such that
⋃n
j=1 Ij = I and f|Ij is continuous
monotone, for every j = 1, . . . , n. We define fj := f|Ij . The inverse branches are denoted by
f−1
j : f(Ij) → Ij . Let χIi be the characteristic function on the interval Ii. The following are
naturally satisfied
f ◦ f−1
j (x) = χf(Ij)(x)x, f−1
j ◦ f|Ij (x) = χIj (x)x.
Let {1, 2, . . . ,m} be the alphabet associated to some partition {J1, . . . , Jm} of open sub-
intervals of I so that
⋃m
j=1 Jj = I, not necessarily I. The address map, is defined by
ad :
m⋃
j=1
Jj → {1, 2, . . . ,m}, ad(x) = i if x ∈ Ji.
We define
Ωf :=
{
x ∈ I : fk(x) ∈
m⋃
j=1
Jj for all k = 0, 1, . . .
}
.
Note that Ωf = I. The itinerary map it : Ωf → {1, 2, . . . ,m}N is defined by
it(x) = ad(x)ad(f(x))ad
(
f2(x)
)
· · ·
and let
Λf = it(Ωf ). (4)
The space Λf is invariant under the shift map σ : {1, 2, . . . ,m}N → {1, 2, . . . ,m}N defined by
σ(i1i2 · · · ) = (i2i3 · · · ),
and we have it ◦ f = σ ◦ it. We will use σ meaning in fact σ|Λf . A sequence in {1, 2, . . . ,m}N is
called admissible, with respect to f , if it occurs as an itinerary for some point x in I, that is, if
it belongs to Λf . An admissible word is a finite sub-sequence of some admissible sequence. The
set of admissible words of size k is denoted by Wk = Wk(f). Given i1 · · · ik ∈ Wk, we define
Ii1···ik as the set of points x in Ωf which satisfy
ad(x) = i1, . . . , ad
(
fk(x)
)
= ik.
Orbit Representations from Linear mod 1 Transformations 5
2.2 Linear mod 1 interval maps
Now, let us consider the family of linear mod 1 transformations as in equation (1). In the sequel
we will denote fβ,α by f . The behavior of the dynamical system (I, f) is characterized by the
sequences it(f(c+
j )) and it(f(c−j )), for each discontinuity point cj , see [14]. Let us consider the
partition of monotonicity I = {I1, . . . , In} of f , with
I1 = ]0, (1− α)/β[ , . . . , Ij = ](j − α)/β, (j + 1− α)/β[ , . . . ,
. . . , In = ](n− 1− α)/β, 1[ , (5)
which is the minimal partition of monotonicity for f (n = [β] + 1 with [β] being the integral
part of β and α > 0. For α = 0: n = [β] if β is an integer, and n = [β] + 1 if β /∈ N).
A characterization of the values of α, for which there is a Markov partition, is partially given
by the following:
Proposition 1. If itf (0) = (ξ1, ξ2, . . . , ξl, . . . ) and itf (1) are periodic (with ξl = ξ1) then
α =
ξl + ξl−1β + ξl−2β
2 + · · ·+ ξ1β
l−1
1 + β + β2 + · · ·+ βl−1
.
In particular α ∈ Q(β).
Proof. See [14, Proposition 2.6] for full details. �
3 Subshift algebras from linear mod 1 transformations
As in [7], we consider the equivalence relation
Rf = {(x, y) : fn(x) = fm(y) for some n,m ∈ N0}.
We write x ∼ y whenever (x, y) ∈ Rf . Consider the equivalence class Rf (x) (=
⋃
j∈Z f
j(x)
also called the generalized orbit of x) and set Hx the Hilbert space l2(Rf (x)) with canonical
orthonormal basis {|y〉 : y ∈ Rf (x)}, in Dirac notation. Note that Hx = Hy (are the same
Hilbert spaces) whenever x ∼ y. The inner product (·, ·) is given by
〈y|z〉 = (|y〉 , |z〉) = δy,z.
Let now f be the linear mod 1 transformation defined in equation (1) and I = {I1, . . . , In}
be the partition of monotonicity as written down in equation (5). For every i = 1, . . . , n, let
fi := f|Ii be the restriction of f to the subinterval Ii of the partition I. For every i ∈ {1, 2, . . . , n}
let us define an operator Ti on Hx defined first on the orthonormal basis as follows:
Ti |y〉 = χf(Ii)(y)
∣∣f−1
i (y)
〉
(6)
and then extend it by linearity and continuity to Hx. Note that χf(Ii)(x) = 1 if and only if there
is a pre-image of x in Ii. The we have
T ∗i |y〉 = χIi(y) |f(y)〉 . (7)
Indeed, on one hand
(|y〉, Ti|z〉) =
(
|y〉, χf(Ii)|f
−1
i (z)〉
)
and on the other hand we have
(T ∗i |y〉, |z〉) = (χIi(y)|f(y)〉, |z〉) = χIi(y)δf(y),z.
So since χf(Ii)|f
−1
i (z) = χIi(y)δf(y),z we have shown that the adjoint of Ti is given by equa-
tion (7). We further remark that Ti is a partial isometry: namely, Ti is an isometry on its
restriction to span{|y〉 : y ∈ f(Ii)} ∩Hx and vanishes in the remaining part of Hx.
6 C. Correia Ramos, N. Martins and P.R. Pinto
Lemma 1. The operators Ti satisfy the relations
n∑
i=1
TiT
∗
i = 1, T ∗µTµTν = TνT
∗
µνTµν and T ∗µTµT
∗
ν Tν = T ∗ν TνT
∗
µTµ,
for µ, ν given admissible words.
Proof. Consider TiT
∗
i acting on a vector |y〉 of the canonical basis of Hx,
TiT
∗
i |y〉 = χIi(y)Ti |f(y)〉 = χIi(y)χf(Ii)(f(y))
∣∣f−1
i ◦ f(y)
〉
= χIi(y) |y〉 ,
since χIi(y)χf(Ii)(f(y)) = χIi(y). Then
(T1T
∗
1 + · · ·+ TnT
∗
n) |y〉 = (χI1(y) + · · ·+ χIn(y)) |y〉 = |y〉 .
Now, consider T ∗µTµTν acting on a vector |y〉 of the canonical basis for some µ = µ1 · · ·µk,
ν = ν1 · · · νr admissible words,
T ∗µTµTν |y〉 = T ∗µTµχfr(Iν)(y)
∣∣f−1
ν1 ◦ · · · ◦ f
−1
νr (y)
〉
= T ∗µχfk+r(Iµν)(y)
∣∣f−1
µ1 ◦ · · · ◦ f
−1
µk
◦ f−1
ν1 ◦ · · · ◦ f
−1
νr (y)
〉
= χfk+r(Iµν)(y)
∣∣f−1
ν1 ◦ · · · ◦ f
−1
νr (y)
〉
.
On the other hand
TνT
∗
µνTµν |y〉 = TνT
∗
µνχfk+r(Iµν)(y)
∣∣f−1
µ1 ◦ · · · ◦ f
−1
µk
◦ f−1
ν1 ◦ · · · ◦ f
−1
νr (y)
〉
= Tνχfk+r(Iµν)(y) |y〉 = χfk+r(Iµν)(y)
∣∣f−1
ν1 ◦ · · · ◦ f
−1
νr (y)
〉
. (8)
Finally since T ∗µTµ|y〉 = χfk(Iµ)(y)|y〉 and T ∗ν Tν |y〉 = χfr(Iν)(y)|y〉 for admissible words µ, ν, we
easily conclude that T ∗µTµT
∗
ν Tν = T ∗ν TνT
∗
µTµ. �
As an immediate consequence of Lemma 1 (and above equation (8)) we obtain the following.
Proposition 2. Let OΛf be the subshift algebra associated to the subshift Λf as defined in (4)
above, then ρx : OΛf → B(Hx) defined by ti → Ti is a representation of OΛf .
We remark here that T ∗i Ti = 1 for all i = 2, . . . , n − 1 and T ∗1 T1 = 1 if and only if α = 0.
Besides T ∗nTn = 1 if and only if β = n ∈ N. Therefore ρx is a representation of a Cuntz algebra
if and only if α = 0 and β is a positive integer. In this case, the interval map f is a Markov map
and moreover the partition with the monotonicity intervals, as in (5), reduces to
I1 = ]0, 1/n[, I2 = ]1/n, 2/n[, . . . , Ij = ](j − 1)/n, j/n[, . . . , In = ](n− 1)/n, 1[
and coincides with the (minimal) Markov partition [7]. From the viewpoint of interval maps,
these Cuntz algebra On representations were treated in [7, Remark 2.9].
We remark that if fβ,α is a linear mod 1 map with α /∈ Q and β = 1 then f is not a Markov
map by Proposition 1 and thus the representation ρx of Proposition 2 is never a representation
of a Cuntz–Krieger algebra.
Orbit Representations from Linear mod 1 Transformations 7
3.1 Irreducibility of the representations
For the linear mod 1 transformation map f and the linear operators T1, . . . , Tn ∈ B(Hx) defined
in equation (6), we may consider the following operator
V = T ∗1 + · · ·+ T ∗n , (9)
which satisfies V |y〉 = |f(y)〉 on every vector basis |y〉. In general V is not unitary (unless β = 1
so that f becomes an invertible function). Let U be the diagonal operator
U |y〉 = e2πiy |y〉 , (10)
which is an unitary operator, with U∗ |y〉 = e−2πiy |y〉 . In order to emphasize that U, V ∈ B(Hx),
we write Ux and Vx for the above operators U and V , respectively.
For a self-adjoint set of operators S ⊆ B(H) on some Hilbert space H, containing the
identity 1, the von Neumann algebra generated by S equals the double commutant S ′′, which
in turn is also equal to the closure of S under the strong operator topology (this is the famous
bicommutant von Neumann theorem e.g. the textbook [17]). We note that S ′ = {t ∈ B(H) :
ts = st, for all s ∈ S} and S ′′ = (S ′)′ and S ′′′ = S ′. Also si converges to s in the strong operator
topology if ||(si − s)ξ|| → 0 for every vector ξ ∈ H.
Let Aβ,α = C∗(U, V ) be the C∗-subalgebra of B(Hx) generated by U and V , and consider
the representation ρx from Proposition 2.
Proposition 3. We have Aβ,α ⊆ ρx(OΛf )′′.
Proof. By definition of the C∗-algebra Aβ,α, we only need to prove that U , V belong
to ρx(OΛf )′′. It is clear that V ∈ ρx(OΛf ) ⊆ ρx(OΛf )′′. We now show that U ∈ ρx(OΛf )′′.
For each µ ∈ Wk, let m(µ) be some point in Iµ ∩ Rf (x). Note that if we have it(y) = (αj)
∞
j=1,
for some point y ∈ Rf (x) then lim
j→∞
m(α1 · · ·αj) = y, and the limit is independent on the
choice of m(α1 · · ·αj) ∈ Iα1···αj , since for each j ∈ N there is r > j so that Iα1···αj ⊃ Iα1···αr .
Let Mk =
∑
µ∈Wk
e2πim(µ)TµT
∗
µ . We can see that lim
k→∞
Mk = U in the strong topology, since
lim
k→∞
‖Mkv − Uv‖ = 0, for every v ∈ Hx. Therefore, U is in the von Neumann algebra generated
by the operators T1, . . . , Tn. �
Lemma 2. Let x ∈ I and f be the linear mod 1 transformation (1) with fixed α and β. Let
Q ∈ B(Hx) be an operator commuting with both U and V , then Q = λI for some λ ∈ C.
Proof. First of all we remark that the
e2πiz are all distinct, (11)
with z ∈ Rf (x), since z ∈ [0, 1]. Let Q ∈ B(Hx) commuting with both U and V . For each
z ∈ Rf (x) let µz := e2πiz. By definition U |z〉 = µz|z〉, i.e., every µz is an eigenvalue of U . We
easily get
UQ|z〉 = µzQ|z〉 (12)
by applying the definitions and the fact that U and Q commute. For every z ∈ Rf (x), set
ξz = Q|z〉. Then we can write equation (12) as follows: Uξz = µzξz. Since {|z〉, z ∈ Rf (x)} is
an o.n. basis of Hx, there are constants cw with w ∈ Rf (x) such that ξz =
∑
cw|w〉. Since
Uξz =
∑
w∈Rf (x)
µwcw|w〉
8 C. Correia Ramos, N. Martins and P.R. Pinto
and
µzξz =
∑
w∈Rf (x)
µzcw|w〉,
we conclude that cw = 0 for all w 6= z, because the µw’s are all distinct by (11) and {|w〉} is
an o.n. basis of Hx. It follows that ξz = cz|z〉 or equivalently Q|z〉 = cz|z〉 for some cz ∈ C.
But V |z〉 = |f(z)〉, so V Q = QV gives cz = cf(z). Therefore Q is a multiple of the identity
operator. �
Theorem 1. The representation ρx of the subshift C∗-algebra OΛf as in Proposition 2 is irre-
ducible. Moreover ρx ∼ ρy if and only if x ∼ y.
Proof. To prove that ρx is irreducible we prove that ρx(OΛf )′ = CI. First note that from
Proposition 3 we have C∗(U, V ) ⊆ ρx(OΛf )′′ and so taking commutant and using von Neumann
bicommutant theorem, we conclude that ρx(OΛf )′ ⊆ C∗(U, V )′.
Now let Q ∈ ρx(OΛf )′. So we can conclude that Q ∈ C∗(U, V )′ and thus Q commutes with
both U and V . By Lemma 2 we conclude that Q = λI for some λ ∈ C. Therefore ρx(OΛf )′ = CI
and so ρx is an irreducible representation of OΛf .
It is clear that if Rf (x) = Rf (y), then ρx and ρy are unitarily equivalent. Notice that Ux
and Uy have the same eigenvalues if and only if x ∼ y. Hence ρx and ρy can be unitarily
equivalent only when x ∼ y. �
Remark 1. If β = n is an integer and α = 0 then the partition of I into monotonicity subinter-
vals of fn,0 is a Markov partition, the subshift Λfn,0 is the full shift and the underlying C∗-algebra
is the Cuntz algebra On. Furthermore we recover in Theorem 1 our previous result obtained
in [7].
Acknowledgment
First author acknowledges CIMA-UE for financial support. The other authors were partially
supported by the Fundação para a Ciência e a Tecnologia through the Program POCI 2010/FE-
DER.
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http://dx.doi.org/10.1007/BFb0082847
1 Introduction
2 Preliminaries
2.1 Symbolic dynamics on piecewise monotone interval maps
2.2 Linear mod 1 interval maps
3 Subshift algebras from linear mod 1 transformations
3.1 Irreducibility of the representations
References
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