Cuntz-Krieger algebras associated with Hilbert $C^*$-quad modules of commuting matrices

0.19 MB·English

by  Kengo Matsumoto

CUNTZ-KRIEGER ALGEBRAS ASSOCIATED WITH HILBERT
C∗-QUAD MODULES OF COMMUTING MATRICES
2
KENGOMATSUMOTO
1
0
2

n
Ja Abstract. Let OHAκ,B be the C∗-algebra associated with the Hilbert C∗-
quadmodulearisingfromcommutingmatricesA,Bwithentriesin{0,1}. We
5 willshowthatiftheassociatedtilingspaceXκ istransitive,theC∗-algebra
A,B

A] NOH,MAκ,,BthisesKim-gprleouapnsdopfutrheelysiimnfipnleitep.urIenlypianrfitniciutelr,Cf∗o-raltgweobrpaoOsitHiv[Ne],i[nMte]gaerres
κ
computed byusingtheEuclideanalgorithm.
O
.
h 1. Introduction
t
a In [11], the author has introduced a notion of C∗-symbolic dynamical system,
m
whichisageneralizationofafinitelabeledgraph,aλ-graphsystemandanautomor-
[ phismofaunitalC∗-algebra. Itisdenotedby(A,ρ,Σ)andconsistsofafinitefamily

1 {ρα}α∈ΣofendomorphismsofaunitalC∗-algebraAsuchthatρα(ZA)⊂ZA,α∈Σ
v and α∈Σρα(1)≥ 1 where ZA denotes the center of A. It provides a subshift Λρ
6 over Σ and a Hilbert C∗-bimodule Hρ over A which gives rise to a C∗-algebra
5 P A
O as a Cuntz-Pimsner algebra ([11], cf. [6], [19]). In [13] and [14], the author
0 ρ
has extended the notion of C∗-symbolic dynamicalsystem to C∗-textile dynamical
1
. system which is a higher dimensional analogue of C∗-symbolic dynamical system.
1 The C∗-textile dynamical system (A,ρ,η,Σρ,Ση,κ) consists of two C∗-symbolic
0
dynamical systems (A,ρ,Σρ) and (A,η,Ση) with a common unital C∗-algebra A
2
1 anda commutationrelationbetween the endomorphismsρ andη througha map κ
: stated below. Set
v
i Σρη ={(α,b)∈Σρ×Ση |η ◦ρ 6=0},
X b α
Σηρ ={(a,β)∈Ση×Σρ |ρ ◦η 6=0}.
r β a
a
We assume that there exists a bijection κ : Σρη −→ Σηρ, which we fix and call a
specification. Then the required commutation relations are
η ◦ρ =ρ ◦η if κ(α,b)=(a,β). (1.1)
b α β a
A C∗-textile dynamical system provides a two-dimensional subshift and a multi
structure of Hilbert C∗-bimodule that has multi right actions and multi left ac-
tions and multi inner products. Such a multi structure of Hilbert C∗-bimodule is
called a Hilbert C∗-quad module, denoted by Hρ,η. In [14], the author has intro-
κ
duced a C∗-algebra associated with the Hilbert C∗-quad module. The C∗-algebra
OHρκ,η has been constructed in a concrete way from the structure of the Hilbert
C∗-quad module Hρ,η by a two-dimensional analogue of Pimsner’s construction of
κ
C∗-algebras from Hilbert C∗-bimodules. It is generated by the quotient images of
creationoperatorsontwo-dimensionalanalogueofFock Hilbert module by module
maps of compact operators. As a result, the C∗-algebra has been proved to have
1
a universalproperty subject to certainoperator relations of generatorsencoded by
structure of the Hilbert C∗-quad module ([14]).
Let A,B be two N × N matrices with entries in nonnegative integers. We
assume that both A and B are essential, which means that they have no rows or
columns identically to zero vector. They yield directed graphs G = (V,E ) and
A A
G = (V,E ) with a common vertex set V = {v ,...,v } and edge sets E and
B B 1 N A
E respectively, where the edge set E consist of A(i,j)-edges from the vertex v
B A i
tothe vertexv andE consistofB(i,j)-edgesfromthe vertexv tothe vertexv .
j B i j
WethenhavetwoC∗-symbolicdynamicalsystems(A ,ρA,E )and(A ,ρB,E )
N A N B
with A =CN. Denote by s(e),r(e) the source vertex and the range vertex of an
N
edge e. Put
ΣAB ={(α,b)∈E ×E |r(α)=s(b)},
A B
ΣBA ={(a,β)∈E ×E |r(a)=s(β)}.
B A
Assume that the commutation relation
AB =BA (1.2)
holds. Wemaytakeabijectionκ:ΣAB −→ΣBA suchthats(α)=s(a),r(b)=r(β)
for κ(α,b) = (a,β), which we fix. This situation is called an LR-textile system
introduced by Nasu ([16]). We then have a C∗-textile dynamical system (see [14])
(A ,ρA,ρB,E ,E ,κ).
N A B
Let us denote by HA,B the associatedHilbert C∗-quad module defined in [14]. We
κ
set
E ={(α,b,a,β)∈E ×E ×E ×E |κ(α,b)=(a,β)}. (1.3)
κ A B B A
Each element of E is called a tile. Let Xκ ⊂ (E )Z2 be the two-dimensional
κ A,B κ
subshift of the Wang tilings of E (cf. [24]). It consists of the two-dimensional
κ
configurationsx:Z2 −→E compatibletotheirboundaryedgesoneachtile,andis
κ
calledthesubshiftofthetilingspaceforthespecificationκ:ΣAB −→ΣBA. Wesay
that XAκ,B is transitive if for two tiles ω,ω′ ∈Eκ, there exists (ωi,j)(i,j)∈Z2 ∈XAκ,B
such that ω =ω,ω =ω′ for some (i,j)∈Z2 with j <0<i. We set
0,0 i,j
Ω ={(α,a)∈E ×E |s(α)=s(a),κ(α,b)=(a,β) for some β ∈E ,b∈E }
κ A B A B
(1.4)
and define two |Ω |×|Ω |-matrcies A and B with entries in {0,1} by
κ κ κ κ
1 κ(α,b)=(a,β) for some β ∈E ,
A
A ((α,a),(δ,b))= (1.5)
κ
(0 otherwise
for (α,a),(δ,b)∈Ω , and
κ
1 κ(α,b)=(a,β) for some b∈E ,
B
B ((α,a),(β,d))= (1.6)
κ
(0 otherwise
for (α,a),(β,d)∈Ω respectively. Put the matrix
κ
A A
H = κ κ . (1.7)
κ B B
κ κ
(cid:20) (cid:21)
2
It has been proved in [14] that the C∗-algebra O associated with the Hilbert
HAκ,B
C∗-quad module HA,B is isomorphic to the Cuntz-Krieger algebra O for the
κ Hκ
matrix H . In this paper, we first show the following theorem.
κ
Theorem 1.1 (Theorem 2.9). The subshift Xκ of the tiling space is transitive
A,B
if and if the matrix H is irreducible. In this case, H satisfies condition (I) in
κ κ
the sense of [2]. Hence if the subshift Xκ of the tiling space is transitive, the
A,B
C∗-algebra O is simple and purely infinite.
HAκ,B
We will second see the following theorem.
Theorem 1.2 (Theorem2.10). If the matrix A or B is irreducible, the matrix H
κ
is irreducible and satisfies condition (I), so that the C∗-algebra O is simple
HAκ,B
and purely infinite.
Let N,M be positive integers with N,M > 1. They give 1 × 1 commuting
matricesA=[N],B =[M]. WewillpresentK-theoryformulaefortheC∗-algebras
O withexchangingspecificationκ. ThedirectedgraphG associatedtothe
Hκ[N],[M] A
matrix A= [N] is a graph consists of N-self directed loops denoted by E with a
A
vertex denoted by v. Similarly the directed graph G consists of M-self directed
B
loops denoted by E with the vertex v. We fix a specification κ : E ×E −→
B A B
E ×E definedbyexchangingκ(α,a)=(a,α)for(α,a)∈E ×E . Wewillhave
B A A B
the following K-theory formulae for the C∗-algebra O . In its computation,
Hκ[N],[M]
the Euclidean algorithm will be used.
Theorem 1.3. For integers 1 < N ≤ M ∈ N and a specification κ of exchanging
directed N-loops and M-loops, the C∗-algebra O is a simple purely infinite
Hκ[N],[M]
Cuntz-Krieger algebra whose K-groups are
K (O )∼=0,
1 Hκ[N],[M]
M−2
K (O )∼=Z/(N −1)Z⊕···⊕Z/(N −1)Z
0 Hκ[N],[M]
z N}|−2 {
⊕Z/(M −1)Z⊕···⊕Z/(M −1)Z
⊕Zz/dZ⊕Z/[k1,k}2|,...,kj+1](M −{1)(M +N −1)Z
where d=(N−1,M−1) the greatest common divisor of N−1 and M−1, and the
sequence k ,k ,...,k is the successive integral quotients of M −1 by N −1 by
0 2 j+1
the Euclidean algorithm, and the integer [k ,k ,...,k ] is defined by inductively
1 2 j+1
[k ]=1, [k ]=k , [k ,k ]=1+k k , ...,
0 1 1 1 2 1 2
[k ,k ,...,k ]=[k ,k ,...,k ]k +[k ,...,k ].
1 2 j+1 1 2 j j+1 1 j−1
We remark that the C∗-algebras studied in this paper are different from the
higher rank graph algebras studied by A. Kumjian–D. Pask [7], G. Robertson–T.
Steger [21], V. Deaconu [3], and etc. .
Throughout the paper, we denote by N and by Z the set of positive integers
+
and the set of nonnegative integers respectively.
3
2. Transitivity of the tilings Xκ and simplicity of O
A,B HAκ,B
Let Σ be a finite set. The two-dimensionalfull shift over Σ is defined to be
Z2
Σ ={(xi,j)(i,j)∈Z2 |xi,j ∈Σ}.
An element x ∈ ΣZ2 is regarded as a function x : Z2 −→ Σ which is called a
configurationon Z2. For a vector m=(m ,m )∈Z2, let σm :ΣZ2 −→ΣZ2 be the
1 2
translation along vector m defined by
σm((xi,j)(i,j)∈Z2)=(xi+m1,j+m2)(i,j)∈Z2.
A subset X ⊂ΣZ2 is said to be translationinvariantif σm(X)=X for all m∈Z2.
It is obvious to see that a subset X ⊂ ΣZ2 is translation invariant if and only if
X is invariant only both horizontaly and vertically, that is, σ(1,0)(X) = X and
σ(0,1)(X)=X. For k ∈Z , put
+
[−k,k]2 ={(i,j)∈Z2 |−k≤i,j ≤k}=[−k,k]×[−k,k].
A metric d on ΣZ2 is defined by for x,y ∈ΣZ2 with x6=y
1
d(x,y)= if x =y ,
2k (0,0) (0,0)
where k = max{k ∈ Z | x = y }. If x 6= y , put k = −1
+ [−k,k]2 [−k,k]2 (0,0) (0,0)
on the above definition. If x = y, we set d(x,y) = 0. A two-dimensional sub-
shift X is defined to be a closed, translation invariant subset of ΣZ2 (cf. [9,
p.467]). A two-dimensional subshift X is said to have the diagonal property if
for (xi,j)(i,j)∈Z2,(yi,j)(i,j)∈Z2 ∈ X, the conditions xi,j = yi,j,xi+1,j−1 = yi+1,j−1
imply x = y ,x = y (see [13]). The diagonal property has the
i,j−1 i,j−1 i+1,j i+1,j
following property: for x∈X and (i,j)∈Z2, the configurationx is determined by
the diagonal line (xi+n,j−n)n∈Z through (i,j).
We henceforth go backto our previoussituation ofC∗-textile dynamicalsystem
(A ,ρA,ρB,E ,E ,κ) coming from N ×N commuting matrices A and B with
N A B
specification κ as in Section 1. We always assume that both matrices A and B are
essential. It yields a two-dimensional subshift Xκ as follows: Let Σ be the set
A,B
E of tiles defined in (1.3). For ω = (α,b,a,β) ∈ E , define maps t(= top),b(=
κ κ
bottom):E −→E and l(=left),r(=right):E −→E by setting
κ A κ B
t(ω)=α, b(ω)=β, l(ω)=a, r(ω)=b
as in the following figure:
α=t(ω)
◦ −−−−→ ◦
a=l(ω) b=r(ω)
 
◦ −−−−→ ◦
y β=b(ω) y
A configuration (ωi,j)(i,j)∈Z2 ∈EκZ2 is said to be paived if the conditions
t(ω )=b(ω ), r(ω )=l(ω ), l(ω )=r(ω ), b(ω )=t(ω )
i,j i,j+1 i,j i+1,j i,j i−1,j i,j i,j−1
holdforall(i,j)∈Z2. LetXAκ,Bbethesetofallpavedconfigurations(ωi,j)(i,j)∈Z2 ∈
EZ2.ItconsistsoftheWangtilingsofthetilesofE (see[24]). Thefollowingpropo-
κ κ
sition is easy.
4
Proposition 2.1. Xκ is a two-dimensional subshift having diagonal property.
A,B
Let eω,ω ∈ Eκ be the standard basis of C|Eκ|. Put the projection Eω = ρBb ◦
ρA(1)(=ρA◦ρB(1))∈A for ω =(α,b,a,β)∈E . We set
α β a N κ
HA,B = e ⊗E A .
κ ω ω N
ωX∈Eκ
ThenHA,B hasanaturalstructureofnotonlyHilbertC∗-rightmoduleoverA but
κ N
also two other Hilbert C∗-bimodule structure, called Hilbert C∗-quad module. By
two-dimensionalanalogueofPimsner’sconstructionofHilbertC∗-bimodulealgebra
([19]), we have introduced a C∗-algebra O (see [14] for detail construction).
HAκ,B
LetΩ bethesubsetofE ×E definedin(1.4). Wedefinetwo|Ω |×|Ω |-matrcies
κ A B κ κ
A and B with entries in {0,1} as in (1.5) and (1.6). The matrices A and B
κ κ κ κ
represent the concatenations of edges as in the following figures respectively:
α δ
◦ −−−−→ ◦ −−−−→
if A ((α,a),(δ,b))=1,
a b κ
 
◦ −−−−→ ◦
y y
and
α
◦ −−−−→ ◦
a
if B ((α,a),(β,d))=1.
 β  κ
◦ −−−−→ ◦
y y
d

Let Hκ be the 2|Ωκ|×2|Ωκ| matrix defined in (1.7). We have provedthe following
y
result in [14].
Theorem 2.2. The C∗-algebra O associated with Hilbert C∗-quad module
HAκ,B
HA,B definedbycommutingmatricesA,B andaspecification κisisomorphic tothe
κ
Cuntz-KriegeralgebraO for thematrixH . ItsK-groupsK (O )arecomputed
Hκ κ ∗ Hκ
as
K (O )=Zn/(A +B −I )Zn,
0 Hκ κ κ n
K (O )=Ker(A +B −I ) in Zn,
1 Hκ κ κ n
where n=|Ω |.
κ
We will study a relationship between transitivity of the tiling space Xκ and
A,B
simplicity of the C∗-algebra O . An essential matrix with entries in {0,1} is
HAκ,B
said to satisfy condition (I) (in the sense of [2]) if the shift space defined by the
topological Markov chain for the matrix is homeomorphic to a Cantor discontin-
uum. Theconditionisequivalenttotheconditionthateveryloopinthe associated
directed graph has an exit ([8]). It is a fundamental result that a Cuntz-Krieger
algebraissimple andpurely infinite if the underlyingmatrixis irreducibleandsat-
isfies condition (I) ([2]). We will find a condition of the two-dimensional subshift
Xκ of the tiling space under which the matrix H is irreducible and satisfies
A,B κ
5
condition (I). Hence the condition yields the simplicity and purely infiniteness of
the algebra O .
HAκ,B
We areassumingthatbothofthe matricesAandB areessential. Thenwehave
Lemma 2.3. Both of the matrices A and B are essential.
κ κ
Proof. For (α,a) ∈ Ω , by definition of Ω , there exist β ∈ E and b ∈ E
κ κ A B
such that κ(α,b) = (a,β). Since A is essential, one may take β ∈ E such that
1 A
s(β ) = r(b)(= r(β)). Hence (b,β ) ∈ ΣBA. Put (α ,b ) = κ−1(b,β ) ∈ ΣAB so
1 1 1 1 1
that (α ,b)∈Ω and A ((α,a),(α ,b))=1 as in the following figure:
1 κ κ 1
◦ −−−α−→ ◦ −−−α−1→ ◦
a b b1
◦ −−−β−→ ◦ −−−β1−→ ◦
y y y
For (δ,b) ∈ Ω there exists α ∈ E such that r(α) = s(δ)(= s(b)) because A
κ A
is essential. Hence (α,b) ∈ ΣAB. Put (a,β) = κ(α,b) so that (α,a) ∈ Ω and
κ
A ((α,a),(δ,b))=1 as in the following figure:
κ
α δ
◦ −−−−→ ◦ −−−−→
a b
 β 
◦ −−−−→ ◦
y y
Therefore one sees that A is essential, and similarly that B is essential. (cid:3)
κ κ
Hence we have
Proposition 2.4. The matrix H is essential and satisfies condition (I).
κ
Proof. Bythepreviouslemma,bothofthematricesA andB areessential. Hence
κ κ
every row of A and of B has at least one 1. Since
κ κ
A A
H = κ κ ,
κ B B
κ κ
(cid:20) (cid:21)
every row of H has at least two 1′s. This implies that a loop in the directed
κ
graphassociatedtothe matrixH musthasanexitsothatthe matrixH satisfies
κ κ
condition (I). (cid:3)
For (α,a),(α′,a′)∈Ω , and C,D =A or B, we have
κ
[C D ]((α,a),(α′,a′))= C ((α,a),(α ,a ))D ((α ,a ),(α′,a′)).
κ κ κ 1 1 κ 1 1
(α1,Xa1)∈Ωκ
Hence [A A ]((α,a),(α′,a′))6=0 if andonly if there exists (α ,a )∈Ω suchthat
κ κ 1 1 κ
κ(α,a ) = (a,β) for some β ∈ E and κ(α ,a′)= (a ,β ) for some β ∈ E as in
1 A 1 1 1 1 A
the following figure:
◦ −−−α−→ ◦ −−−α−1→ ◦ −−−α−′→
a a1 a′
◦ −−−β−→ ◦ −−−β1−→ ◦
y y y
6
And also [A B ]((α,a),(α′,a′)) 6= 0 if and only if there exists (α ,a ) ∈ Ω such
κ κ 1 1 κ
that κ(α,a ) = (a,β) for some β ∈ E and κ(α ,b ) = (a ,α′) for some b ∈ E
1 A 1 1 1 1 B
as in the following figure:
◦ −−−α−→ ◦ −−−α−1→ ◦
a a1 b1
 β  α′ 
◦ −−−−→ ◦ −−−−→ ◦
y y y
a′

Similarly [B A ]((α,a),(α′,a′)) 6= 0 if and only if there exists (α ,a ) ∈ Ω such
κ κ y 1 1 κ
that κ(α,b) = (a,α ) for some b ∈ E and κ(α ,a′) = (a ,β ) for some β ∈ E
1 B 1 1 1 1 A
as in the following figure:
α
◦ −−−−→ ◦
a b
◦ −−−α1−→ ◦ −−−α−′→
y y
a1 a′
◦ −−−β1−→ ◦
y y
And also [B B ]((α,a),(α′,a′)) 6= 0 if and only if there exists (α ,a ) ∈ Ω such
κ κ 1 1 κ
thatκ(α,b)=(a,α )for some b∈E andκ(α ,b )=(a ,α′) forsome b ∈E as
1 B 1 1 1 1 B
in the following figure:
α
◦ −−−−→ ◦
a b
◦ −−−α1−→ ◦
y y
a1 b1
 α′ 
◦ −−−−→ ◦
y y
a′

Lemma 2.5. A B =B A . 
κ κ κ κ y
Proof. For (α,a),(α′,a′) ∈ Ω , we have [A B ]((α,a),(α′,a′)) = m if and only if
κ κ κ
there exist (α ,a′)∈Ω ,i=1,...,m such that κ(α,a′)=(a,β ) for some β ∈E
i i κ i i i A
and κ(α ,b )=(a′,α′) for some b ∈E as in the following figure:
i i i i B
◦ −−−α−→ ◦ −−−α−i→ ◦
a a′i bi
◦ −−−β−i→ ◦ −−−α−′→ ◦
y y y
a′

7
y
Put (a ,β′)=κ(β ,a′). We then have (β ,a )∈Ω as in the following figure:
i i i i i κ
α
◦ −−−−→ ◦
a a′
i
◦ −−−β−i→ ◦ −−−α−′→
y y
ai a′
 β′ 
◦ −−−−i→ ◦
y y
If(β ,a )=(β ,a )inΩ ,thenwehaveβ =β sothata′ =a′ andhenceα =α .
i i j j κ i j i j i j
Therefore we have [B A ]((α,a),(α′,a′))=m. (cid:3)
κ κ
Lemma 2.6. The following four conditions are equivalent.
(i) The matrix H is irreducible.
κ
(ii) For (α,a),(α′,a′)∈Ω , there exist n,m∈Z such that
κ +
A (A +B )n((α,a),(α′,a′))>0, B (A +B )m((α,a),(α′,a′))>0.
κ κ κ κ κ κ
(iii) The matrix A +B is irreducible.
κ κ
(iv) For (α,a),(α′,a′) ∈ Ωκ, there exists a paved configuration (ωi,j)(i,j)∈Z2 ∈
Xκ such that
A,B
t(ω )=α, l(ω )=a, t(ω )=α′, l(ω )=a′
0,0 0,0 i,j i,j
for some (i,j)∈Z2 with j <0<i.
Proof. (i) ⇐⇒ (ii): The identity
A (A +B )n A (A +B )n
Hn = κ κ κ κ κ κ (2.1)
κ B (A +B )n B (A +B )n
κ κ κ κ κ κ
(cid:20) (cid:21)
implies the equivalence between (i) and (ii).
(ii)=⇒(iii): Supposethatfor(α,a),(α′,a′)∈Ω ,thereexistsn∈Z suchthat
κ +
A (A +B )n((α,a),(α′,a′))>0 so that
κ κ κ
(A +B )n+1((α,a),(α′,a′))>0.
κ κ
Hence the matrix A +B is irreducible.
κ κ
(iii) =⇒ (ii): As A and B are both essential, for (α,a),(α′,a′) ∈ Ω there
κ κ κ
exists (α ,a ),(α ,a )∈Ω such that
1 1 2 2 κ
A ((α,a),(α ,a ))=1, B ((α,a),(α ,a ))=1.
κ 1 1 κ 2 2
Since A +B is irreducible, there exist n,m∈Z such that
κ κ +
(A +B )n((α ,a ),(α′,a′))>0, (A +B )m((α ,a ),(α′,a′))>0.
κ κ 1 1 κ κ 2 2
Hence we have
A (A +B )n((α,a),(α′,a′))>0, B (A +B )m((α,a),(α′,a′))>0.
κ κ κ κ κ κ
(ii) =⇒ (iv): For (α,a),(α′,a′) ∈ Ω , take (α ,a )∈ Ω and β ∈ E such that
κ 1 1 κ A
κ(α,a )=(a,β). By(ii),thereexistsm∈Z withB (A +B )m((α,a),(α′,a′))>
1 + κ κ κ
0. One may take b′ ∈ E and β′ ∈ E satisfying κ(α′,b′) = (a′,β′), so that there
B A
8
exists a paved configuration (ωi,j)(i,j)∈Z2 ∈XAκ,B such that ω0,0 =(α,a1,a,β) and
ω =(α′,b′,a′,β′) for some (i,j)∈Z2 with j <0<i as in the following figure:
i,j
◦ −−−α−→ ◦ −−−α−1→ ◦
a a1
y◦ −−−β−→ y◦ −−−−→ .y..


y
...
α′
◦ −−−−→ ◦
a′ b′
 β′ 
◦ −−−−→ ◦
y y
(iv) =⇒ (ii): The assertion is clear. (cid:3)
Definition. A two-dimensional subshift Xκ is said to be transitive if for two
A,B
tiles ω,ω′ ∈ Eκ there exists a paved configuration (ωi,j)(i,j)∈Z2 ∈ XAκ,B such that
ω =ω and ω =ω′ for some (i,j)∈Z2 with j <0<i.
0,0 i,j
Theorem 2.7. The subshift Xκ of the tiling space is transitive if and only if the
A,B
matrix H is irreducible.
κ
Proof. Assume that the matrix H is irreducible. Hence the condition (iv) in
κ
Lemma2.6holds. Letω =(α,b,a,β),ω′ =(α′,b′,a′,β′)∈E betwotiles. SinceA
κ
is essential, there exists β ∈ E such that r(β)(= r(b)) = s(β ), so that (b,β ) ∈
1 A 1 1
ΣBA. Onemaytake(α ,b )∈ΣAB suchthatκ(α ,b )=(b,β )andhence(α ,b)∈
1 1 1 1 1 1
Ω as in the following figure:
κ
◦ −−−α−→ ◦ −−−α−1→ ◦
a b b1
◦ −−−β−→ ◦ −−−β1−→ ◦
y y y
For (α1,b),(α′,a′) ∈ Ωκ, by (iv) in Lemma 2.6, there exists (ωi,j)(i,j)∈Z2 ∈ XAκ,B
suchthatt(ω )=α ,l(ω )=b,t(ω )=α′,l(ω )=a′ forsome(i,j)∈Z2 with
0,0 1 0,0 i,j i,j
j < 0 < i. Since Xκ has diagonal property, there exists a paved configuration
A,B
(ωi′,j)(i,j)∈Z2 ∈XAκ,B such that ω0′,0 =ω,ωi′,j =ω′. Hence XAκ,B is transitive.
Conversely assume that Xκ is transitive. For (α,a),(α′,a′) ∈ Ω , there exist
A,B κ
b,b′ ∈ E and β,β′ ∈ E such that ω = (α,b,a,β),ω′ = (α′,b′,a′,β′) ∈ E . It is
B A κ
clearthatthe transitivity ofXκ implies the condition(iv) inLemma 2.6,sothat
A,B
H is irreducible. (cid:3)
κ
Lemma 2.8. If A or B is irreducible, Xκ is transitive.
A,B
9
Proof. Suppose that the matrix A is irreducible. For two tiles ω =(α,b,a,β),ω′ =
(α′,b′,a′,β′) ∈ E , there exist concatenated edges (β,β ,...,β ,α′) in the graph
κ 1 n
G forsomeedgesβ ,...,β ∈E . SinceXκ hasdiagonalproperty,thereexists
A 1 n A A,B
a configuration (ωi,j)(i,j)∈Z2 ∈ XAκ,B such that ω′ = ωi,j for some i > 0,j = −1.
Hence Xκ is transitive. (cid:3)
A,B
Since the C∗-algebra O is isomorphic to the Cuntz-Krieger algebra O by
HAκ,B Hκ
[14], we see the following theorems.
Theorem 2.9. The subshift Xκ of the tiling space is transitive if and if the
A,B
matrix H is irreducible. In this case, H satisfies condition (I). Hence if the
κ κ
subshift Xκ of the tiling space is transitive, the C∗-algebra O is simple and
A,B HAκ,B
purely infinite.
By Lemma 2.8, we have
Theorem 2.10. If the matrix A or B is irreducible, the matrix H is irreducible
κ
and satisfies condition (I), so that the C∗-algebra O is simple and purely infi-
HAκ,B
nite.
3. The algebra O for two positive integers N,M
Hκ[N],[M]
Let N,M be positive integers with N,M > 1. They give 1 × 1 commuting
matricesA=[N],B =[M]. WewillpresentK-theoryformulaefortheC∗-algebras
O with exchanging specification κ. In the computations below, we will use
Hκ[N],[M]
EuclideanalgorithmtofindorderofthetorsionpartoftheK -group. Thedirected
0
graph G for the matrix A=[N] is a graph consists of N-self directed loops with
A
a vertexdenotedby v. The N-selfdirectedloops aredenoted by E . Similarly the
A
directed graph G for B = [M] consists of M-self directed loops denoted by E
B B
with the vertex v. We fix a specification κ : E ×E −→ E ×E defined by
A B B A
exchanging κ(α,a) = (a,α) for (α,a) ∈ E ×E . Hence Ω = E ×E so that
A B κ A B
|Ω | = |E |×|E | = N ×M. We then know A ((α,a),(δ,b)) = 1 if and only if
κ A B κ
b = a, and B ((α,a),(β,d)) = 1 if and only if β = α as in the following figures
κ
respectively.
α
◦ −−−−→
α δ
◦ −−−−→ ◦ −−−−→ a
and
 α=β
a a=b ◦ −−−−→
y
 
  d
y y

In[14],the K-groupsforthe caseN =2andM =3havebeencomputedsuchthat
y
K (O )∼=Z/8Z, K (O )∼=0.
0 H[κ2],[3] 1 H[κ2],[3]
We will generalize the above computations.
Let I be the n×n identity matrix and E the n×n matrix whose entries are
n n
all 1′s. For an N ×N-matrix C =[c ]N and an M×M-matrix D =[d ]M ,
i,j i,j=1 k,l k,l=1
10