1001.0037.txt

C-ALGEBRAS AND FELL BUNDLES ASSOCIATED TO A TEXTILE SYSTEM
VALENTIN DEACONU
Abstract. The notion of textile system was introduced by M. Nasu in order to analyze endomorphisms and
automorphisms of topological Markov shifts. A textile system is given by two nite directed graphs GandH
and two morphisms p;q:G!H, with some extra properties. It turns out that a textile system determines
a rst quadrant two-dimensional shift of nite type, via a collection of Wang tiles, and conversely, any such
shift is conjugate to a textile shift. In the case the morphisms pandqhave the path lifting property, we
prove that they induce groupoid morphisms ;: (G)!(H) between the corresponding  etale groupoids
ofGandH.
We dene two families A(m;n) and A(m;n) ofC-algebras associated to a textile shift, and compute
them in specic cases. These are graph algebras, associated to some one-dimensional shifts of nite type
constructed from the textile shift. Under extra hypotheses, we also dene two families of Fell bundles which
encode the complexity of these two-dimensional shifts. We consider several classes of examples of textile
shifts, including the full shift, the Golden Mean shift and shifts associated to rank two graphs.
1.Introduction
In dynamics, the time evolution of a physical system is often modeled by the iterates of a single trans-
formation. However, multiple symmetries of some systems lead to the study of the join action of several
commuting transformations, where new and deep phenomena occur.
The classical shift of nite type from symbolic dynamics was studied with powerful tools from linear
algebra and matrix theory. The number of period npoints, the zeta function and the entropy can all be
simply expressed in terms of the kktransition matrix A. The Bowen-Franks group BF(A) =Zk=(IA)Zk
is invariant under 
ow equivalence, and it was recovered in the K-theory of the Cuntz-Krieger algebra OA
generated by partial isometries s1;s2;:::;sksuch that
1 =kX
i=1sis
i; s
jsj=kX
i=1A(j;i)sis
ifor 1jk:
The algebraOAis simple and purely innite if and only if Ais transitive (for every i;jthere exists msuch
thatAm(i;j)6= 0), andAis not a permutation matrix. These C-algebras can also be understood as graph
algebras, which were studied and generalized by several authors, see [R].
The higher dimensional analogue of a shift of nite type consists in all d-dimensional arrays of symbols
from a nite alphabet subject to a nite number of local rules. Such arrays can be shifted in each of the d
coordinate directions, giving dcommuting transformations. There are also dtransition matrices, which in
Date : April 5, 2019.
1991 Mathematics Subject Classication. Primary 46L05; Secondary 46L55.
Key words and phrases. textile system, shift of nite type, graph C*-algebra, Fell bundle.
Research partially supported by a UNR JFR Grant.
1arXiv:1001.0037v1  [math.OA]  30 Dec 20092 VALENTIN DEACONU
general do not commute. There are deep distinctions between the case d= 1 andd2: for example, it is
easy to describe the space of such arrays in the rst case, but there is no general algorithm which will decide,
given the set of local rules, whether or not the space of such arrays is empty in the second case.
Although the general theory of multi-dimensional shifts of nite type is still in a rudimentary stage, there
are particular classes where signicant progress was made, and where graphs and matrices play a useful role.
These include the class of algebraic subshifts, see [S1, S2] and the class of two-dimesional shifts associated
totextile systems , or to Wang tilings . For these classes, some of the conjugacy invariants, like entropy (the
growth rate of the number of patterns one can see in a square of side n), the number of periodic points and
the zeta functions were computed.
In the literature, there are some papers relating higher dimensional shifts of nite type and C-algebras.
For example, the particular case of shifts associated to rank dgraphs was studied by A. Kumjian, D. Pask and
others. In this case, the translations in the coordinate directions are local homeomorphisms, and there is a
canonical  etale groupoid and a C-algebra associated to such a graph, which is Morita equivalent to a crossed
product of an AF-algebra by the group Zd. Under some mild conditions, the groupoid is essentially free
and theC-algebra is simple and purely innite. For more details, see [KP]. Also, in [PRW1] and [PRW2],
the authors analyze the C-algebra of rank two graphs whose innite path spaces are Markov subgroups of
(Z=nZ)N2, like the Ledrappier example, see also [KS] and [LS]. In all these examples, the entropy is zero.
The connections between higher dimensional subshifts of nite type and operator algebras remains to be
explored further, and we think that this is a fascinating subject.
In this paper, in an attempt to apply results from operator algebra to arbitrary two-dimensional shifts of
nite type supported in the rst quadrant, we construct two families of C-algebras, dened using some one-
dimensional shifts associated to a textile shift as in [MP2]. The K-theory groups of these algebras provide
invariants of the two-dimensional shift. We also construct groupoid morphisms and families of Fell bundles
associated to some particular textile systems. We consider several examples of textile shifts, related to rank
two graphs, to the full shift, to the Golden Mean transition matrices and to cellular automata.
Acknowledgements . The author wants to express his gratitude to Alex Kumjian, David Pask and
Aidan Sims for helpful discussions.
2.Textile systems and two-dimensional shifts of finite type
Throughout this paper, we consider nite directed graphs G= (G1;G0), whereG1is the set of edges, G0
is the set of vertices, and s;r:G1!G0are the source and range maps, which are assumed to be onto.
Denition 2.1. A textile system (see [N]) is a quadruple T= (G;H;p;q ), whereG= (G1;G0),H=
(H1;H0) are two nite directed graphs, and p;q:G!Hare two surjective graph morphisms such that
(p(e);q(e);r(e);s(e))2H1H1G0G0uniquely determines e2G1. We have the following commutative
diagram:C-ALGEBRAS AND FELL BUNDLES ASSOCIATED TO A TEXTILE SYSTEM 3
H0p G0q!H0
"r"r"r
H1p G1q!H1
#s#s#s
H0p G0q!H0
The dual textile system T= (G;H;s;r ) is obtained by interchanging the pairs of maps ( p;q) and (s;r).
The new graphs G= (G1;H1) and H= (G0;H0) have source and range maps given by pandq, ands;rare
now graph morphisms. Note that, even if the initial graphs GandHhave no sinks, the new graphs Gand
Hmay have sinks (vertices vsuch thats1(v) =;).
A rst quadrant textile weaved by a textile system Tis a two-dimensional array ( e(i;j))2(G1)N2, such
thatr(e(i;j1)) =s(e(i;j)) and such that q(e(i1;j)) =p(e(i;j)) for alli;j2N. It is clear that
(e(i;j))j2N2G1(the innite path space of G) for alli2N. In some cases, the set of such arrays may be
empty (see Example 3.1 in [A]).
Remark 2.2. A textile system associates to each edge e2G1a square called Wang tile with bottom edge
s(e), top edge r(e), left edge p(e), and right edge q(e):
r(e)
p(e)eq(e):
s(e)
If we letX=X(T) to be the set of all textiles weaved by T, thenXis a closed, shift invariant subset
of (G1)N2, and we obtain a two-dimensional shift of nite type, dened below. Alternatively, if we use
Wang tiles, we get a tiling of the rst quadrant. We will describe in Proposition 3.1 the connection between
two-dimensional shifts of nite type and textile systems.
Denition 2.3. LetSbe a nite alphabet of cardinality jSj. The fulld-dimensional shift with alphabet S
is the dynamical system ( SNd;), where
m(x)(n) =x(n+m); x2X; n;m2Nd:
A subsetXSNdwhich is closed in the product topology and which is -invariant is called a d-dimensional
shift of nite type or a Markov shift if there exists a nite set (window) FNdand a set of admissible
patternsPSFsuch that
X=X[P] =fx2SNdj(mx)jF2Pfor everym2Ndg:
Many times F=f(0;0;:::;0);(1;0;:::;0);(0;1;:::;0);:::;(0;0;:::;1)g. A shift of nite type has dtransition
matrices of dimension jSjwith entries inf0;1g, which in general do not commute.
Denition 2.4. LetS1andS2be alphabets, let FNdbe a nite subset, and let  : SF
1!S2be a map.
A sliding block code dened by  is the map
:SNd
1!SNd
2;(x)n= (xjF+n);n2Nd:4 VALENTIN DEACONU
Ford= 1 we recover the notion of cellular automaton. Two shifts of nite type X[P1];X[P2] are conjugate
if there is a bijective sliding block code :X[P1]!X[P2]. In this case, the dynamical systems ( X[P1];)
and (X[P2];) are topologically conjugate (see [LS]).
Ford= 2, any Markov shift can be specied by two transition matrices. Such shifts are investigated by
N.G. Markley and M.E. Paul in [MP1]. Two kktransition matrices AandBwith no identically zero rows
or columns are called coherent if
(AB)(i;j)>0 i (BA)(i;j)>0 and (ABt)(i;j)>0 i (BtA)(i;j)>0;
whereBtis the transpose. If AandBare coherent, it is proved that
X(A;B) =fx2SN2:A(x(i;j);x(i+ 1;j)) = 1 and B(x(i;j);x(i;j+ 1)) = 1 for all ( i;j)2N2g
becomes a two-dimensional shift of nite type, where S=f0;1;2;:::;k1g.
For more about multi-dimensional shifts of nite type, we refer to [S1, S2] and [L, LS]. We illustrate now
with some examples of textile systems and their associated two-dimensional shifts.
Example 2.5. LetG1=fa;bg;G0=fu;vgwiths(a) =u=r(b);s(b) =v=r(a);and letH1=fxg;H0=
fwgwithp(a) =p(b) =x=q(a) =q(b):
� ��
���
� ��
�
Figure 1.
The corresponding two-dimensional shift has alphabet S=fa;bgand transition matrices
A="
1 1
1 1#
; B="
0 1
1 0#
:
We will see later that this shift is a particular case of a cellular automaton, obtained from the automorphism
of the Bernoulli shift ( fa;bgN;) which interchanges aandb. It also corresponds to a rank two graph,
because the transition matrices commute and the unique factorization property is satised (see [KP] section
6).
Example 2.6. LetG1=fa;b;cg;G0=fug;H1=fe;fg;H0=fvgwithp(a) =p(b) =e;p(c) =f;q(a) =
f;q(b) =q(c) =e.
Then the corresponding two-dimensional shift of nite type has alphabet fa;b;cgand transition matrices
A=2
6640 0 1
1 1 0
1 1 03
775; B =2
6641 1 1
1 1 1
1 1 13
775:
Note thatAandBare coherent in the sense of Markley and Paul, but do not commute, so this shift is not
associated to a rank two graph.C-ALGEBRAS AND FELL BUNDLES ASSOCIATED TO A TEXTILE SYSTEM 5
��
��� � �� �
�
�
Figure 2.
Example 2.7. LetG1=fa;b;cg;G0=fu;vg;s(a) =s(b) =r(c) =r(b) =u;r(a) =s(c) =v;H1=
feg;H0=fwg;p(a) =p(b) =p(c) =q(a) =q(b) =q(c) =e.
� � ��
�� �
� ��
�
Figure 3.
This textile system is isomorphic to the dual of the previous one. The corresponding two-dimensional
shift of nite type has the same alphabet, but the transition matrices are interchanged.
3.Textile systems associated to a two-dimensional shift of finite type
From a two-dimensional shift of nite type Xwe will construct a double sequence of textile systems
T(m;n), considering higher block presentations of Xsuch thatXand the shift determined by T(m;n) are
conjugated. Recall
Proposition 3.1. (see [JM] ) Let (X;)be a two-dimensional shift of nite type with alphabet S. Then,
moving to a higher block presentation of Xif necessary, there exists a textile system Tsuch thatXis
determined by T.
Proof. ConsiderB=B(2;2) the set of 22 admissible blocks =a b
c dinX, and construct a graph G
withG0labeled by the rows of the blocks in B,G1=B; s() =c d,r() =a b, and a graph HwithH0=S
andH1labeled by the columns of the blocks in B. Dene graph morphisms p;q:G!Hbyp() =a
c,
q() =b
d. It is clear that T= (G;H;p;q ) is a textile system such that Xis the set of textiles weaved by
T. 6 VALENTIN DEACONU
Corollary 3.2. Form;n1, letB(m;n)denote the set of mnadmissible blocks in X, and forn2
dene a graph G(m;n)withG0(m;n) =B(m;n1)andG1(m;n) =B(m;n). For2G1(m;n), lets() =
the lowerm(n1)block ofand letr() = the upperm(n1)block of. Then for m2there
are graph morphisms p;q:G(m;n)!G(m1;n)dened byp() =the left (m1)nblock of,q() =
the right (m1)nblock of, where2G1(m;n). ThenT(m;n) := (G(m;n);G(m1;n);p;q)for
m;n2are textile systems, and Xis determined by T(m;n). The shift Xis also determined by the dual
textile system T(m;n) := ( G(m;n);G(m;n1);s;r), where G1=B(m;n),G0(m;n) =B(m1;n)and the
source and range maps are given by pandqas above.
We illustrate with some two-dimensional shifts of nite type and their associated textile systems. In each
case, the morphisms p;qare dened as in 3.1.
Example 3.3. (The full shift). Let S=f0;1gand letX=SN2. In the corresponding textile system T=
T(2;2), the graph G=G(2;2) is the complete graph with 4 vertices. Indeed, G1=(
a b
c dja;b;c;d2S)
andG0=f0 0;0 1;1 0;1 1g:The graph H=G(1;2) is the complete graph with 2 vertices. Indeed,
H1=(
0
0;1
0;0
1;1
1)
andH0=f0;1g.
Example 3.4. (Ledrappier). Let S=Z=2Z, and letXSN2be the subgroup dened by x2Xi
x(i+ 1;j) +x(i;j) +x(i;j+ 1) = 0 for all ( i;j)2N2:
We haveG0(2;2) =H1=SS, andG1(2;2) has 8 elements, corresponding to the 2 2 matrices ( a(i;j))
with entries in Ssuch thata(1;1) +a(2;1) +a(2;2) = 0. The Ledrappier shift is associated to a rank two
graph, and if we consider the new alphabet
0
0 0;1
0 1;1
1 0;0
1 1;
then the transition matrices are
A=2
666641 1 0 0
0 0 1 1
1 1 0 0
0 0 1 13
77775; B =2
666641 1 0 0
0 0 1 1
0 0 1 1
1 1 0 03
77775;
see [PRW1].
Example 3.5. (Golden Mean). Let S=f0;1g, with transition matrices
A=B="
1 1
1 0#
:
Then in the corresponding textile system T=T(2;2), the graphs G=G(2;2) andH=G(1;2) have
G0=f0 0;0 1;1 0g;
G1=(
0 0
0 0;0 1
0 0;0 0
0 1;1 0
0 0;1 0
0 1;0 0
1 0;0 1
1 0)
;C-ALGEBRAS AND FELL BUNDLES ASSOCIATED TO A TEXTILE SYSTEM 7
H0=f0;1g; H1=(
0
0;1
0;0
1)
:
Example 3.6. (Cellular automata). Let k1 and letYf0;1;:::;k1gNbe a subshift of nite type. It
is known that a continuous, shift-commuting onto map ':Y!Yis given by a sliding block code. Given
such a', dene a closed, shift invariant subset
X=f(ym)2YNjym+1='(ym) for allm2Ngf 0;1;:::;k1gN2:
In a natural way, Xbecomes a two-dimensional Markov shift. In the corresponding textile system T=
T(2;2), we have G0f0;1;:::;k1gf 0;1;:::;k1g,G1= the set of admissible 2 2 blocksa b
c dwith
a;b;c;d2f0;1;:::;k1g,H0=f0;1;:::;k1g, andH1= the set of admissible columnsa
c.
Fork= 2,Y=f0;1gNand'dened by interchanging the letters 0 and 1, we recover the textile system
from example 2.5.
Recall that many rank two graphs can be obtained from two nite graphs G1andG2with the same set
of vertices such that the associated vertex matrices commute, and a xed bijection :G1
1G1
2!G1
2G1
1
such that if (;) = (0;0), thenr() =r(0) ands() =s(0). Here
G1
1G1
2:=f(;)2G1
1G1
2js() =r()g;
ands;rare the source and range maps. This rank two graph is denoted by G1G2. The innite path
space is a rst quadrant grid with horizontal edges from G1and vertical edges from G2. Each 11 square
is uniquely determined by one horizontal edge followed by one vertical edge.
Proposition 3.7. Any rank two graph of the form G1G2determines a textile system.
Proof. Indeed, let Hi=Gop
i, the graph Giwith the source and range maps interchanged for i= 1;2. The
mapinduces a unique bijection H1
1H1
2!H1
2H1
1, where
H1
1H1
2=f(;)2H1
1H1
2jr() =s()g:
We letGwithG1=H1
1H1
2identied with H1
2H1
1by the map ,G0=H1
1, and we let H=H2. Dene
s(;) =; r (;) =0; p(;) =0, andq(;) =, where0;0are uniquely determined by the
bijection(;) = (0;0). 
Remark 3.8. For a cellular automaton with 'as in 3.6 dened by an automorphism of a rank one graph
G, in [FPS] the authors associated a rank two graph whose C-algebra is a crossed product C(G)oZ, and
they computed its K-theory.
4.C-algebras associated to a two-dimensional shift of finite type
Recall that in Corollary 3.2 we constructed a family T(m;n) = (G(m;n);G(m1;n);p;q) of textile
systems from a two-dimensional shift of nite type. This denes a family of graph C-algebrasA(m;n) :=
C(G(m;n)) form;n2. The dual textile system T(m;n) = ( G(m;n);G(m;n1);s;r) determines another8 VALENTIN DEACONU
family A(m;n) :=C(G(m;n)), where G(m;n) is the graph with source and range maps given by pandq,
described in Corollary 3.2.
Remark 4.1. We haveA(m;n)=A(m;2) for alln2 and A(m;n)=A(2;n) for allm2. Indeed, the
graphG(m;n) is a higher block presentation of G(m;2) and the graph G(m;n) is a higher block presentation
ofG(2;n) (see [B]).
For matrix subshifts, we can be more specic. Consider A;B two coherent kktransition matrices
indexed byf0;1;:::;k1gas in [MP2], and let X(A;B) be the associated matrix shift.
Theorem 4.2. For a matrix shift X(A;B)we have A(2;n)=OAnandA(n;2)=OBnforn2. The
transition matrices AnandBncan be constructed inductively as in [MP2] , and they dene two sequences
(Y(An))n1and(Y(Bn))n1of one-dimensional shifts of nite type associated to X(A;B).
Proof. Consider the strip
Kn=f(i;j)2N2: 0jn1g
and the alphabet
Qn=B(1;n) =f:is a 1nblock occuring in X(A;B)g;
ordered lexicographically starting at the top. Dene Y(An) =fxjKn:x2X(A;B)gto be the Markov shift
with alphabet Qnand transition matrix An, obtained by restricting elements of X(A;B) to the strip Kn.
The shiftY(Bn) is dened similarly, considering strips
Ln=f(i;j)2N2: 0in1g
and alphabets
Rn=B(n;1) =f:is an1 block occuring in X(A;B)g:
Clearly,A1=AandB1=B. Forn2,Anis aknknmatrix, where knis the sum of all entries in
Bn1. Supposeand0are 1nblocks inX(A;B) andj;j02f0;1;2;:::;k1g. Then
An+1 
j
;j0
0!
= 1
if and only if the 1 (n+1) blocksj
andj0
0occur inX(A;B) andA(j;j0)An(;0) = 1. By Proposition
2.1 in [MP2], the matrix An+1is the principal submatrix of A
Anobtained by deleting the mth row and
column ofA
Anif and only if B(i;j) = 0, where m=jkn+h;0h<kn, and
i1X
l=0k1X
t=0Bn1(t;l)h<iX
l=0k1X
t=0Bn1(t;l):
The matrix Bn+1is constructed similarly, by deleting rows and columns from B
Bn.

Recall that the dynamical system ( X(A;B);) is (topologically) strong mixing if given any nonempty
open setsUandVinX(A;B), there isN2N2such thatn(U)\V6=;for allnN(componentwise
order).C-ALGEBRAS AND FELL BUNDLES ASSOCIATED TO A TEXTILE SYSTEM 9
Corollary 4.3. Assume that the transition matrices An;Bnare not permutation matrices. Then the C-
algebras A(2;n)andA(n;2)are simple and purely innite if and only if (X(A;B);)is strong mixing.
Proof. Apply Proposition 2.2 in [MP2]. 
Example 4.4. For the full shift described in 3.3, we have A=B="
1 1
1 1#
=A1=B1andAn+1=
Bn+1=A
Anforn1. The corresponding C-algebras are A(2;n)=A(n;2)=O2n.
Example 4.5. For the shift associated to the textile system in 2.5, we have
A1=A="
1 1
1 1#
; A2=2
666641 1 1 1
1 1 1 1
1 1 1 1
1 1 1 13
77775;:::
B1=B="
0 1
1 0#
; B2=2
666640 0 0 1
0 0 1 0
0 1 0 0
1 0 0 03
77775;:::
with corresponding sequences of C-algebras A(2;n)=O2nandA(n;2)=C(T)
M2n, sinceAnis the
2n2nmatrix with all entries 1, and Bnis a 2n2npermutation matrix. Note that the dynamical system
(X(A;B);) is not strong mixing.
Example 4.6. Consider the Golden Mean shift X(A;A) withA="
1 1
1 0#
. Since the number of n-words
inY(A) is a Fibonacci number, the dimension knofAn=Bnis also a Fibonacci number, where k1= 2 and
k2= 3. It is easy to see that to get An+1fromA
Anwe have to remove the last 2 knkn+1rows and
columns. Thus
A1=A; A 2=2
6641 1 1
1 0 1
1 1 03
775; A3=2
666666641 1 1 1 1
1 0 1 1 0
1 1 0 1 1
1 1 1 0 0
1 0 1 0 03
77777775;
A4=2
6666666666666641 1 1 1 1 1 1 1
1 0 1 1 0 1 0 1
1 1 0 1 1 1 1 0
1 1 1 0 0 1 1 1
1 0 1 0 0 1 0 1
1 1 1 1 1 0 0 0
1 0 1 1 0 0 0 0
1 1 0 1 1 0 0 03
777777777777775
etc, which are transitive and not permutation matrices. The sequence of simple purely innite C-algebras
A(2;n)=A(n;2)=OAnencodes the complexity of the Golden Mean shift.10 VALENTIN DEACONU
Remark 4.7. We have natural projections Y(An+1)!Y(An) andY(Bn+1)!Y(Bn) such that
X(A;B) = lim Y(An) = lim Y(Bn):
The families of C-algebrasA(m;n) and A(m;n) can be thought as C-bundles overf(m;n)2N2jm;n
2g, and we can interpret the corresponding section C-algebras as other algebras associated to the shift
X(A;B). In the case X(A;B) is constructed from a rank two graph, the relationship between the graph
C-algebra and the above C-algebras remains to be explored.
5.Groupoid morphisms and Fell bundles from textile systems
Denition 5.1. A surjective graph morphism :G!Hhas the path lifting property for s(oris an
s-bration) if for all v2G0and for all b2H1withs(b) =w=(v) there isa2G1withs(a) =vwith
(a) =b. Similarly, we dene an r-bration. If the morphism has the path lifting property for both sand
r, we say that is a bration. The morphism is a covering if it has the unique path lifting property for
bothsandr.
Remark 5.2. The morphisms pandqin the textile systems from examples 2.6 and 2.7 are brations. The
morphismp=qin example 2.5 is a covering. The canonical morphisms pandqfor the full shift (see 3.3) are
covering maps, but the full shift does not dene a rank two graph, because the unique factorization property
fails. Also, note that in this case, the horizontal and vertical shifts are not local homeomorphisms.
In general, the morphisms pandqin a textile system don't have the path lifting property: let G1=
fa;b;cg;G0=fu;vg;s(a) =r(a) =u;s(b) =r(c) =u;s(c) =r(b) =v;H1=fe;fg;H0=fwg;p(u) =
p(v) =q(u) =q(v) =w;p(a) =p(b) =e;p(c) =f;q(a) =e;q(b) =q(c) =f.
� ��
��� � �
��
��
Figure 4.
Then foru2G0andf2H1withs(f) =p(u) =wthere is no edge x2G1withs(x) =uandp(x) =f.
Also, forv2G0ande2H1withs(e) =w=q(v) there is no x2G1withs(x) =vandq(x) =e. Note also
that the graph G= (G1;H1) from the dual textile system has sinks.
Proposition 5.3. Consider any rank two graph of the form G1G2with the corresponding textile system
described in Proposition 3.7. Then the morphism qhas the unique path lifting property for s, and the
morphismphas the unique path lifting property for r.
Proof. Indeed, given 2G0=H1
1and2H1=H1
2withq() =s(), there is a unique ( ;)2G1=
H1
1H1
2such thats(;) =andq(;) =. The proof for pis similar. C-ALGEBRAS AND FELL BUNDLES ASSOCIATED TO A TEXTILE SYSTEM 11
Remark 5.4. For the textile system T(2;2) associated to a two-dimensional shift, we can characterize the
(unique) path lifting property for the morphisms p;qin terms of lling a corner of a 2 2 block. For example,
phas the (unique) path lifting property for sif for any admissible columna
cand for any admissible row
c d, there is a (unique) bwhich completes the admissible block =a b
c d:Similarly, we can characterize
the path lifting property for the morphisms p;qinT(m;n).
For a topological groupoid , we denote by sandrthe source and the range maps, by 0the unit space,
and by 2the set of composable pairs.
Denition 5.5. Let ; be topological groupoids. A groupoid morphism : ! is a continuous map
which intertwines both the range and source maps and which satises
(
1
2) =(
1)(
2) for all (
1;
2)22:
It follows that
ker:=f
2j(
)20g
contains the unit space 0. Agroupoid bration is a surjective open morphism : ! such that for any
2 andx20with(x) =s() there is
2 withs(
) =xand(
) =. Note that, using inverses, a
groupoid bration also has the property that for any 2 andx20with(x) =r() there is
2 with
r(
) =xand(
) =. If
is unique, then is called a groupoid covering .
ForGa nite graph without sinks, let G1be the space of innite paths, and let :G1!G1be the
unilateral shift (x1x2x3


) =x2x3


. Let
(G) =f(x;mn;x0)2G1ZG1jm(x) =n(x0)g
be the corresponding  etale groupoid with unit space ( G)0=f(x;0;x)jx2G1gidentied with G1.
Proposition 5.6. LetG;H be nite graphs with no sinks. Then any morphism :G!Hwith the path
lifting property for sinduces a surjective continuous open map
':G1!H1; ' (x1x2x3


) =(x1)(x2)(x3)



and a groupoid bration
: (G)!(H);given by(x;k;x0) = ('(x);k;' (x0))
with kernel 
 =f(x;0;x0)2(G)j'(x) ='(x0)g:Ifis a graph covering, then is a groupoid covering.
Proof. Lety1y2


2H1beginning at w12H0. Sinceis onto, there is v12G0with(v1) =w1. By the
path lifting property, there is x12G1with(x1) =y1. Continuing inductively, there is x1x2


2G1such
that'(x1x2


) =y1y2


, and therefore 'is onto. Consider a cylinder set
Z=fa1


anx1x2


2G1jx1x2


2G1g:12 VALENTIN DEACONU
By the path lifting property, '(Z) is the cylinder set in H1determined by the nite path (a1)


(an).
Hence':G1!H1is continuous and open. We have
((x;k;x0)(x0;l;x00)) =(x;k;x0)(x0;l;x00) = ('(x);k+l;'(x00));
andis a groupoid morphism.
Since'is surjective and takes cylinder sets into cylinder sets, is surjective, continuous and open. To
show thatis a bration, consider = (y;k;y0)2(H) andx02(G)0=G1with'(x0) =s() =y0.
Since'is onto and intertwines the shift maps, we can nd 
= (x;k;x0)2(G) with(
) =. Hence
is a groupoid bration. In the case is a covering, let's show how we can nd 
in a unique way. We have
k=mnandmy=ny0;mx=nx0. Forymandv=s(xm+1) =s(x0
n+1) withr(ym) =(v) there is a
uniquexmwithr(xm) =vand(xm) =ym. We can continue inductively to nd a unique xwith'(x) =y,
and it follows that is a groupoid covering. Now ( x;k;x0)2keri'(x) ='(x0) andk= 0. 
Corollary 5.7. Given a textile system (G;H;p;q )such thatG;H have no sinks and p;qhave the path lifting
property, we get two groupoid brations ;: (G)!(H). Ifpandqare coverings, we get two groupoid
coverings;: (G)!(H).
Example 5.8. Consider the coverings p=q:G!Hin the textile system of the full shift as in Example
3.3. We obtain a covering =: (G)!(H) of Cuntz groupoids.
Recall that a (saturated) Fell bundle over a groupoid  is a Banach bundle :E! with extra structure
such that the ber E
=1(
) is anEr(
){Es(
)imprimitivity bimodule for all 
2. The restriction of E
to the unit space 0is aC-bundle. The C-algebraC
r(;E) is a completion of Cc(;E) inL(L2(;E)).
For more details, see [DKR], where the following result is proved.
Theorem 5.9. Given an open surjective morphism of  etale groupoids : !with amenable kernel

 :=1(0), there is a Fell bundle E=E()over such thatC
r()=C
r(E).
Using Corollary 3.2 and Theorem 4.2, we get
Theorem 5.10. Given a matrix shift X(A;B)such that in the associated family of textile systems T(m;n)
the morphisms pandqhave the path lifting property, there are two families of Fell bundles E(m;n)(p)and
E(m;n)(q)over (G(m1;n))such that
A(m;n)=C
r(E(m;n)(p))=C
r(E(m;n)(q)):
Example 5.11. Consider the textile system ( G;H;p;q ) from example 2.5. In this case ( G)0has two points,
andC(G)=C(T)
M2. The maps pandqare coverings, they both induce the morphism
: (G)!(H)=Z; (x;k;x0) =k;
and the two Fell bundles E(2;2)(p) andE(2;2)(q) overZcoincide. The ber over 0 2Zis isomorphic to M2.
Remark 5.12. For a rank two graph G1G2, since the map qin the corresponding textile system is a
covering, we get a goupoid covering : (G)!(H). Recall that G1=H1
1H1
2;G0=H1
1andH=H2,
whereHi=Gop
i. In particular, ( H) acts on ( G)0and (G)=(H)n(G)0(see [DKR] Proposition 5.3).C-ALGEBRAS AND FELL BUNDLES ASSOCIATED TO A TEXTILE SYSTEM 13
Example 5.13. Consider the textile system from example 2.6. Here G1=fa;b;cgN;H1=fe;fgNare
Cantor sets, C(G)=O3andC(H)=O2, the Cuntz algebras. The morphisms pandqare brations
and induce dierent groupoid morphisms ;: (G)!(H). The bers of the Fell bundle E(2;2)(p) over
y2(H)0=H1are isomorphic to M2n, wherenis the number of e0siny. Forn=1,M21is the
UHF-algebra of type 21.
Example 5.14. Let (G;H;p;q ) be the textile system from example 2.7. The space G1fu;vgNis dened
by the vertex matrix
A="
1 1
1 0#
andC(G)=OA. The space H1has one point, and ( H)=Z. Bothpandqinduce the same morphism
: (G)!Z; (x;k;x0) =kas in Example 5.11, and the Fell bundle E(2;2)(p) =E(2;2)(q) corresponds to
the grading ofOA.
Example 5.15. The full shift X=f0;1gN2determines a sequence of textile systems T(n;2) = T(2;n),
whereG(n;2) = G(2;n) is the complete graph with 2nvertices, and G(n1;2) = G(2;n1) is the complete
graph with 2n1vertices. The two families of Fell bundles over the Cuntz groupoid ( G(n1;2)) have
C*-algebras isomorphic to C(G(n;2))=O2n.
Example 5.16. For the Golden Mean shift with transition matrices
A=B="
1 1
1 0#
;
the corresponding graphs G(n;2) = G(2;n) have vertex matrices as in Example 4.6.
The morphisms pandqinT(2;n) =T(n;2) are brations and determine dierent groupoid morphisms
(G(2;n))!(G(2;n1)) and two Fell bundles E(2;n)(p) andE(2;n)(q) over (G(2;n1)). It would be
interesting to calculate the bers of these Fell bundles.
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Department of Mathematics, University of Nevada, Reno NV 89557-0084, USA
E-mail address , Valentin Deaconu: [email protected]