Groupoid C*-algebras of the pinwheel tiling

Groupoid C*-Algebras of the Pinwheel Tiling

Michael Fredrick Whittaker B.Sc., University of Victoria, 1999

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of

MASTER OF SCIENCE

in the Department of Mathematics and Statistics

@ Michael Fredrick Whittaker, 2005 University of Victoria

All rights reserved. This thesis may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author.

Supervisor: Dr. Ian F. Putnam

Abstract

Anderson and Putnam, and Kellendonk discovered methods of defining a C*- algebra on a noncommutative space associated with a tiling. The method em- ployed was to use Renault's theory of groupoid C*-algebras of an equivalence relation on the tiling metric space. C*-algebras of a tiling have two purposes, on one hand they reveal information about the long range order of the tiling and on the other hand they provide interesting examples of C*-algebras. However, the two constructions do not include tilings such as the pinwheel tiling, with tiles appearing in an infinite number of orientations. We rectify this deficiency, with many interesting results appearing in the process.

Contents

A b s t r a c t 11

. .

Table of C o n t e n t s iii List of Figures v Acknowledgement v i 1 I n t r o d u c t i o n 1 2 T h e G e o m e t r y of Tilings 4

2.1 The Tiling Metric Space . . . .

.

. .

.

.

.

. .

.

. .

.

.

.

. . 4 2.2 Substitution Tiling Systems . . . .

.

. . . .

.

11 2.3 The Pinwheel Tiling . .

.

. .

.

.

. .

.

.

. .

.

. . . .

.

. . . .

.

. 14

3

t tale

Equivalence Relations a n d C*-algebras 20 3.1 Principal

tale

Topological Groupoids . . . .

.

. . . .

.

. . 21 3.2 C*-algebras of

tale

Groupoids . . .

.

. .

.

. . . 24

4 A C*-algebra of a Tiling w h e n

r

= Rd 38

4.1 An

tale

Equivalence Relation of the Tiling Space for

r

= Rd . . 39 4.2 The Reduced C*-algebra of a Tiling for

r

= Rd . . . .

.

. . . 47 5 A C*-algebra of a Tiling w h e n

I?

=

S E ( d )

53

5.1 An

tale

Equivalence Relation of the Tiling Space for

r

= S E ( d ) 54 5.2 The Reduced C*-algebra of a Tiling for

I?

= S E ( 2 ) . . . .

.

. 62

6 An AT-subalgebra of a tiling for I? = S E ( 2 ) 72

6.1 An Inductive Limit C*-subalgebra of C;Jh&,,) for

I?

= S E ( 2 )

.

73 6.2 A C*-subalgebra of

Ain

. . . .

.

. . . .

.

. . . .

.

. . . .

.

. 87

7 Conclusion 96

List

of Figures

2.1 The pinwheel substitution . . . 15

2.2 The first two rotations of po . . . 16

2.3 A patch of the pinwheel tiling . . . 17

2.4 All edge patterns in the pinwheel tiling . . . 18

Acknowledgement

First and foremost I would like to thank Chelsea Lowe for continually inspir- ing and supporting me in all aspects of my life. I would also like to thank the members on my thesis committee; Thierry Giordano, Bob Miers, Marcelo Laca, and Ian Putnam. In particular, Ian Putnam has given so much time, thought, and inspiration into this thesis that I am eternally grateful to him. My family has always been there for me, and I have benefitted from each of them in many ways. Finally, I would not be who I am today without my friends, thanks.

Chapter

1

Introduction

In 1961 the philosopher Hao Wang introduced the notion of aperiodic tilings. One of his graduate students managed to find an example of an aperiodic tiling, however, the tiling required over 20000 different proto-tiles. Tiling theory was all but forgotten about for over a decade when Roger Penrose invented a tiling that required only two different shapes that could only be put together t o form a tiling in an aperiodic fashion. In the 80s there was a massive revival of tiling theory due to the discovery of quasi-crystals. Quasi-crystals refer to a crystalline structure found in various alloys that do not follow the symmetries found in every other crystalline object. In fact, crystallographers have a theorem stating that crystals could only possess two, three, four, and six fold symmetry. The discovery of ten fold symmetry in certain alloys sent shock waves through the crystallographic community. The answer to the problem was found in the Penrose tiling, which also displays ten fold symmetry.

The notion of a tiling refers to the covering of the d-dimensional real vector space with polygons such that the polygons only intersect on their borders. In

CHAPTER 1. INTRODUCTION 2 addition we also insist that each polygon is homeomorphic to the closed unit ball of IRd and the polygons meet edgeto-edge and vertex-to-vertex. Given a particular tiling of Rd we refer to the distinct polygons, disregarding their position in Rd, as the proto-tiles of the tiling. A simple observation reveals that covering the plane with squares satisfies the definition of being a tiling with a single square as the proto-tile of the system. We would like to rule out such cases because in our case the tiling of the plane with squares is not very interesting. An aperiodic tiling refers to a tiling T of Rd such that for every non-zero z in Rd, T

+

x

#

T.

The geometry of aperiodic tilings with certain other key assumptions falls into the class called the mathematics of long range order. The study of long range order is primarily focused on trying to ascertain how much order a system exhibits. We remark that chaos theory falls into the same class of objects.

Tilings of the plane can be viewed as dynamical systems by creating a tiling metric space, which consists of all translations of a particular tiling completed in a tiling metric. With such a tiling metric space, Rd acts on points (tilings) via translation and this situation can be viewed as a dynamical system. The connections between topological dynamical systems and operator algebras are very well established and both fields have profited from the other. As such, tiling theory is a wonderful marriage of the two diverse fields and benefits from the mixed viewpoint. Using tilings to create a C*-algebra began with a paper by Bellissard. Some time later two papers were written using C*-algebra techniques on a non-commutative space associated with a tiling; one by Jared Anderson and Ian Putnam [I] and the other by Johannes Kellendonk [9]. Both papers used the tiling metric space in different ways to produce a C*-algebra, however, the two algebras appeared t o be quite different. Later it was discovered that the two different C*-algebras are strongly Morita equivalent (a notion introduced by Marc Rieffel) .

CHAPTER 1. INTRODUCTION 3

The goal of this thesis is to construct a C*-algebra on the pinwheel tiling. The pinwheel tiling has a structure that is different from most other aperiodic tilings in the literature. In particular, the pinwheel tiling has tiles appearing in an infinite number of distinct orientations. The pinwheel tiling was invented by John Conway in the early 90s and has commanded attention ever since. Shortly after the discovery of the pinwheel tiling Charles Radin wrote a paper introducing various wonderful properties of the pinwheel tiling [18]. However, the pinwheel tiling has never had a C*-algebra associated to its tiling space. With this fact in mind we extend the C*-algebra of a tiling presented in both [8] and [9] to include tilings like the pinwheel tiling.

To begin the construction of a C*-algebra of a tiling we need to assign some conditions to both the tiling and the tiling metric space. The definitions and results in the second chapter appear in a variety of papers on the subject, [I], [8],

[9], and [18], to name a few. In the third chapter we introduce Renault's theory of groupoid C*-algebras in the case of an 6tale relation, [19]. Chapter four is focused on the results found in [8] and [9] with respect to constructing a C*-algebra of a tiling. In particular, we show that tiling metric space is a topological Cantor set and we define an equivalence relation that preserves this structure in a nice way. Using the equivalence relation, which is &ale, we construct a C*-algebra using Renault's approach. Chapter five extends the results of chapter four to include tilings like the pinwheel. We find that the Cantor set from chapter four is replaced by the product of a Cantor set and a circle (in two dimensions). Finally, in chapter six we define a nested sequence of C*-subalgebras to the C-algebra presented in chapter 5. This situation allows us to find the inductive limit of the system, which is also a C*-subalgebra of the C*-algebra presented in chapter 5 . Moreover, the

the inductive limit of the system for the pinwheel tiling is an AT-algebra, a class of C*-algebras that have been studied extensively in George Elliott's classification program.

CHAPTER 2. THE GEOMETRY O F TILINGS

Chapter

2

The Geometry

of Tilings

2.1

The Tiling Metric Space

Let IRd denote the d-dimensional Euclidean vector space with the usual norm and metric, denoted by 1x1 and (x - y( respectively for x, y in IRd. An open ball in Rd is defined as B(x, r ) = {y((x - y(

<

r ) , which is the ball centered at x with radius r. Given a subset X contained in Rd, we can translate the subset X by a

vector y, written formally, X

+

y = {x

+

y(x E

X}. We

now have all the necessary requirements to define a tiling.

Definition 2.1. A tiling T of Rd is a countable collection of closed subsets

{ t l ,

t2, tS,

- -

)

of Rd such that:

03

U

ti =

Btd,

i=l

Int(ti)

n

I n t ( t j ) =

Q)

whenever i

#

3

where Int denotes the interior of a tile. We further insist that each ti for i =

CHAPTER 2. THE GEOMETRY O F TILINGS 5 Tilings are collections of subsets of Rd. So if T is a tiling and x is in Rd, the translation of

T

by x is defined as T

+

x = {t

+

x ( t E T), which is also a tiling of Rd. Further, we will look a t the set of all translations of a tiling T which we denote T

+

Rd =

{T

+

x ( x E Rd). For a given tiling T we will define a metric on T

+

Rd. The process of defining a metric on T

+

Rd begins by choosing a group

I?

which is fundamental to the construction of this metric. This group I? will be a closed subgroup of S E ( d ) such that &Id

C

I?

C

SE(d) where Rd is viewed as the group of translations and S E ( d ) is the group of orientation preserving isometries of Rd.

We proceed by describing SE(d) as a semi-direct product, allowing us to define a metric on SE(d). The semi-direct product SE(d) consists of the group of translations, Rd, and the special group of orthogonal matrices of dimension dl SO(d). The group SO(d) is defined as the square d dimensional matrices such that for 0 in SO(d); 80T =

I

and det 0 = 1. We define the semi-direct product

SE (d) =

Etd

K S O (d)

with group operations as follows, let (xl, 61) and (x2, 82) be in S E ( d ) ,

See [2] for the construction of a semi-direct product. To define a metric on SE(d) we define a metric on Rd and SO(d) independently which may be added together to define a metric on SE(d). Endow Rd with the usual metric and define a metric on SO(d) as follows; for O1 = (aij) and

e2

= (bij) in SO(d),

CHAPTER 2. T H E GEOMETRY OF TILINGS 6

We may define a metric on I' by restricting the SE(d) metric to

r.

For example,

Rd may be defined as the subgroup {(x, I)

I

x E Rd and I is the identity of S O ( d ) ) , and the SE(d) metric is equal to the usual IRd metric.

Given a subset X in Rd an element (y, 8) in

r

acts by multiplication on the left as follows

(y,8) . X = {y

+

8(x)

I

x E X ) .

Therefore (y, 8) acts on a tiling, T, by multiplication on the left since a tiling consists of a collection of subsets of Rd; i.e.

if T =

U

ti then _{(y, 6 ) T} =

U(y

+

8(ti)).

i=l i=l

All the tilings presented in this paper are two dimensional, so the group I? is defined such that R2

2 I'

SE(2). The two dimensional special orthogonal group, S 0 ( 2 ) , is isomorphic to rotations around the origin in R2. Hence, we may view the two dimensional euclidean group, SE(2), as the semi-direct product of translations and rotations in R2

Before we define the metric on the space T

+

IRd, the idea of a proper subset of a tiling must be made precise.

Definition 2.2. A patch

P

in a tiling T is a finite subset of tiles in T; i.e. P = i t l , t 2 ,

- -

. ,

t n )

such that ti is in T , for i = 1 , 2 ,

-

. -

,

n.

The following definition requires a special type of patch in a tiling

T,

for x in Rd

and R

>

0, let T

n

B(x, R) = {t E T

I

t

c

B ( x , R)). Note that the intersection given in the patch T

n

B(x,

R)

is an abuse of notation, as

T

is a collection of subsets and B(x, R) is a set. So for the remainder of this paper the reader is reminded that intersections of this form are defined as {t E T

I

t

c

B(x,

R)).

C H A P T E R 2. T H E G E O M E T R Y OF T I L I N G S 7

set of tilings by A. However, the set of tilings that we will be most interested in is

T

+

Rd. The metric we present requires us to have chosen the group

r,

however, we note that any choice of

I?

will work in the definition.

Definition 2.3. Given tilings TI and T2 in a set of tilings A , we define the

tiling metric, d r , o n the set A as follows. For 0

<

E

<

1 we say the distance

between

TI

and T2 is less than E if we can find elements yl and 7 2 i n I? such that

~ s E ( Y ~ , (071))

<

€ 7 ~ s E ( Y ~ , ( 0 , I ) )

<

6 , a n d y l ( T l ) n B ( O , ~ - ~ ) = 7 2 ( T 2 ) n B ( O , ~ - ~ ) .

Now d r ( T l , T 2 ) is the infimum over the set consisting of each E that satisfies the

above hypothesis. If n o such 6 exists we define d r ( T l , T 2 ) t o be 1.

The above definition of the tiling metric combines the two current approaches of defining a metric on the space T

+

Rd. If

I'

= Rd then the definition is the same as the definition presented in [I], [8], and [9]. If l? = S E ( d ) then the definition is similar to [12] and [l7].

The distance between two tilings is small when the tilings have the same pattern on a large ball about the origin, up to a small orientation preserving isometry in

r.

For example suppose that there are vectors x and y in Rd so that the pattern at T - x matches the pattern at T - y on a large ball about the origin, up to a small orientation preserving isometry in l?. Then d r ( T -

x ,

T

-

y ) is small regardless of the size of 1x1 and

1

y

1.

Definition 2.4 ([8]). Given a tiling T , we let StT denote the completion of the

metric space (T

+

Rd, d r ) . flT is called the continuous hull of T .

Each element in the continuous hull flT can be represented as a tiling of Rd and, by definition, the metric dr extends to CIT. Moreover, for every tiling T' in the continuous hull, T'

+

x is also in OT for a11 x in Rd. Thus, Rd is a continuous group action on flT and we are interested in the dynamics of the system

(aT,

Rd),

CHAPTER 2. THE GEOMETRY OF TILINGS 8

see [8].

So far we have been looking at the global structure of the continuous hull OT. We would now like to concentrate on the local structure. Not surprisingly, the local structure of a single tiling, or more accurately the patterns in each tiling, play a huge role in the global structure.

Definition 2.5 ([8]). Suppose A i s a collection of tilings, we say A has finite &l

complexity with respect to the group if, for all R

>

0, the set {T

n

B(x, R)

I

T E

A

and x E Rd)/I' is finite. W e denote finite local complexity with F L C .

Lemma 2.6. T h e continuous hull of a tiling OT has finite local complexity with

respect t o

I'

if for all R

>

0, the set (T' f l B(0, R)

I

T i E RT)/I' is finite.

proof: The continuous hull of a tiling, by definition, is the completion of the set of all translates of the tiling T. Hence, for T' E

nT,

R

>

0, and every patch

T'

n

B(x, R) there is a tiling T" E RT such that T'

n

B(x, R) - x is contained in

T". Therefore,

{TI

n

B(x, R)

I

T' E RT and x E R d ) / r = {Ti

n

B(0, R)

I

T'

E

R T ) / ~ I

'Finite local complexity' is possibly the most important property for a tiling to possess, we will see that in one way or another almost every result in this thesis requires FLC. One may view F L C as regulating the aperiodicity of a tiling T when T is aperiodic (see definition 2.10). Radin and Wolff, [16], have the most important result that follows directly from F L C .

Theorem 2.7 ([16]). If T is a tiling satisfying F L C then the metric space (OT, do) i s compact.

CHAPTER 2. THE GEOMETRY OF TILINGS 9

At this point it seems reasonable to make some comments about the group

I'

because the three previous definitions and Radin and Wolff's theorem are all contingent on

r.

The question remains, how do we choose

I'

from a tiling T? Actually, given a tiling T, we may create the continuous hull QT using any I' such that EXd

5

r

5

SE(d). However, whether a particular continuous hull, QT, satisfies FLC with a particular

I'

is the crux of the matter. We would like to choose

I?

to be as 'small' as possible so that the continuous hull QT satisfies FLC.

For example, the Penrose tiling will exhibit FLC when the group

I'

is equal to

R2.

The predominant example in this paper is the pinwheel tiling. The continuous hull of a pinwheel tiling has FLC if

I?

is equal to SE(2), and fails to have FLC

for

r

equal to any proper subgroup of SE(2). An example of a pinwheel tiling and the pinwheel tiling space are presented explicitly in section 2.3, and we further show that the pinwheel tiling space exhibits FLC if

I'

= SE(2). Therefore, the continuous hull of both a Penrose tiling and a pinwheel tiling will be compact if we choose the group

I'

with FLC in mind. In section 2.2 we will see another reason why I' must be suitably chosen for the continuous hull of a tiling to satisfy FLC.

One can now ask how the points in QT behave under the action of translation. The action of translation is continuous in the tiling metric dn, and therefore extends continuously to all of StT. Since the continuous hull of a tiling was created by looking at all translations of a particular tiling then completing in the tiling metric, one may ask how many extra elements were added to QT in the completion.

In particular, one can ask whether a point T' in f l T is dense under the action of

Rd

or if every point in QT is dense under the action of Rd. After we have defined what is meant by the orbit of a point under a group action, we formalize the above questions with a definition.

C H A P T E R 2. T H E GEOMETRY OF TILINGS

space X and let x be in

X .

T h e orbit of x under q5 is

Definition 2.9 ([l]). If there i s a dense orbit of a point i n SZT then we say

(aT,

R d )

i s topologically transitive and if every point i n

(aT,

Rd) has a dense orbit we say

that ( a T , Rd) is minimal.

Notice that by the construction of flT, the orbit of T is dense under the action of translation, so (OT, R d ) is topologically transitive. Whether ( a T , R d ) is minimal under the action of translation is a much more difficult question and is addressed in [I]. Notice that the above definition is really a question about a dynamical system, and this is the context that the continuous hull of a tiling under the action of Rd should be viewed. We will see, in chapter 4 and 5, that the action of translation becomes critical to creating a C*-algebra of a tiling. In [I] the dynamical system (OT, Rd) is shown to have a deep mathematical structure with a small number of assumptions.

One of the assumptions in [I] asks for a tiling to be 'irregular' in a formal manner. Although some beautiful artwork has been created using periodic tilings, most notably by M.C. Escher, periodic tilings are not incredibly interesting in the context of this paper. However, if we add the hypothesis that the original tiling T is aperiodic, we will see that every tiling in the continuous hull may also be aperiodic as follows.

Definition 2.10 ([8]). W e say that a tiling T i s aperiodic if

T

+

x

#

T

for

every non-zero x in Rd. Furthermore, we say that the continuous hull OT is

strongly aperiodic if QT contains n o periodic tilings.

For the remainder of this paper we will assume that every continuous hull presented is strongly aperiodic, and for examples we will show strong aperiod-

CHAPTER 2. THE GEOMETRY OF TILINGS 11

icity. The following result, which appears in [8], provides a useful criterion for determining whether

OT

is strongly aperiodic.

Proposition 2.11 ([8]). If the tiling T is aperiodic and minimal, t h e n flT is strongly aperiodic.

2.2

Substitution Tiling Systems

In this section we will look at one way to construct a tiling

T.

The method of construction is called a substitution system. We will define substitution rules that break up each tile into smaller tiles and then expands the smaller tiles back up to the original size. The substitution rules must be defined on each tile and the best way to begin is to define the 'proto-tiles':

Definition 2.12. If { p l , pa,

. .

-

,

p,) i s a finite and non-empty collection of subsets of Rd each of which i s homeomorphic t o the closed unit ball i n Rd, we say that

{pl, pz,

. . ,

p,) is a set of proto-tiles for a given collection of tilings A if for each

tile

t

in T such that T i s in A we have

t

= y(pi) for some y in the group I?,

i E { 1 , 2 , . . . , n ) .

In the previous definition the group I? is critical. For the Penrose tiling we will have a finite number of proto-tiles when

I?

is translation by vectors in Rd. However, in the case of the pinwheel tiling (see section 2.3), we have tiles appearing in an infinite number of orientations, therefore,

I?

must be equal to S E ( 2 ) because we require a finite number of proto-tiles. In both cases the group

I'

required coincides with the group

I'

required for the continuous hull to exhibit FLC.

Sometimes we will require two tiles that look alike to be different tiles, when such a need arises we refer to the tiles as having labels. Quite often the set of

C H A P T E R 2. T H E GEOMETRY OF TILINGS 12 proto-tiles will require labels because two proto-tiles may divide differently but have the same shape and orientation. Another useful idea is a partial tiling. We define a partial tiling as a set of tiles in Rd that overlap only on the boundary of individual tiles and the union of the tiles is a proper subset of

Rd. A

partial tiling differs from a patch in that a partial tiling need not be a subset of a tiling.

We are now in a position to define what is meant by a substitution rule on the proto-tiles {pl,p2,. 0. ,p,). A substitution rule consists of a scaling factor

X

>

1 and a substitution w, such that for each p in {pl, p2,

.

.

-

,

p,), w(p) is a finite collection of tiles {tl,

.

-

,

tl) such that:

Int(tj)

n

Int(tk) =

0,

1

(J

t, = Xp. j=1

Basically each proto-tile is being divided up into smaller versions of proto-tiles that have been moved by the group

I?

and then expanded by the scaling factor

X

so as to be the same size as the original proto-tiles. So for each proto-tile pi we apply w and receive a partial tiling of moved proto-tiles. Now extend the definition of w to a moved proto-tile t. We may be rewrite t as

t

= y(pi) for some y in

I',

pi in {pl

,

p2,

-

. ,

p,) and define w(t) = w(y (pi)) = y(w(pi)). We may therefore define w(T) = {w(t)

I

t E T) for a tiling T and w(P) = {w(t)

I

t E P) for any finite (or infinite) partial tiling P .

Since we are now able to apply a substitution w to partial tilings, we may iterate w to create arbitrarily large partial tilings. For example, given a proto-tile p we may build a sequence of partial tilings, wYp) for k = 1 , 2 ,

. .

.

,

which becomes arbitrarily large provided we add the following additional hypothesis.

CHAPTER 2. THE GEOMETRY OF TILINGS 13 Definition 2.13 ([I]). A substitution rule w o n a set of proto-tiles {pl

,

pz, _.

_,

_p,)

is said t o be primitive if, for some k 2 1, a n image under I' of each proto-tile pi appears inside the patch w ~ ( ~ ~ ) for each pair i , j i n {1,2,

- - ,

n ) .

The following quote appears in [8], Kellendonk and Putnam sketch the proof that a substitution system will produce a tiling.

We will assume that our substitution is primitive [ a . . ] . One can find a translate (euclidean motion) of one of the proto-tiles

t

and a k

>

1 so that the sequence of patches wkn(t), for n = 1,2,3, .

- -

,

grows to cover the plane and is consistent in the sense that that any two agree where they overlap. We let T denote the union of these patches which is a tiling.

Therefore we now have a way t o create a tiling T , and this shows that the tiling metric space

RT

is non-empty for substitution tilings. With the additional as- sumption that w is primitive, the tiling metric space

RT

does not depend on the initial choice of T , so we may write

R

rather than

RT,

see [8] for details.

We conclude this section with two results appearing in [I] and a theorem appearing in in 1231.

Theorem 2.14 ([I]). Consider the continuous hull of a tiling created from a primitive substitution tiling system. T h e n

2. w :

R

H S-2 is continuous,

CHAPTER 2. THE GEOMETRY OF TILINGS 14

Theorem 2.15 ([I], [23]). Consider the continuous hull of a tiling created from

a primitive substitution tiling system. T h e continuous hull 0 contains n o periodic

tilings if and only if w restricted t o i s injective.

proof: (+): See [23]

( e ) :

See [I].

2.3

The Pinwheel Tiling

The main example the present text is the pinwheel tiling. We will see that there is an incredible amount of structure in the pinwheel tiling, so much that the tiling space is not well understood. We begin by describing the substitution for the pinwheel tiling.

The substitution for the pinwheel tiling begins with a (1,2, fi)-right triangle and it's mirror image. We will denote these two tiles by po and pl where the ver- tices of po have (x, y)-coordinates (0,O)

,

(2, O), ( 2 , l ) and pl has vertex coordinates (0, O), (2,O)

,

(2, - 1). An important observation for the pinwheel tiling is that the smallest angle in the triangle above is equal t o arctan(l/2). It is well known that arctan(l/2) is irrational, as stated in [18].

The pinwheel substitution w is depicted in figure 2.1, with the scaling factor X =

&.

The substitution w divides each of the two tiles into five isometric copies of the tiles p~ and pl modulo rotation. Observe that the tile in the center of the patch w(po) is isometric to po and has been rotated by 8 = arctan(l/2). With this in mind, define

Re

to be the rotation operator defined on the proto-tiles that rotates each proto-tile by 6 in the counter-clockwise direction; i.e. the rotation

CHAPTER 2. THE GEOMETRY OF TILINGS

Figure 2.1 : The pinwheel substitution

operator in SO(2) is defined by

So the tile in the center of the patch w(po) is the tile Re(po) modulo translation. One may now observe that the tile in the center of the patch w ~ ( ~ ~ ) is the tile Ri(po) modulo translation and the tile in the center of the patch wn(pO) is the tile Rt(po) modulo translation (see figure 2.2 for the first two iterations). The point is that because 6 = arctan(l/2) is an irrational number and the irrational rotations are dense in the circle we will have tiles appearing in an infinite number of orientations (see [4]). Moreover, because our next goal is to look at the continuous hull for the pinwheel tiling (which will be complete) we will have tiles appearing in an uncountably infinite number of orientations. Similarly, pl will have tiles in an infinite number of orientations after applying w an infinite number of times. The above argument implies that in order for a pinwheel tiling to have F L C the group

r

must be at least SE(2), proposition 2.16 shows that the continuous hull of a pinwheel tiling does satisfy

FLC

if

r

= SE(2).

CHAPTER 2. THE GEOMETRY OF TILINGS 16

Figure 2.2: The first two rotations of po

As in section 2.2, we can produce a tiling by applying the substitution a finite number of times and taking the union, this is a pinwheel tiling. A finite patch of a pinwheel tiling is depicted in figure 2.3. Using a pinwheel tiling, T , we build the continuous hull of T, QT. The set of proto-tiles for a pinwheel tiling is {pa, pl).

The set of proto-tiles being {po, pl) is convenient because the substitution is primitive under one iteration of w. Indeed, both po and pl modulo euclidean motion appear in each of w(po) and w(pl), giving primitivity for k = 1. We finish this section by showing that the pinwheel tiling satisfies F L C .

We now wish to show that a pinwheel tiling satisfies F L C if

I'

=

SE(2),

but we must first introduce the notion of the edge patterns of a tiling. Given a tile t in a tiling T we define the edge pattern o f t to be all tiles

t'

in T that share a common edge with

t.

Therefore, we may look at the set of edge patterns of a tiling. In particular, if we have a substitution tiling we may find all the edge patterns by iterating the substitution on each of the proto-tiles until no new patterns arise. If the substitution is primitive, we may iterate a single proto-tile until each proto-tile occurs in the partial tiling and then begin the process of finding edge patterns. Once an iteration of the substitution produces no new edge patterns, there cannot possibly be any remaining edge patterns. The pinwheel tiling has 24 edge patterns; figure 2.4 shows all edge patterns for any pinwheel tiling. No new edge patterns

CHAPTER 2. THE GEOMETRY OF TILINGS

CHAPTER 2. THE GEOMETRY OF TILINGS

Figure 2.4: All edge patterns in the pinwheel tiling

arise in the sixth iteration of a proto-tile, but there are new edge patterns in the fifth iteration.

Proposition 2.16. If

r

= SE(2) and T is a pinwheel tiling then the continuous hull, flT, of a pinwheel tiling has FLC.

proof: Assume

r

= SE(2) and fix R

>

0. We must show that {B(O, R)

n

T

I

T' E flT)/SE(2) is finite. Our first observation is that because

R2

is contained in

SE(2)

we may assume that the origin is in the interior of a tile. Moreover,

CHAPTER 2. THE GEOMETRY OF TILINGS 19 since SO(2) is also contained in S E ( 2 ) the tile at the origin may be assumed to be one of po or pl. We proceed with a combinatorial construction to get an upper bound on the number of possible patterns in {B(O, R)

n

TI

1

TI E R T ) / S E ( 2 ) .

The fact that there are only a finite number of distinct edge patterns shows that the set {B(O, R ) fl TI

I

T'

E f l T ) / S E ( 2 ) is finite. Indeed, the origin is contained in the interior of one of two proto-tiles which each have 24 possible edge patterns. Each of the tiles in the edge pattern of the proto-tile has only 24

possible edge patterns. And so on. After three steps in this process the patch created necessarily covers all points within 1 of the tile containing the origin. After six steps the patch will contain all points within 2 of the tile containing the origin,

and so on. Therefore, after a finite number of iterations we will have covered the ball B(0, R ) with only a finite number of possible patterns modulo S E ( 2 ) .

Notice that the pinwheel tiling will not have FLC if

r

=

R2.

Indeed, if

I'

=

R2

then we have an infinite number of possible rotations of po or pl sitting on the origin. Thus, FLC cannot be satisfied.

Chapter

3

Et ale Equivalence Relations and

In this chapter, we step away from the theory of tilings and look at the construc- tion of a C*-algebra from an dtale equivalence relation. We define the notion of an dtale equivalence relation on a locally compact metrizable space. An dtale equivalence relation can be used to create a C*-algebra by applying Renault's theory of groupoid C*-algebras [19]. In fact one can apply Renault's theory to a much larger class of equivalence relations, however, if we assume the structure of an &ale relation the groupoid C*-algebra becomes much more tractable. In the two subsequent chapters we show that one can form an &ale relation on the tiling space, and therefore using the results in this chapter we form a C*-algebra based on a tiling.

3.1

Principal

tale

Topological Groupoids

In this section we introduce principal 6tale topological groupoids. Putnam intro- duces the term local action as an alternative to the term principal Stale topological

groupoid to describe the dynamical flavor of the relationship. In this section we

will give definitions of these terms and observe many nice consequence of &ale relations.

Definition 3.1.

A

groupoid is

a

set G together with a subset G ( ~ ) of G x G, a

product m a p (x, y) H xy : G ( ~ ) G, and a n involution x H x-' : G t G that

satisfy the following properties:

1. if (x,

Y)

and (y, z) are in ~ ( ~ 1 , t h e n (xy, z) and (x, yz) are in G ( ~ ) with

( X Y ~ = x(yz);

2. (x, x-') and (x-l, x) are always in G ( ~ ) for all x E G with x-' (xy) = y and

(zx)x-I = z whenever (x, y) and (z, x) are in G ( ~ ) .

If (x, y) is in G(2) we say that x is composable with y, and we call the set G ( ~ ) the set of composable pairs of G. Note that if (x, y) is in G(2) it is not necessarily true that (y, x) is in G ( ~ ) , usually it is not. Basically a groupoid is a set with inverses and a partially defined associative product. Observe that any group satisfies the requirements of being a groupoid and the set of composable pairs G ( ~ ) is G x G.

Definition 3.2. Given any groupoid G we m a y define t w o maps called the range

and source maps given by:

r : G t G defined by r(x) =ax-',

One can observe that r ( G ) = s ( G ) and we call G(O) = r ( G ) = s(G) the set of units

of G because r ( x ) x = x . s ( x ) = x for all x in G.

In the present paper we will focus entirely on the groupoid obtained from an equivalence relation. Given a set

X

and an equivalence relation

R

on

X

x X,

the equivalence relation R has a natural groupoid structure such that:

One can now apply the definition of the range and source maps which yields r(x, y) = (x, x) and s(x, y) = ( y

,

y) for all (x, y) in

R.

Thus, the unit space, R(O) is given by the diagonal Ax = { ( x , x )

I

x E X). We may identify the diagonal with X and observe that the range and source maps are the projections onto the first and second coordinate respectively. Throughout this section we will make the identification between Ax and X without any further explanation.

Definition 3.3. If a groupoid

R

has a topology,

I ,

and

R ( ~ )

has the relative

topology from R x

R

then if

(x,

y ) H xy and x H x-I are continuous then we call

R

a topological groupoid.

The following consequences are immediate from the above definition. The range and source maps are continuous and x H x-I is a homeomorphism. If R is Hausdorff, then R(O) is closed in R. If R(O) is Hausdorff, then

R ( ~ )

is closed in

R

x

R. Moreover,

R(')

is both a subspace of

R

and a quotient of R by the map r.

We also observe that in a Hausdorff topology a single point { x ) is closed for all

x E R ( O ) , from this we have r-'{x) is closed in

R.

The following definition requires some additional notation. If R is an equiva- lence relation on a set

X ,

with

U

and

V

subsets on

R

we define:

CHAPTER 3. ETALE EQUIVALENCE RELATIONS AND C*-ALGEBRAS 23 UV = {(x, z )

I

there exists y E X such that (x, y ) E U , ( y , z ) E V), and

U-' = {(x, y)

I

(y, x) E U). Also note that there are some redundancies in the definition of an &ale topology in the case of a topological groupoid.

Definition 3.4. Let R be a n equivalence relation o n a compact metrizable space

X.

W e say the topology 7 o n R is a n &tale topology for

R

if the following conditions

are satisfied.

1. (R, 7 ) is u-compact;

2. A x = {(x, x)

I

x E X) is open in (R, 7 ) ;

3. every point (xo, yo) i n R has a n open neighborhood U in (R,

7 )

such that

r ( U ) and s ( U ) are open i n X and r : U -+ r ( U ) and s : U + s ( U ) are

homeomorphisms;

4.

if U and V are open sets i n (R,

I ) ,

then UV i s open in (R, I ) ;

5. if

U

is a n open set i n (R,

I ) ,

then

U-'

is open i n (R, 7 ) .

Definition 3.5. Let R be a n equivalence relation o n a compact metrizable space

X . W e say R i s a n ttale equivalence relation o n X if there exists a n Ctale topology

for R.

Given an &ale equivalence relation the range and source maps actually deter- mine the topology, each equivalence class is countable, and the diagonal

Ax

is also closed in R. Therefore the diagonal Ax is a clopen set. Condition 3 is by far the strongest assertion in the definition of an &ale equivalence relation, as the following lemma shows.

Lemma 3.6 ([19]). T h e product map from R ( ~ ) to R is a local homeomorphism

CHAPTER 3. ETALE EQUIVALENCE RELATIONS AND C*-ALGEBRAS 24 proof: For ((2, y), (y, z)) in R ( ~ ) , we may choose a compact neighborhood U of (x, y) and a compact neighborhood V of (y, z) such that rlu and slv are homeo- morphisms onto their images. Then (U x V)

n

R ( ~ ) is a compact neighborhood of

((2, y), (y, z)) on which the product map is injective, indeed

(x'

,

y') ( y', z') = (x", y") (y", z")

=+

r((xf

,

y')) = r((xf'

,

y")) and s ( ( ~ ' , z')) = s((yf', z"))

+

(XI, y') = (x", y") and (y', z') = (y", z"). Therefore the product map is a local homeomorphism.

In the literature, &tale relations are examples of r-discrete groupoids, however the name r-discrete carries some ambiguity as different authors use the name for slightly different ideas. Putnam goes on to rename the triple (X, R,

7)

a local

action for two reasons; every 6tale relation looks like a group action on some space

X

and condition 3 actually shows that locally R is homeomorphic to X . Definition 3.7 (Putnam). Let

X

be a compact metrzzable space, let

R

be an

equivalence relation on X, and let 7 be an ktale topology for R. We say that

(X,

R ,

7)

is a local action.

3.2

C*-algebras

of Etale Groupoids

In this section, we show how one can construct a C*-algebra from an &ale relation. Throughout, we will assume that R is an &ale relation with corresponding &ale topology 7. The construction is an application of the seminal work of Renault on groupoid C*-algebras 1191. In subsequent chapters we will apply the construction of the groupoid C*-algebra to equivalence relations that we define on a subset of the continuous hull. One can view the groupoid C*-algebra as a direct generaliza- tion of a group C*-algebra and the construction can be better understood in this

CHAPTER 3. ETALE EQUIVALENCE RELATIONS AND C*-ALGEBRAS 25

context. We avoid proofs until we arrive at the representation theory, however we give references and encourage the interested reader to peruse them.

As an aside, it should be noted that we may form a C*-algebra based on a larger class of equivalence relations on locally compact spaces. The construction is basically the same, however, the groupoid C*-algebra requires a left invariant measure (Haar measure) and the convolution is an integral rather than the sum defined below.

Definition 3.8. Given a n ;tale relation R the continuous functions _- of compact

support, denoted by Cc(R), is defined by:

Cc(R) = {f : R + C

1

f is continuous, and vanishes off a compact set K

C

R}.

W e denote the support o f f by s u p p ( f )

C

K . T h e n Cc(R) is a vector space with

operations defined by:

(Af

+

pg)(x,y) =

A f

(2, Y)

+

w ( x , Y) for all f > 9 E Cc(R); (x,Y) E R; A& E

c.

We aim to give Cc(R) a *-algebra structure, so we must define a product and involution on C,(R). We begin by introducing notation for the equivalence class of a point in

X.

Notation 3.9. For a n equivalence relation

R

and

x

i n

X ,

the equivalence class

of x, denoted 1x1, is defined as

Definition 3.10. If R is a n ttale relation we define a product and involution o n

Remark 3.11. From this point forward, whenever we refer to Cc(R) we will be assuming the above *-algebra structure.

The symmetric property of equivalence relations and continuity of the inverse implies that f * is always an element of Cc(R) (we will have a topology on Cc(R) shortly). To see that the product is an element of C,(R) we require the following two lemmas.

Lemma 3.12. Given a locally compact ;tale equivalence relation R, for any func-

tion f in CJR) there is a positive integer N such that for every x i n X the number

of non-zero elements i n the set

{

f

(x, y)

1

(x, y) E R) is bounded by N . Similarly,

#{f (y, x)

1

(y, x) E R)

<

M

for some constant M .

proof: Since f is in Cc(R), the support of f is contained in a compact subset of R, say K. We can find a cover of K by open sets of the type defined in part 3 of definition 3.4. Since K is compact there is a finite subcover of such sets,

say U l , . 0 .

,

UN. By definition the maps r and s are homeomorphisms on such

sets, so in particular r and s are injective. Thus for fixed x in X and each i in {I,

,

N ) , the set {y E X ( (x,y) E Ui) has at most one element. So the set {y E X

I

(x, y) E K} has at most

N

elements which we denote by {yl, ya .

-

,

yN}. Thus

Lemma 3.13. Given an ;tale equivalence relation R the product off and g is an element of C,(R), f o r all f and g in Cc(R).

proof: By lemma 3.12 f

*

g has finite support (hence compact). To show that f

*

g is continuous we begin by assuming that supp( f )

G

U and supp(g)

G

V such that U and

V

are open sets satisfying part 3 of definition 3.4. Then for any point

(x, z) in R

In the proof of lemma 3.12 we have that if x E r(U) then there is a unique y so that (x, y) E U and if z E s(V) then there is a unique y' so that (y', z) E V. Furthermore, by the definition of the product if y

#

y' then f

*

g(x, z) = 0. Therefore, if the product is non-zero we may assume there is a unique y so that

which is continuous since it is a pointwise product and composition of continuous functions on the same element (x, y).

For the general case, since f and g have compact support (say

Kf

and

Kg)

we may find a cover of their respective supports with open sets satisfying part 3 of definition 3.4. Further we can find a finite subcover of such sets, say

Kf

G

u z l U i and Kg

G

uEIV,

such that f can be written as f =

xi

fi and g can be written as g = C i g i , where supp(fi)

C

Ui and supp(gi)

E

V,. By the pasting lemma (see

[ll]) and continuity on each Ui, respectively V,, f

*

g is continuous. Hence f

*

g is in Cc(R).

The above definition of the product and involution gives us a *-algebra that reflects the groupoid structure of the &ale equivalence relation R. Observe that the product and involution looks alot like matrix multiplication and matrix in- volution, this is not an accident as will be apparent in the next example. The matrix model is a wonderful tool to help us understand the algebra Cc(R). We can 'almost7 view discrete countable equivalence relations as giving rise to generalized matrix algebras.

The following example makes this idea more precise. Consider the space X = {1,2,3,4,5, 6) and the equivalence relation R on X defined by 1 w 2 , 3 w 4

-

5 , 6. Then the map which sends f in Cc(R) to the 6

x

6 complex valued matrix

defines an isomorphism between Cc(R) and M,(C) @ M3(C) G3 C.

We would like the *-algebra on

R

t o be a C*-algebra. In particular, a C*- algebra that preserves the structure of our equivalence relation. First we must put a topology on the space Cc(R).

Definition 3.14. Given a locally compact Hausdorfl space X , the inductive &t

topology can be defined o n Cc(X) as follows. A sequence { f,) converges to f in

Cc(X) i j and only if there exists a compact subset K of

X

such that s u p p ( f ) is i n

K , supp({ f,)) i s eventually in K , and for suficiently large N , {

fnInLN

converges unijormly t o f o n K .

(2.e: the product and involution in C,(R) are continuous in the inductive limit topology).

proof: Bures [5] p.40.

We are almost in a position to build the C-algebra C*(R) created by com- pleting the *-algebra Cc(R) in a suitable norm. However, we are still in need of a norm, in particular, a C*-norm. We begin the process of defining a C*-norm by

defining a *-algebra norm on Cc(R).

Definition 3.16. W e define three n o r m s o n the *-algebra Cc(R) by:

Proposition 3.17.

(1

(1,

is a *-algebra n o r m o n Cc(R).

proof: To see that

(1

-

(1,

is a *-algebra norm we refer the reader t o either Bures [5] p.42 or Renault [19] prop. 1.4 p.50.

r

Lemma 3.18. If a sequence {fn)2=1 in Cc(R) converges in the inductive limit topology t o f in C,(R) t h e n {f,);=, converges t o f in

1)

.

11,.

proof: Since Cc(R) has the inductive limit topology; { f , ) +

f

if and only if

there exists an integer Nl such that ~ u p p { f , ) , ~ ~ ~ K for some compact set

K

in

R,

and

there exists an integer Nz such that { f n ) n 2 N z converges uniformly to f on

K .

Since K is compact in R, by lemma 3.12 we have for any fixed x in X there is a constant M such that the set { ( x , y ) E

R

I

r - l { x ) E K ) has at most M elements which we denote { ( x , y l ) ,

. .

.

,

( x , y M ) ) . Now let E

>

0 , by uniform convergence

for n

2

N 2 we may choose N

2

m a x { N l , N 2 ) sufficiently large so that for all

( x , y) in K we have ] f n ( x , y ) - f ( x , y ) )

<

E / ~ M for a11 n

>

N. Therefore, since

~ u p ~ { f , ) , ~ ~ and s u p p ( f ) are contained in

K

we have for every x in X M

This implies that

5

<

e for all n

2

N

2

Thus, { f,) converges to f in

(1.

(I,,

and similarly i f n ) converges to f in

(1

.

(1,.

Now since ( ( g ( ( , = m a x { ( ( g ( l T , ( ( g ( ( , ) , we also have that { f , ) converges to f in

1)

.

11,. I

To find a C*-norm we must represent Cc(R) on a suitable Hilbert space. The representation will depend on the above *-algebra norm

(1

.

(I,,

in particular we will only use representations that are bounded by the *-algebra norm.

Definition 3.19 ( [ 1 9 ] ) . A representation of C,(R) is a nondegenerate *-homomorphism

rr : C c ( R ) -+

B(3-I)

that is continuous from the inductive limit topology of

Cc(R)

t o

t h e weak operator topology of

B(3-I). W e call a representation

rr of Cc(R) bounded

CHAPTER 3.

TALE

EQUIVALENCE RELATIONS AND C*-ALGEBRAS 31 Remark 3.20. Recall that a representation is non-degenerate in the above defin- ition if the linear span {n(f)[

I

f E Cc(R) and

<

E 7f) is dense in 7f.

We have finally arrived at the point where we can define a C*-algebra norm on the *-algebra Cc(R). In fact, one has a choice between two norms, namely the full and reduced C*-algebra norm. In this thesis we will make use of only one of the norms, the reduced norm, however we include the full norm for completeness. For an &ale relation, R , the two norms will actually be equal in many cases. The two norms below were both introduced by Renault [19], as well as the corresponding C*-algebras.

Definition 3.21 ([19]). The full C*-algebra norm is defined as:

11

f

)Ic*

= sup{lln( f )

llop

I

i.r is a bounded representation of Cc(R)).

Proposition 3.22 ([19]). For an ttale relation the norm

(1.

(Ic*

is a C*-norm and is bounded above and below as follows. Given a representation n : Cc(R) 4

B('FI),

we have ((.lr(f)ll,

I

Ilf

((c*

<

(If

(I*

for all

f

in Cc(R).

proof: See Renault [I91 or Bures [5] for a rather general proofs. We note that the proofs in both are quite lengthy and span over several pages.

The remainder of this section is devoted to introducing the reduced norm and the reduced C*-algebra. Because we will be using the reduced C*-algebra to find C*-algebras of a tiling we do not omit any details. To introduce the reduced norm, we look at a special set of representations that are indexed by the unit space. Recall that the unit space of an equivalence relation is the set r ( R ) = s(R) = R(') which may be identified with the compact metrizable space X. We begin by introducing the Hilbert space of square summable functions on the equivalence class of xo in X, and proceed by defining the induced representations from the unit space.

CHAPTER 3.

TALE

EQUIVALENCE RELATIONS AND C*-ALGEBRAS 32 Definition 3.23 ( [ 1 9 ] ) . For every xo i n X , the Hilbert space e2([xo]) i s defined as

Definition 3.24 ( [ 1 9 ] ) . Given a n &tale relation R, for eve? xo in

X

the induced

representation from the unit space .rr,, : Cc(R) -+ B(e2([xo])) is given by

such that x is in [xo] and J i s in e2([x0]).

The induced representations from the unit space are special examples of in- duced representations of the *-algebra Cc(R), for more on induced representations see [19], page 81. The representations given above are usually called the reduced

representations of Cc(R). We proceed by showing that the reduced representations

are indeed bounded representations from Cc(R) into the C*-algebra D(e2([xo])).

Proposition 3.25. For a n &tale relation R, xo in X , and

<

in e2([xo]) the reduced

representation

rxO

given by

is a bounded representation of Cc(R).

proof:

3.12 and using the notation in the lemma we have

where M and N arise in lemma 3.12. Since the double sum is merely a finite sum of complex numbers, it is finite. So ? r x o ( f ) < is in Y2([xo]) for every

C

in

Y2([x0]) and is therefore well defined.

I I ~ ~ ~ ( f ) l l ~ ~ is bounded b y

11

f

[Ir:

Let f be in C,(R) and let

C

and q be in

Y2([x0]). Then again making use of lemma 3.12 we have

I

I

Il

f

ll*

11t112 llqll2 by Cauchy-Schwartz.

Again, note that M and N arise in lemma 3.12. Thus, l l ? r x o ( f ) l l w

<

11

f

11.

for all f in C,(R). Furthermore this also show that ? r x o ( f ) is continuous in

rxo is a *-homomorphism: Let

f

and g be in C,(R). On one hand,

On the other hand,

The sums in the above calculations are finite due to lemma 3.12, implying that any change of order is justified. Thus rxo

(f

*

g) = rxo

(f

)nxo (9). To see that nxo is *-preserving we make use of the inner product on !2([xo]).

where again the change of order in the sums is justified by lemma 3.12. Thus, .~r,~(f *) = nxo(

f)*

and nxo is a *-homomorphism.

0 nxo is nondegenerate on L2([xo]): Fix q in e 2 ( [ x o ] ) and suppose that ( s l r x o ( f ) J , q ) =

0 for every f in C c ( R ) and every

5

in e 2 ( [ x o ] ) . For each ( a , z') in R one can find an open set, U(,,,t), defined in part 3 of definition 3.4. Since both the maps r and s are bijective on the set U(,,,t), one can observe that for fixed

y' in U(,,,I) the set { ( y , y')

I

( y , y') E U(,,,I)) has exactly one element (see lemma 3.12). Now U(,,,t) is open so we can define a function fU(,

,,

which

,I

takes the value 1 at ( z , z')

,

the value 0 outside of U(,,,O, and is in C c ( R ) . i.e:

1 i f ( y , y ' ) = ( z , z 1 )

fu,,,*I) (Y 7 37 = _{y , )} if (Y 7 Y 1 ) E U(z.2)

0 otherwise. One can also define a sequence (,I in e 2 ( [ x 0 ] ) by

Then for all z and z' in [xo] we have:

Y € I Z O I

q ( z ) = 0 for all z E [ x O ] .

Thus, nxo is non-degenerate. I

We may now define the regular representation of C c ( R ) . Define a Hilbert space 3-t as follows

H

=

$

e2(ko])

xoEX

The reduced representations are bounded operators on 7-l in the natural way. Therefore, we may define a non-degenerate representation T R as

TQ : Cc(R) +

B ( H )

defined by T R =

@

rX0.

20EX

which is called the regular representation of Cc(R). One observes from the above proposition that the regular representation is indeed a groupoid representation in the sense of definition 3.19.

Definition 3.26. The reduced C*-algebra norm is defined as:

11

f

[ITed

= sup{117i( f )

[lop

I

T is a reduced representation).

Proposition 3.27. For an ttale relation the norm

(1.

(ITed

is a C*-norm and

11.

[ITed

is dominated by the full C*-algebra norm

11

.

Ilc*

proof:The fact that

11

.

)ITed

is dominated by the full C*-algebra norm is clear because the sup is over fewer elements. Now

(1

.

(ITed

is a C*-algebra norm because for every xo in X, t2([x0]) is a Hilbert space, so it follows that B(t2([xo])) is a C*-algebra. Therefore,

The full (?*-algebra and the reduced C*-algebra are now defined as follows

Definition 3.28. The full C*-algebra of an ttale relation, denoted by C*(R), is

T h e reduced C*-algebra of a n btale relation, denoted by C;JR), i s the completion of

C,(R)

in the metric induced by the reduced C*-algebra n o r m

11 [ITed.

As noted earlier, we will only make use of the reduced C*-algebra. The reason for using only one of the C*-algebras is because our main goal is to get our hands on a C*-algebra, rather than get into the business of comparing the two C*-algebras.

CHAPTER 4. A C*-ALGEBRA OF A TILING WHEN I' = RD

Chapter

4

A

C*-algebra of a

Tiling

when

r

=

Itd

In chapter 2 we defined the continuous hull, QT, of a tiling created by defining a metric on the space T

+

Rd and completing in metric. The metric on T

+

Rd was created by using a group

I'

such that Rd

C

I'

S E ( d ) and the group

I'

carried through to be used for many other definitions in chapter 2. In particular, we choose

r

so that the continuous hull of a tiling will exhibit FLC, which implies the continuous hull is a compact metric space. The goal of the present chapter is to sufficiently simplify the continuous hull Q T , allowing us to exploit the relationship

between flT and

r

more explicitly. Once we have simplified the continuous hull we will be able to define an &ale relation that corresponds with translation.

Given a tiling T , the running assumptions in this chapter are as follows

1. The group

r

is equal to Rd, the group of translations on Rd.

CHAPTER 4. A (7-ALGEBRA OF A TILING WHEN F =

RD

39

3. The continuous hull CIT is strongly aperiodic.

The continuous hull of a tiling with the aforementioned assumptions has been extensively studied in [ I ] , [9], and [8]. The results and definitions appearing in this chapter are essentially a survey of the relevant results provided in these papers. The goal is to extend these definitions and results to the case where

Rd

is properly contained in

r,

which occurs in the next chapter.

Notation 4.1. W h e n F is equal t o Rd we m a y use additive notation for elements

of l? applied t o tilings. A s defined in chapter 2,

I'

=

Rd

i s the subgroup of S E ( d )

defined by { ( x , I )

I

x E

Rd

and I i s the identity of S O ( d ) ) . Elements of

r

are multiplied by elements of CIT o n the left as follows,

(x,

I ) T 1 = x

+

I ( T 1 ) = x

+

T' such that T' is in CIT and x i s i n

Rd.

I n this chapter we denote ( x , I) T 1 by TI

+

x , which simplifies the notation considerably.

4.1

An

tale

Equivalence Relation of the Tiling

Space for

I?

=

EXd

In this section we will define an &ale relation on a subset of the continuous hull of a tiling. We begin by defining the punctured hull of a tiling.

Definition 4.2. A punctured tiling, T , is defined as follows. Choose a point i n the interior of each proto-tile that breaks the symmetry of the proto-tile, which i s

called a puncture. For each proto-tile pi i n { p l , p2,

-

.

,

p,), denote the puncture

by x ( p i ) . Since each tile i n T i s the translate of a proto-tile, we m a y extend

CHAPTER 4. A C-ALGEBRA OF A TILING WHEN

r

=

RD

40

(tl = t2

+

x for somex E I?) then

I?

relates their punctures in the same way

( ~ ( t l ) = x(t2)

+

x ) .

In the previous definition we say that a puncture breaks the symmetry of a tile t if for all non-zero y in the isometry group of R d , y(tpunc)

#

t,,,,, for any y

which leaves the puncture fixed. We may now define a subset of the continuous hull as follows.

Definition 4.3 ([g]). Given a continuous hull of a tiling Q T , the discrete hull of

Q T , denoted by Q,,,,, is defined to be all punctured tilings T in QT such that the

origin of Bd is a puncture of some tile

t

in Q T . Recalling the notation that T ( 0 )

is the tile in T that contains the origin, 0 in R d , we have

Q,,, = { T E $2,

1

x ( t ) = 0 for t = T ( 0 ) )

Moreover, since punctures are in the interior of tiles, the tile t such that x ( t ) = 0

is unique.

We now observe some facts about the punctured hull, R,,,,, with respect to the group of translations

I?

= R d .

Lemma 4.4. Given a continuous hull QT with

I?

=

Rd,

the discrete hull R,,,, satisfies the following:

1. If T' is in Q T , then T'

+

x is in Qpunc for some x in

I?.

2. Q,,, is closed in f l T .

3. There is an E

>

0 such that for any tiling T in Qpunc, T

+

x is not in R,,,

for any x with 0

<

1x1

<

E .