OPERATOR ALGEBRAS AND DYNAMICAL SYSTEMS 13

Ñ é Ñ é

R [Ì ) = {(÷, ÷ ) G R : D{x, ÷ ) Ì} c × ÷ × is closed and hence com-

pact. Furthermore, if M{ M2, then

RP(Ml)

c

RP(Af2),

and

RP(MX)

is

a compact, open topological subspace of R (Af2). This allows us to furnish

RP

=

\JM0RP(M)

with that topology in which every

RP(M)

is compact

and open, and R is locally compact, second countable space in this topol-

ogy. Let

[[RP]]

be the

group18

of all homeomorphisms V of Xp which

satisfy the following conditions: (1) {(Vx, x):x G X} C

RP(M)

for some

Ì 1; (2) the set {x: Fx = x} c Xp is open. Then

[[RP]]x

=

RP(x)

for

every X G I . If

SP

c

RP

is the subrelation introduced in Example 1.2(3)

Ñ é Ñ é

then S (Af) = {(÷, ÷ ) G S :D(x, ÷ ) Ì) c × ÷ X is an equivalence

relation for every Ì 0, and

[SP](M)

= {F G

[[SP]):(Vx)k

= xk for all

xeXp and |fc| M} is a finite group with

[SP]{M)x

=

Sp(M){x)

for every

÷ e. Xp. We topologize

SP

=

\JM0SP(M)

as above so that each

SP(M)

is

a compact, open subset of

SP

, set

[[SP]]

= 1)Ì ï\$

Ñ]{Ì) c

[ [

R P

] ]

a n d n o t e

that

[[SP]]x

=

SP(x)

for every xeX.

We begin by associating a C*-algebra with the equivalence relation

SP.

For every Af 0 we write C(Xpy ' for the Space of complex valued func-

tions on Xp which depend only on the coordinates x_M, ... , xM. Let

C(Xp) be the set of continuous, complex valued functions on Xp, and

ñ ñ

let Cc(S ) denote the continuous functions with compact support on S .

Every V G

[[SP]]

and h G C(Xp) defines a linear Operator on

CC(SP)

by (L

F

/)(x,x') =

f{V'lx9x')

and (Af

A

/)(x,;0 = Ä(x)/(x, ÷'), / e

CC(SP).

Consider the algebras

f(Sp)

and ^(S

P

)

( M )

c ^ ( S

P

) generated

by {Lv,Mh:V

G

[[SP]],/z

G C(*

p

)} and {L

K

,M^:F

G

[SP]{M)

, Á G

C(J\fp)(

' } , respectively, where Af 0. As we have seen in Example

(2),

^(SPYM)

is a direct sum of certain finite dimensional matrix alge-

bras Mk(C). The completion of ^~(S ) in the Operator norm || · \\p on

thethe

Hubert space

p

= L

2 P

, (ì»)*? ) is th desired

C*-algebra19

g?(Si

P)

o f equivalenceHrelation(S S . The algebrae

U A / O ^ ^

)

*s

dens e n

&(SP),

and ^(S

P

) is an

^F-algebra.20

For the corresponding one-sided algebras ^(S

/ P

) and

W(S,P)

(cf. ex-

ample 1.2(4)) we set D(y, / ) = max{ra, n} if (y, y) G S/P and J»0) =

18

countable group [[RP]] is an ample group in the sense of [Kri7]. 19The

If â i s an arbitrary stochastic matrix compatible with Ñ , and if || · |L· is the Operator

norm on

&(SP)

acting on the Hubert space HQ =

L2(SP

,

(//ß)gL)

, then || · \\p = \\ · \\Q , and

the completions of ^ ( S

P

) in the two norms || · ||p and || • \\Q coincide (cf. [Dix])

Á C*-algebra sf is an AF-algebra if there exists an increasing sequence {s/n , ç 1) of

finite-dimensional subalgebras of sf such that \Jn srfn is dense in sf (cf. [Bra]— AF Stands

for approximately finite dimensional).