OPERATOR ALGEBRAS AND DYNAMICAL SYSTEMS 13
Ñ é Ñ é
R ) = {(÷, ÷ ) 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).
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