14 OPERATOR ALGEBRASAND DYNAMICAL SYSTEMS
Gp{y') for some (smallest) m , « €N , p u t
S,P(M)
= {(y, y) e
SfP:
D(y, / )
Ì} , Ì 0, and denote by
[[SfP]]
the group of homeomorphisms V of
yp for which {(Vy, y):y e Ãñ} c S (Af) for some Ì 0 (in particular,
/P
{y G y^: Vy - y} is open for every V e [[S ]]). By following the construc-
tion just described we obtain a C*-algebra W(S' ) which is again AF , since
the union of the finite-dimensional algebras ^(S
/ P
)
( M )
(defined as above)
t P
lies dense in ^( S ).
If in the above discussion we replace
SP
and
S/P
by
RP
and
R,P
we
obtain algebras
f(Rp), 3r{R!P)
and C*-algebras
^(R7*), &(R,P)
associ-
ated with the equivalence relations
RP
and
R,P
. However, the sets
RP(M)
and
R,P(M)
= {(x, x') G
R/P:
D(X , x') Af} , Ì 0, are not equivalence
relations, and the resulting algebras are not AF . Furthermore,
&(RP)
and
&(Rip)
contain C*-subalgebras isomorphic to
W(SP)
and
&{S,P),
respec-
tively.
The algebras W{RP), W(SP), ^(R/jP), and ^(S /P ) are contained in the
corresponding von Neumann algebras
sf(RP),
J / ( S
P
)
,
ja^(R,p),
and
ä?(S,P).
The restriction of the State r\u defined by (1.8) on any one of
these von Neumann algebras to the relevant C*-algebra is again denoted by
çç . If we are dealing with the measure of maximal entropy the State w is
H-p tri ñ
ç
fp
ñ
éñ
a trace on w(S ) and &(& ) (since mp is invariant under S and S ),
but not on
W{RP)
and
W(R,P).
There are other C*-algebras which can be associated with Markov shifts,
like the Cuntz-Krieger algebra Op in [CunK], which take into account the
isotropy groups arising from the shift-action on the (shift-)periodic points in
Xp . These can be viewed as C* -algebras arising from groupoids rather than
equivalence relations and will not be discussed here (cf. [Ren]).
If R and S are isomorphic equivalence relations on the measure Spaces
(X, S?, ì) and (Y, ^ , v), respectively, then it is easy to see that there ex-
ists an isomorphism of their von Neumann algebras J / ( R ) and sf{$) which
carries Jf(R) onto ^#(S) ([Dyel]). Conversely, if there exists an isomor-
phism
0 : J / ( R )
-
J / ( S )
such that Ö(^Ã(Ê)) = J?(S), then a result due
to von Neumann [Neu] implies that R and S are isomorphic. However,
it is possible that the algebras J / ( R ) and J/(S) are isomorphic, but that
no isomorphism between them carries ^#(R) onto Jf(S). In this case we
are outside the realm of ergodic theory, and the Situation becomes much
more difficult to analyze. In view of this very serious problem it is remark-
able that—for a particular dass of von Neumann algebras—the isomorphism
problem can be reduced to the isomorphism of equivalence relations, and that
the equivalence relations giving rise to this dass of von Neumann algebras
can be classified completely.
1.4.
DEFINITION
[Ziml], [ConFW]. Let R be a nonsingular equivalence
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