12
OPERATOR ALGEBRAS AND DYNAMICAL SYSTEMS
Ô be a nonsingular, free action of a countable group G on a measure
space (X, S?, ì) . Then we can write R as G ÷ X and Ç as L2(G ÷
I , l x / i ) , where ë is the counting measure on G. The algebra J / ( R ) is
generated by the Operators (Mhf)(g, x) = h{Tgx)f(g ,x), h e L°°(X, ì) ,
and (L ,f)(g, x) =
f{g'~Xg,
x), g £ G. This is the Murrayvon Neumann
group measure space construction in its original setting [MurNl].
(2) 77*e vo/t Neumann algebra of á transitive equivalence relation. Let
AE = {0, ... , ç  1}, ì({é}) = 1 //é, / = 0, ... , Ë  1, and let R = I x
* . Then
/4L)(/,7)
= */" for all (i,j) Å R, Ç =
Cn\
and J / ( R ) is
generated by the Operators {MJ^j, fc) = S^j/U, k) and (£,·/)(./, k) =
f(j  i (modn), k), i = 0, ... ,n  1. In order to realize J / ( R ) explicitly
we dehne basis elements vt
}
e Ç, 0 /, j ç by vt .(/', / ) = S^
y)
(./ ^
and write et for the ith unit vector in
Cn
. The map vt — • et g e. extends
linearly to an isomorphism of Ç and C"(gC" and sends s/(R) to Afrt(C)®
ln , where Mn(C) denotes the algebra of all complex ç ÷ ç matrices and ln
is the nxn identity matrix. In this picture ç is the trace Agln — • tr(A)/n .
If Ë" = Í , //({/}) = 1 for all / € AE , and R = NxN, the above construction
gives sf(R) =
B(/2(N))
g 1 c
B(/2(N)
®
/2(N))

B(/2(N2)),
where 1 denotes
the identity in
B(/2(N)).
Finally, let X be a finite set, ì({÷}) =
\X\~l
for every ÷ e X, and let
R c X x X be an equivalence relation. If B{9 ... , Bk denotes the set of
distinct Requivalence classes in X (on each of which R is transitive), then
sf(R)~s/(RB)e®sf(RB)~Mn(C)eeMn (C), where ni = \Bt\
for / = 1, ... , k , and ç is the normalized trace on j / (R).
(3) C*algebras associated with Markov shifts. Éú R is a nonsingular
equivalence relation on a measure space (X, S?, ì) , where X is a com
pact, metrizable space and S? is the Borel field of X, and if [R] has a
distinguished countable subgroup of homeomorphisms of X (which hap
pens, for example, if R = R
r
for a nonsingular action Ô of a count
able group G by homeomorphisms of X), then we can associate a sepa
rate C*algebra ^(R) c J/(R) with the equivalence relation R. From
the point of view of dynamics, this construction has been particularly use
ful in the context of Markov shifts (a general construction of C*algebras
from topological equivalence relations is described in [Ren]). Let Xp and
RP
be defined as in example 1.2(3). For every {x,x) G
RP
satisfying
(1.4) we choose the integers m, m , n , n occurring there as small as pos
sible and put D(x, x) = max{m, m , n, n}. For every Ì 0, the set
Á nonsingular action Ô of G on (X, S?, ì) is free if ì({÷ e ×: Ô ÷ = ÷})  0 for
every g Ö 1 in G. It is an open problem whether every nonsingular, ergodic equivalence
relation R on (X, S*, ì) is of the form R = R (ôçïÜì) for a free action Ô of some
countable group G on (X, SP, ì).