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) =
x), g £ G. This is the Murray-von 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
= */" for all (i,j) Å R, Ç =
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^
(|./ ^
and write et for the ith unit vector in
. 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) =
g 1 c
where 1 denotes
the identity in
Finally, let X be a finite set, ì({÷}) =
for every ÷ e X, and let
R c X x X be an equivalence relation. If B{9 ... , Bk denotes the set of
distinct R-equivalence classes in X (on each of which R is transitive), then
sf(R)~s/(RB)e--®sf(RB)~Mn(C)e---eMn (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
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
be defined as in example 1.2(3). For every {x,x) G
(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).
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