preserves ì : take X E
, ì - Lebesgue measure on R
, and let
V: R+ -+ R+ be defined by setting
F0 = 0, and Vx = ^
for every
Ó **(*)· 10~*€R+,
where ak ,x){x) 0 and 0 ak(x) 9 for every k G AE . In other words, V
replaces the leading digit in the decimal expansion of a point ÷ by 0. Then
V is nonsingular and ergodic on (×, ì) , and R preserves ì .
This kind of pathological behaviour can obviously not occur if an endo-
morphism V of (X, ¥*, ì) can be
to a nonsingular, (properly)
ergodic automorphism W of a measure space (Y, ZT, v). There are many
unresolved problems and phenomena in this area, and we refer to [EigS] for
some recent results.
In order to describe how a nonsingular equivalence relation gives rise to
a von Neumann algebra we consider a complex Hubert Space Ç with inner
product (·, ·) and denote by B(H) the algebra of all bounded linear Opera-
tors A:H Ç with norm || · ||. The adjoint A* of an Operator Á e B(H)
is defined by (Av, w) = (v, A*w), í ,w e Ç. Á set J / c B(i/) is se//"
adjoint if A* e 3? for every ^4 e J / . Á C*-algebra is a norm closed, seif
adjoint subalgebra of B(H) which contains the identity Operator 1 on Ç.
Á C*-algebra sf c B(7/) is a von Neumann algebra if J / is closed in the
strong topology, i.e. in the smallest topology on B(H) in which all the maps
Á —• ||Ëí|| , ü G ff, are continuous. If stf c B(H) is a self-adjoint sub-
set then its commutant stf' = {B e B(H):AB = 2?.4 for every y 4 e J / } is
a von Neumann algebra, and $/ is a von Neumann algebra if and only if
Á von Neumann algebra J / is afactor if J / HJ/' = C · 1 ,
and J / =• B(//) if and only if s/' = C · 1 .
The most simple-minded construction of a von Neumann algebra $f from
a nonsingular equivalence relation R on a measure Space (X, S*, ì) is ob-
tained by setting Ç =
ì ) and j * = {{Uv: V e [R]} U L°°(A\ ì)) " c
B(//), where (7F is the unitary Operator Uvf = {Üìí/Üì) ' · ( / · V),
f £ Ç, F e [ R ] , and where every Á Å / ^ ( X , ì ) is regarded as a multipli-
cation Operator in B(H). It is not difficult to verify that L°°(X, ì) c B(i/)
is a maximal Abelian
and that sf nsf' = {/ e L°°(X, ì) :
An automorphism W on (F, ^ , v) extends V if there exists a nonsingular, surjective
map ø: Õ —• X such that ø W -V - ø V-SL.Q.
12An Operator U B(//) is ww/ary if U* = U~l (or, equivalently, if UU* = U*U = 1) .
If J / C B(//) is a von Neumann algebra, a subalgebra 38 c «flf is maximal Abelian if
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