10 OPERATOR ALGEBRASAND DYNAMICALSYSTEMS

Rv

preserves ì : take X — E

+

, ì - Lebesgue measure on R

+

, and let

V: R+ -+ R+ be defined by setting

F0 = 0, and Vx = ^

ak(x)•I0~k

kk0(x)

for every

x=

Ó **(*)· 10~*€R+,

kk0(x)

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

extended11

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 =

L2(X,

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

subalgebra13,

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