Fields

Institute

Volume 20, 1998

Communications

Right Inverse of the Module of Approximately Finite
Dimensional Factors of Type III and Approximately
Finite Ergodic Principal Measured Groupoids
Colin E. Sutherland
Department of Mathematics
University of New South Wales
Kensington, NSW 2052 Australia

Masamichi
Department
University

Los Angeles,

Takesakiof Mathematics
of California

California 90095-1555 USA

Introduction
It is well known from the work of Connes [2, 3], Krieger [10, 11], and Haagerup[6], that the classifications of ergodic transformations up to orbit equivalence, ofapproximately finite dimensional (AFD) factors up to isomorphism, and of ergodic
flows up to conjugacy are all equivalent problems. The proof proceeds by constructing maps from transformations to flows (the stable range or Poincaré flow
construction),from transformations

to factors(the group-measure-spaceconstruc-tion) and from factors to flows (the flow of weights construction) and showing thatthese maps implement the desired equivalence.
Notwithstanding its beauty, the proof is unsatisfactory in at least two respects.
Firstly, it is far from clear what the inverse of the various maps above (particularly
those to flows) are. In fact surjectivity of the stable range map involves constructing for each flow an action of Z? with the given flow as stable range and using thefact that all such actions are singly generated up to orbit equivalence. Secondly,
each of the maps involved is in fact a covariant functor, transforming orbit equivalence and automorphisms to self-conjugacies, automorphisms and self conjugacies
respectively; the proof fails to show whether or not these various homomorphisms
are split.
Our purpose in this paper is to remedy this by constructing explicitly for each
ergodic flow an AFD factor and an ergodic transformation which have the given flow
as flow of weights and stable range respectively. These constructions are readily
1991 Mathematics Subject Classification. Primary 46L10 Secondary 28D99, 46140, 46M 15.
This
research has been supported in part by the Australian Research Council and the
National Science Foundation.

149

150

Colin

E. SutherlandandMasamichi

Takesakiseen to be functorial, thus proving in particular that the map mod of [4] fromautomorphisms of AFD factors to automorphisms of the corresponding flows of

weights is in fact split.

Our results are also in the spirit of [7, 8], [14] and [5],emphasizing the necessity of adopting a functorial point of view in analyzing nonsmooth classification problems.
In the first section, we construct a functorial inverse to the flow of weights
construction. Section Two contains work based on unpublished results of Robert
Wong [15] on inverting the stable range map, generalising earlier work of [9] on
measure preserving flows. Although Wong's results are now quite old, they havenever been circulated owing to his departure from mathematics and we feel theyshould be made available to the mathematical community.
1

Inverting the Flow of Weights and Moduleof an AFD

FactorFirst we want to construct an explicit functor from the category of ergodic flows
to that of AFD factors of type III. We fix a separable abelian von Neumann algebra
C and a o—weakly dense separable C*—algebra subalgebra B of C. To each ergodic
one parameter automorphism group 8 of C we associate the C*—algebra A generatedby {0f(a) : a € B, f € L*(R)} where 6;(a) = [; f(t)6:(a)dt.

Then {4,R,0} is aseparable C* —covariant system. Set X = Sp(A), a compact metrizable space. The
action 6 of R on A gives rise to a topological transformation group {X,R, F} suchthat

0:())(z) = f(F'w),

zeX,feAteR

Let 6,(C) be the set of all normal states on C and &%(C) be the set of all faithfulnormal states on C. To each ¢ € &,.(C) there corresponds a Radon measure py,
on X such that p(f) = [ f(z)duy(z),f €Aand if p € G9 then C is canonically
identified with L>(X, p,,). To save notations, we will identify the measure pi, with
© € 6,(C). We want to find a functorial way of constructing an AFD factor R oftype III whose flow of weights is precisely the given flow {C,R,6}. First observethat the C*—algebra A, and therefore X as well, depend on 8, so it is appropriate

to write them as A(f) and X (6). Let G(8) = R x X (6) be the associated groupoid
so that{seo =arta=Fa(2) co0)(s, Fyz)(t,z)= (s+ t,z),s,teR,zeX(9).

”)

Let {Ro1,R,a} be a fixed AFD factor of type IL, equipped with a trace scaling
one parameter automorphism group {a : t € R}, whose existence is guaranteed by
the existence of an AFD factor of type III;, and setbo(t,z)= eet (2),p € GY0s)= Cttlog(s (ta)12)(6) €G(0).

We then integrate the above system:

R= xs) Ro dp(x) =C ® Ro 1;
Te(a) = Tx

7(a(z))dp(z),©
~a= Ix) a(z)dp(z) € R;

(6¢ (@)(Fea) = 0%,, (ala), t€ R, o € X(6),(1.3)

where 7 means a fixed faithful semi-finite trace on Ro 1 which is unique up toscalar multiples by positive numbers. It is straightforward to check that the one

Approximately

FiniteMeasuredGroupoids151

parameter automorphism group 6% of R scales the trace T, and extends the given
one parameter automorphism group on the center C of R. Finally set

R(6, 9) = R ge R.

(1.4)

By the structure theorem for factors of type III, [13], R(6,¢) is a factor of type
IIT with core R except for the case that {C,R,0} = {L*°(R),R, translation} inwhich case R(6,¢) = Ro,1. Thus the flow of weights of R(6,p) is conjugate to
the given flow {C,R,6}. The construction of R(6,¢) is canonical except for the
dependence on the state ¢. So we need to look at the dependence of R(6,) on

p € 8% = 8%(C).
have

Choose another ¥ € &%. With f(z) = (2) (z),z € X(6), weSy(t, 2) = f(Fix)b,(2)f(2)7!,z€X(0);

Blt) = Ctog(f(Fi2)) © 0 0) © Aros @)s(15)(7) € G(B).

Therefore with &(y,,) = Jx0) Qlog(f(x)) d(x) € Aut(R), we have0) = agp o0f 0a, yn tER, p19 ey
Typ

=

Top

©

Qy,0)5(1.6)

8(pp) = (pp) © Byp)s
PY, € BL.
It is easy to check that the canonical extension ay.) of Gy.) gives an isomorphism
R(0,¢) onto R(6,1) and that the system {oye : ¥,¢ € G9} satisfies the chain
rule.
To establish the functoriality of the construction of R(8, ¢}, we need to get ridof the dependence on ¢. So set

RO) =o ={z}e J] :2v =awn(@,),

vpe&lo.

(17)

PESY

We

now moveon to the question as to how the association of R(f)reactsto a conjugation of the flow 6. Let o be an element in Aut(C) and set 76, =gofi0o07t,
teR. Then we have o(A(0)) = A(°6). The isomorphism o of A(6)onto A(?8) gives rise to a homeomorphism § : X (8) — X(°6), so that (¢(f))(z) =f(S™z),z € X (96), f € A(6). When we need to indicate the dependence of S on
0, we write S(o) and observe that S(o1 0 02) = S(01) 0 S(02), 01,02 € Aut(C),i.e., S is covariant in o. The flow °F on X(°f) associated with ?8 is then given
by: °F; = So Fy 087 1,t € R. The modules of G(°6) and G(6) are linked in thefollowing way:

6y(t, 8%) = byes (t,x),
Hence(1,2) € G(I) =Rx X(§),9 € &2.

(1.8)

00 = (“0say (£3) € G(O).

Therefore the automorphismo of C extends to an automorphism

6 = oc ® id €Aut(C ® Ro,1) such that(0) =G06Y 0571, t eR;

Typos = Tp 0G,(1.9)

9h € GY;5084) 08" = B(pog1 gos-1)s$Y € EL.

152

Colin

E. SutherlandandMasamichi

Takesaki

The correspondence o € Aut(C) — & = 0 ® id € Aut(C ® Ro,1) obviously satisfies
the chain rule:
01002=gy,003,

01,02€

Aut(C).

(1.10)

The conjugating map & is then canonically extended to an isomorphism, still denoted by &, of R(8, 00) onto R(°8,) for each 9 € GY. The last equality in (1.9)

allows us to define an isomorphism & of R(#) onto R(%6) by:d({zp}) = {0(xpoo-1)},{zp} € R(D).

(1.11)Let Erg(C,R) be the set of ergodic actions 6 of R on a separable abelian von

Neumann algebra C which are not conjugate to L>°(R) with translation.
Theorem 1.1

The correspondences 6 € Erg(C,R) — R(6) and o € Aut(C) —& define a functor from the category of ergodic flows to that of approximately finite
dimensional, factors of type III such that the flow of weights of R(6) is conjugate
to 0. In particular, if § € Erg(C,R) is fized, then the map: 0 € Auty(C) = {a €Aut(C) : a0, = 0; 0 a} — & € Aut(R(0)) is an injective homomorphism which is
the right inverse of the module of R(0).
Remark1.2

The von Neumann algebra R(#) in Equation 1.7 is of course

isomorphic to R(#, ¢) for any ¢ € GY; the construction is given solely to eliminate
the identifications ay ,) of R(6, p) and R(8,v) as p and ¢ vary.
It is possible to avoid this construction as follows. Fix ¢ € GY; it then follows

from the above discussion that for any o € Aut(C), & is an isomorphism of R(6, poo)to R(76,¢), so that G © Ayes.)onto R(°8, ¢), and satisfles= Q(p,pee—1) © 7 is an isomorphism from R(6,¢)(01002) © Apo(51002),0) = F1 © Apoarg) © 02 © Agorap)
for 01 0 02 € Aut(C).

Thus,with 6,(1.12)= G © O(y00,,0), We Obtain a functor 8 €Erg(C,R) — R(0,p),0 € Aut(C) — &,,.
Corollary 1.3 If R is an approximately finite dimensional factor of type III,the short exact sequence

1 — Int(R) —Aut(R) —Aute(C) —1,(1.13)

is split, where {C,R, 8} is the flow of weights of R.
2

Inverting the Stable Range

Map

To set notation, we first recall the construction of the stable range or Poincaré
fiow of an ergodic action of a discrete group G on a standard measure space (X, us).
We writeppog
plg,z) == log log —=2
= (z)

2.1
(2.1)

and observe that p satisfies the cocycle identityp(gh,u) = p(g, hz)p(h,z)

ae. in z for gh eG.

(2.2)

The group G acts on (X,i) = (X x R, uu x edt) via the maps
(x,t)

-_—

(gz,t

— p(g,T)),

ge

G;

(2.3)

Approximately

FiniteMeasuredGroupoids

153

evidently, this action of G commutes with the action of R defined on (X , 1) by

Ss: (x,t) —(x,t +3),

seR,

(24)

and hence defines an action {6; : t¢ € R} of R on the algebra C, of G-invariant
elements of L>(X, i). This action {C,,6;,R}, or any of its point realizations, iscalled the Poincaré flow of the original action of G on (X, it), or the stable range(of the Radon-Nikodym cocycle) - see [12] for an equivalent discussion of terms ofgroupoids.
If v is another measure on X equivalent to py, anddv1)z)=log= tog | —(z
(0)

2.5
(25)

Wou(z,t) = (z,t + f(z)(2.6)

then the map W, , on X x R

carries fi to U, commutes with the action {S; : t € R} in (2.4), and intertwines
the action of G on (X, ft) and (X,7). Consequently W, ,, defines an isomorphism
wy yu : Cy — C, which intertwines the two Poincaré flows. Since wy, ow, = wa4,
for any equivalent measures A, v, pu on X, the flows {C,,,0;,R} are all isomorphic ina coherent manner. In addition, if H is another countable group acting on (Y,v)
and A is an orbit equivalence from G on (X, pu) to H on (Y,v),sothat
A: X —»Yis a Borel isomorphism, carries G orbits to H orbits and px to v (or, more correctly,
to a measure equivalent to v), then the map A: X — Y defined byA(x,t) = (Az,t)(2.7)

carries G orbits on X to H orbits on Y, [i to 7, and intertwines the R-actions on(X, i) and (Y, 7). This then gives rise to an isomorphism a of the flows (C,,0,,R)and (C,, 6;, R) which depends covariantly on A.
In summary, we have
Theorem 2.1 The construction of the stable range flow is a functor fromthe category of (ergodic) actions of discrete groups on standard measure spaces,with orbit equivalence as morphisms, to the category of (ergodic) flows on standard
measure spaces with measure-class preserving conjugacies as morphisms.
The case where G = Z is of particular interest.
will be devoted to proving

The remainder of this section

Theorem 2.2 There is a functor from the category of ergodic flows on standard
measure spaces with measure class preserving conjugacies as morphisms to ergodic
tranformations with measure class preserving conjugacies as morphisms which is aright inverse to the stable range construction.
Note that, rather surprizingly, the morphisms between transformations are conjugacies rather than merely orbit equivalences.
The proof is somewhat analagous to that of Theorem 1.1, but carried out at
the measure space level. To do so, we first need some auxilliary constructions.

Let Zs = Z/5Z be the cyclic group of order 5, and let X = ZY = []2, Zs;
be the cartesian

product

space.

Let

;,

B2,

B3

and

(3; be positive

numbers

154

Colin

E. SutherlandandMasamichi

such that log 81, log B2, log Bs and log (3; are rationally independent.
(ag, 1, 2, 23, tq) be the probability distribution on Zs determined by
Bi

=

o—1/

0,

i=

Takesaki

Let a =

1, 2, 3,4,

(2.8)

and let
[eo]

P=]?

(2.9)

i=1

be the product measure on X. The odometer transformation T is defined as follows.
For x = {z;} € X, let io(z) be the first index ¢ such that z; # 4. Then we set(Tz); =

0,

1 <i

+1,

i= io(z),<ip(x),

Ti,

i> io(x) .

(2.10)In the above definition of (T'z);, 40(z) can be infinite. In this case, z; = 4 forevery i € N. Except for this isolated case, we have x; = (Tz); for large enoughi € N. Since the measure P on X is non-atomic, we can exclude the point zg =(4,4,...,4,...) and its orbit {T"zo : n € Z} from X.
Lemma

2.3

There exist Z-valued Borel functions n;(z), i = 1,2,3,4,

such

that
dPoT
Sr
(z) = B®) gale)©

gra@gna®

ae

re X.

(2.11)

These functions n;’s do not depend on the choice of B;’s as long as the log 3;’s arerationally independent.
Proof
Fix an z € X. Let Ax = (z1,%2,...,%k) X [1541 Zs be the cylinder
set determined by the first k coordinates of z. We then havek

P(A) = [J ee;kPTA) = [] era), -

j=1

j=t

For each j, we have
Oy,
pom
x)

=

’

1

if

B(Tz);ifj=> do(z);B1B2fBaBs

if.

J <io(x).

ee.

io(x);.

Therefore, there exist Z-valued functions n1(z),...,ns(z) such that

P(TAx) =_ 11T 8; gmi(aPlay)

for largek.

Then we have44pray) = Pa i=1TL 00 = [ Ar T]j=1 8 apia).

Approximately

FiniteMeasuredGroupoids

155

Since these A’s are closed under finite intersections and generate the Borel fieldof X, we conclude that T leaves the measure P quasi-invariant, and that
dPoT

-

“dP

ni (x)(z) = 11 pit
i=1

In the above proof, the functions n;(z), 1 <i < 4, do not depend on the choiceof 3;’s, so that they are intrinsic to the odometer.
O
We now let H be the group of all finite permutations on N, i.e. the groupof permutations on N which change only a finite number of elements of N. We
then let H act on X in the canonical way, i.e., if g € H, then (gx); = x41; forz = {z;} € X. We observe that H C [T], the full group of T, and H leaves themeasure P invariant.

Lemma 2.4

Let x € X. If k is a positive integer such that T*(z) is a finite

permutation of x, then we havek—1

D> ni(T(x))=0

for

i=1,2,3,4.

(2.12)

j=0

Proof

Choose h € H with h(z) = T*(z).

For j = 0,1,...,k, T?x differsfrom z in only a finite number of coordinates, so that there exists n € N such that(Tz); = x; for every j > n. Set
ox

Ao

=

(21,2,

yn)

Xx II

Zs,

n+1

A; =T7(Ao),37=0,1,....k.

Then we haveP(A341) _ T

————

P(4;)

—

aT @)

G8;

11

'

5

;

i=0,1,....,k—1.

Since P(Ag) = P(h(Ap)) = P(Ay), we havel= P(A)k—1

4_ 11 P(Aj41) _ II gi

maT (=)

Since 83;’s are multiplicatively independent we obtain the formula (2.12).

O

Let {, u, F;} be a properly ergodic flow on the standard measure space {Q, pt}.
Setdp

p(t,w) = “

oF

Ww),

(tw)eRxQ.

(2.13)

We further introduce notations:{ a(z) = ni (z) log B1 + n2(z) log Ba,b(z) = n(x) logB3 + na(z) log Bs,(2.14)z€X.

We now prove a result which is the key step in proving Theorem 2.2.

Colin156

Sutherland

Takesaki

Then the flowS

ergodic single transformationof the following

is the Poincaré flow

Masamichi

andand {F;,Q, pu} be as above.

Let {T,X,P}

2.5

Theorem{F;,Q, u}E.

on the space {X x 2 x R,P@ pu ® e* ds}:

S(z,w,r) = (Tz, Foz) (w), 7 + log p(a(z),w) + b(z)) Proof
LetY = X x 2 xR
Poincaré flow of S, we set

and dv = dP ® du ® edvoS=S(y,t)(y,1) = {( Sy,tys ++ log
log ———

dir(y,t) =dv®e

w)

(yt)
(y,1)

(2.15)*ds. To find out the
xR=Y,

eY

(2.16)

dt.

Let A be a cylinder subset of X:

A=(21,225.

..,%n)

x II

Zs

,

n41

such that some z;, 1 <j <n, is less than 4, so that n,(z),1 <7 < 4, are constanton A. Let a = a(x) and b = b(z) for all x € A. Set{ Zap(w,r) = (Faw, + logp(a,w) +b), (w,r) €dip =du®e "dr.

2 xR,

(2.17)’

It then follows that Zg; is the composition of a measure preserving transformationand the translation which scales the measure by eb.

ThusZ,; scales the measure[i by the factor e~®. For any Borel set C C Q = Q x R, we have

USA % ON) = [ Xstaser(®) ds) = [ Xaxe(570) dirty)= [| xaxolF-ae).r + logp-a,w) ~b) dv(a,er)
XxQ= /. xr(a)(z) dP(z) / XZap(c)(W,T) dfi(w,T)= P(T(A))i(Za(C)) =e P(A)e™" i(C)=e*P(A)i(C) = /e*® du(y).
AxC

Since the collection of such A x C is closed under finite intersections and generates
the o-fleld of Borel sets in Y = X x §2, we conclude thatd

BOS (y) = ee,~(z,o)=yeY =Xx0.

(2.18)

Hence we haveS(z,w,r,t) = (S(z,w,7),t + a(x), (z,w,1,t) € Y.

(2.19)

Approximately

FiniteFor each h € H,

MeasuredGroupoids157

let k(h, x), or simply k(z), be the integer valued function suchthat hz = T*®)z, z € X. For abbreviation, setstk,z) =k

i=0a(T?x)

k>0,

.

k

.

tk,z) => b(T’z).
§=0

If k(z) = k(h,z) > 0, then we have

85 (2,00,7, 8) = (SK)(2, 0,7), t + s(k(z) — 1,3)
By Lemma 2.4, we know s(k(x) — 1,2) = 0 and t(k(z) — 1,2) = 0, so thatSE)
(2, w, 7, t) = (8%) (z,w,7),1).
Furthermore, we have
SKE(z, Ww, T) =

(TF)

zw,

T+

log

p(s(k(z)-1, w))+ t(k(z)

-1,

z))

=

(hz, w, 7) :If k(x) = k(h,z) < 0, then we consider h~! applied to h(z) to conclude that55) (gw,r,t) = (ha,w, Tt).

Therefore, the full group [3] of § on Y contains the subgroup H xid on X x OxR =Y. Hence any S-invariant measurable subset of Y is of the form X x B for somemeasurable subset B C © x R x R, which is invariant under the transformation:(w, r,t) + (Fo(e)(w), 7 + log p(a(z), w) + b(z),t + a(z))
for all z € X.
In X, the following four sets all have nonzero measure:{re X:a(x)=logh
and b(z)=0}{ze X :a(x)=1logBs and b(z)=0}{reX:a(x)=0
and b(z)=Ilogpls};{re X:a(x)=0
and b(z)=logfls}.
Therefore, the above set B must be invariant under the transformations:(w, 7,1) => (Faw, + log pla,w) + b,¢ + a)
for all a € G; and b € Gg, where G; and G3 are the subgroups of R generated bylog B31, log 82 and log Bs, log Bs respectively. Since these subgroups are both densein R, there exists a measurable subset C' C 2 x R such that

B={(w,nt) eAxRxR:(wt) eC}

and C is invariant under the transformations:R, : {(w,t) — (Fow,t

+a),

a€Gy.

The density of Gy in R implies that C is invariant under the flow

R,: (w,t) — (Fw,t +s),
We now defineW(w,t) = (Fiw,t).

seR.

158

Colin

E. SutherlandandMasamichi

Takesaki

Then we have

WRW
Therefore W(C)(w,t) = (w,t +5).

is invariant under

idx translation on @ x R.

Hence, withW(z,w,r,t) = (z, Fw, rt), we have

WL2Y)SW=C®
where we view W as a unitary on L(Y).
of § is conjugate to F under W.

LQ,

eC&C,

Tt now follows that the associated flow

[OO

To complete the proof of Theorem 2.2, we need to consider the effects of conjugacies of flows. So consider ergodic flows {F;,Q, pu} and {G;,T,v} and suppose

0: (Qu) — (I'v) is a measure space isomorphism with 8 o Fy = G; 0 8 for all ¢t.
Let(0) =XxQUxR,

Pxpxe td)

and analogously for (T, 7); also let Sr, Sc be the transformations on (Q, 1) and(T, 7) as in Equation 2.15, and define 8 :— I" by~dvof

B(x, w,r) = (z,0w,r + log “gp
Lemma

2.6

(2.20)

With notation as above, we have(a) pr (t, w) 42 (Faw) = pat, Ow) 428 (w) a.e. for each t;
(b)

Sr

=

Sq;

(c) 8 depends covariantly on 6.
Proof(a) follows routinely from the Radon-Nikodym Theorem(b) We calculate
80 Sp(z,w,r) = (Tz,0Fyq)(w),r + b(z) + log pr(a(z),w)+

dvod
log log ———(F,du( a(z)(w))),

while

Sch(z,w,r) = (T2,Gomyf(w),r + b(z) + log pc (a(x), O(w))
drolog ——— (w)) .
+log—
The equality of these two expressions follows from a) and the assumptionthat

8 o Fy = Go

8 for all ¢.

(c) is routine, and left to the reader.

O

Remark 2.7
With Lemma 2.6, the proof of Theorem 2.2 is complete. We
also note that composing Theorem 2.2 with the group measure space construction,
which is known to be covariantly functorial for orbit equivalences, yields an alternate
proof of Corollary 1.2.

Approximately

FiniteMeasuredGroupoids

159

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Symposia

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Krieger factors