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1997
Factors from trees
Jacqueline Ramagge
University of Wollongong, [email protected]
A G. Robertson
Newcastle University (UK)
Publication Details
Ramagge, J. & Robertson, A. G. (1997). Factors from trees. Proceedings of the American Mathematical Society, 125 2051-2055.
Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library:
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Factors from trees
Abstract
We construct factors of type III,/n for n E N, n > 2, from group actions on homogeneous trees and their
boundaries. Our result is a discrete analogue of a result of R.J Spatzier, where the hyperfinite factor of
type III1 is constructed from a group action on the boundary of the universal cover of a manifold.
Keywords
factors, trees
Disciplines
Physical Sciences and Mathematics
Publication Details
Ramagge, J. & Robertson, A. G. (1997). Factors from trees. Proceedings of the American Mathematical
Society, 125 2051-2055.
This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/1237
Factors from Trees
Author(s): Jacqui Ramagge and Guyan Robertson
Reviewed work(s):
Source: Proceedings of the American Mathematical Society, Vol. 125, No. 7 (Jul., 1997), pp.
2051-2055
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2162308 .
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PROCEEDINGS
OF THE
AMERICAN
MATHEMATICAL
SOCIETY
Volume 125, Number 7, July 1997, Pages 2051-2055
S 0002-9939(97)03818-5
FACTORS FROM TREES
JACQUI RAMAGGE AND GUYAN ROBERTSON
(Communicated by Palle E. T. Jorgensen)
We construct factors of type III,/n for n E N, n > 2, from group
actions on homogeneous trees and their boundaries. Our result is a discrete
analogue of a result of R.J Spatzier, where the hyperfinite factor of type III1
is constructed from a group action on the boundary of the universal cover of
a manifold.
ABSTRACT.
1. INTRODUCTION
Let r be a group acting simply transitively on the vertices of a homogeneous
tree T of degree n + 1 < o. Then, by [FTN, Ch. I, Theorem 6.3],
where there are s factors of Z2, t factors of Z, and s + 2t = n + 1. Thus
presentation
r = (a,v. ... Xa,+t : a? = 1 for i E { 1, ,s)
r has a
we can identify the Cayley graph of r constructed via right multiplication with T
and the action of r on T is equivalent to the natural action of r on its Cayley graph
via left multiplication.
We can associate a natural boundary to T, namely the set Q of semi-infinite
reduced words in the generators of r. The action of r on T induces an action of r
on Q.
For each x E r, let
Qx= {w E Q : w = x ..
be the set of semi-infinite reduced words beginning with x. The set {Qx}xEr is a
set of basic open sets for a compact Hausdorff topology on Q. Denote by ixi the
length of a reduced expression for x. Let Vm = {x Er
Ixi = m} and define
Nm = IVm . Then Q is the disjoint union of the Nm sets Qx for x E Vm.
We can also endow Q with the structure of a measure space. Q has a unique
distinguished Borel probability measure v such that
r e
n+a1
Tn
e
for every nontrivial x E r. The sets Qx, x E r, generate the Borel a-algebra.
Received by the editors January 26, 1996.
1991 Mathematics Subject Classification. Primary 46L10.
(c)1997
2051
American
Mathematical
Society
2052
JACQUI RAMAGGE
AND GUYAN ROBERTSON
This measure v on Q is quasi-invariant under the action of F, so that r acts on
the measure space (Q, v) and enables us to extend the action of r to an action on
L??(Q,v) via
g f(w) =f(g 1 w)
for all g E r, f E L??(Q, v), and w E Q. We may therefore consider the von
Neumann algebra L? (Q, v) X r which we shall write as LI (Q) >i r for brevity.
2. THE FACTORS
We note that the action of F on Q is free since if gw = w for some g E F and
and
or w = g- g-g-g
w E Q then we must have either w= ggg.
v {ggg
,g1gg
*...
*}
= 0.
The action of F on Q is also ergodic by the proof of [PS, Proposition 3.9], so
that LI (Q) >i F is a factor. Establishing the type of the factor is not quite as
straightforward. We begin by recalling some classical definitions.
Definition 2.1. Given a group F acting on a measure space Q, we define the full
group, [F], of F by
[F] = {T E Aut(Q) Tw E Fw for almost every w E Q}.
The set [F]o of measure preserving maps in [F] is then given by
[F]o = {T E [F] :Tov = v}.
Definition 2.2. Let G be a countable group of automorphisms of the measure
space (Q, v). Following W. Krieger, define the ratio set r(G) to be the subset of
[0, oo) such that if A > 0 then A E r(G) if and only if for every e > 0 and Borel
set E with v(EF)> 0, there exist a g E G and a Borel set F such that v(F) > O,
F U gY C E and
dvogO()-A<e
dv-
for all w E F.
Remark 2.3. The ratio set r(G) depends only on the quasi-equivalence class of the
measure v; see [HO, ?1-3, Lemma 14]. It also depends only on the full group in the
sense that
[H] = [G]
r(H) = r(G).
The following result will be applied in the special case where G = F. However,
since the simple transitivity of the action doesn't play a role in the proof, we can
state it in greater generality.
Proposition 2.4. Let G be a countable subgroup of Aut(T) < Aut(Q). Suppose
there exist an element g E G such that d(ge, e) = 1 and a subgroupK of [G]o whose
action on Q is ergodic. Then
r(G) = {nk :k E Z} U {O}.
Proof. By Remark 2.3, it is sufficient to prove the statement for some group H such
that [H] = [G]. In particular, since [G] = [(G, K)] for any subgroup K of [G]o, we
may assume without loss of generality that K < G.
By [FTN, Chapter II, part 1)], for each g E G and w E Q we have
dvog () E n k E Z} U {O
dv
FACTORS
2053
FROM TREES
Since G acts ergodically on Q, r(G) \ {O} is a group. It is therefore enough to show
that n E r(G). Write x = ge and note that vx = vog-1. By [FTN, Chapter II, part
1)] we have
dvx (w) = n, for all w E QeX
(1)
~~~~dv
Let E C Q be a Borel set with v(EF)> 0. By the ergodicity of K, there exist
ki, k2 E K such that the set
JF= {w E E: k1w E Qx and k2g-1klw E
has positive measure.
Finally, let t = k2g-1ki E G. By construction, F U tY
is measure-preserving,
E}
C
E. Moreover, since K
dvot
dvog-1
dv,,
(k1w)
(k1w) = n for all w EF
=d
dv(W)=
dv
dv
dv
~
by (1), since k1 E Qe. This proves n E r(G), as required.
El
Corollary 2.5. If, in addition to the hypotheses for Proposition 2.4, the action of
G is free, then L??(Q) >i G is a factor of type 11l/n.
Proof. Having determined the ratio set, this is immediate from [Cl, Corollaire 3.3.4].
L
Thus, if we can find a countable subgroup K < [F]o whose action on Q is ergodic
we will have shown that L? (Q) >i r is a factor of type I11/n. To this end, we prove
the following sufficiency condition for ergodicity.
Lemma 2.6. Let K be group which acts on Q. If K acts transitively on the collection of sets {QX: x E F, lxl = m} for each natural number m, then K acts ergodically on Q.
Proof. Suppose that Xo C Q is a Borel set which is invariant under K and such that
v(Xo) > 0. We show that this necessarily implies v(Q \ Xo) = 0, thus establishing
the ergodicity of the action.
Define a new measure ,t on Q by ,u(X) = v(X n Xo) for each Borel set X C Q.
Now, for each g E K,
,u(gX)
=
=
v(gX n Xo) = v(X n g-1X)
v(Xn X)
=
Au(X),
and therefore ,u is K-invariant. Since K acts transitively on the basic open sets Qx
associated to words x of length m this implies that
,t(QX) = H(QY)
whenever lxI= lyl Since Q is the union of Nm disjoint sets QX, x E vm, each of
which has equal measure with respect to ,u, we deduce that
,H(QX)
for each x E Vm, where c
x E F.
=
= ,a(Xo) = v(Xo)
C
> 0. Thus
,a(QX)
= cv(QX)
for every
2054
JACQUI RAMAGGE
AND GUYAN ROBERTSON
Since the sets Qx, x E r generate the Borel o-algebra, we deduce that ,u(X)
cv(X) for each Borel set X. Therefore
v(Q \ XO)
=
c-1,u(Q \ XO)
=
c-1v((Q\xo)nlxo)=o,
=
thus proving ergodicity.
In the last of our technical results, we give a constructive proof of the existence
of a countable ergodic subgroup of [r]o.
Lemma 2.7. There is a countable ergodic group K < Aut(Q) such that K <
[r]o.
Proof. Let x, y E vyi. We construct a measure preserving automorphism kx,y of Q
such that
(1) kxly is almost everywhere a bijection from Qx onto QY,
(2) kxly is the identity on Q \ (Qx U QY).
It then follows from Lemma 2.6 that the group
K = (kx
{x, y} C Vm, m E N)
acts ergodically on Q and the construction will show explicitly that K < [r]o.
Fix x, y E Vm and suppose that we have reduced expressions x = xl ... xm, and
Y = Y1i..Ym
Define kx,y to be left multiplication by yx-v on each of the sets Qxz where Izl = 1
and z 0 {x-1, y 1}. Then kx,y is a measure preserving bijection from each such set
onto Qyz. If Ym= Xm then kx,y is now well defined everywhere on Qx.
Suppose now that Ym#7Xm. Then kx,y is defined on the set Qx \ QxYrn , which
it maps bijectively onto Qy \ QYXm'. Now define kx,y to be left multiplication by
on each of the sets Qxnyrnz where lzi = 1 and z 0 {Xm,Ym}. Then kxy
is a measure preserving bijection of each such Qx'Ymrz onto Q'YXmZ
yxrn ymx-'
Thus we have extended the domain of kx,y so that it is now defined on the set
Qx \ QxY'mlxm which it maps bijectively onto Q' \ Q9yx1 Ym.
Next define kxX to be left muliplication by yx-1ymxlymx-1
QxymlxnmZ where Izi = 1 and z 0 {xX1,y1Y}.
on the sets
Continue in this way. At the jth step kx,y is a measure preserving bijection from
Qx \ Xj onto Qy \ Yj where v(Xj) -* 0 as j -ox so that eventually kx,y is defined
almost everywhere on Q. Finally, define
kx,y(xyjXmYXmYmm...)
= YXmnYmXMnmYmXYm
.Y ..
thus defining kx,y everywhere on Q in such a way that its action is pointwise approximable by F almost everywhere. Hence
K = (kx{y
x, y} C Vm, m
E
N)
is a countable group with an ergodic measure-preserving action on Q and K
[r]o.
<
We are now in a position to prove our main result.
Theorem 2.8. The von Neumann algebra L??(Q) xr is the hyperfinite factor of
type III /n
FACTORS
FROM TREES
2055
Proof. By applying Corollary 2.5 with G = r, g E r any generator of r, and K as
in Lemma 2.7 we conclude that LI?(Q) x r is a factor of type III,/,.
To see that the factor is hyperfinite simply note that the action of r is amenable
as a result of [A, Theorem 5.1]. We refer to [C2, Theorem 4.4.1] for the uniqueness
?
of the hyperfinite factor of type III,/,.
Remark 2.9. In [Spl], Spielberg constructs III, factor states on the algebra 02.
The reduced C*-algebra C(Q) X r Fis a Cuntz-Krieger algebra OA by [Sp2]. What
we have done is construct a type III,/, factor state on some of these algebras OA*
Remark 2.10. From [C2, p. 476], we know that if r = Q X Q* acts naturally on 1p,
then the crossed product L0?(Qp) x r is the hyperfinite factor of type III,/p. This
may be proved geometrically as above by regarding the the boundary of the homogeneous tree of degree p + 1 as the one point compactification of Q?pas in [CKW].
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DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NEWCASTLE, CALLAGHAN, NEW SOUTH WALES
2308, AUSTRALIA
E-mail address: jacquiOmaths .newcastle.
E-mail address: guyanOmaths
. newcastle.
edu. au
edu. au