3 MÀ? vời © Copyright by mcrest, 1986
ON THE COMPUTATION OF INVARIANTS FOR ITPFI FACTORS
T GIORDANO, G SKANDALIS, E J WOODS
Krieger’s theorem [11] states, in part, that the flow of weights considered as a mapping from type II, Krieger factors (see terminology) with algebraic isomorphism as the equivalence relation to strictly ergodic flows with conjugacy as the equivalence relation, is one-to-one and onto between equivalence classes The simplest flows are the pure point spectrum flows ([{5]) The corresponding Krieger factors are known to be ITPF1 (4]), and the motivation for the present work was to obtain explicit eigenvalue list constructions of these factors
This problem leads naturally to the construction of Section 1 where we introduce an invariant @(M, T) (see below) which can be computed much more easily than the flow of weights, and seems to be very useful The main result of this section is Theorem 1.10 which is basic for Section 2 and is also used in Sections 3 and 4 This invariant can be understood in terms of the flow of weights as follows (see Remark 1.11) Let M be a factor, (Q, P, F,) its flow of weights, and 7 a subgroup of the Connes invariant T(M) (which is also} the L°-point spectrum of (Q, P, #2) Let (foloer be a multiplicative choice of eigenfunctions of (Q, P, F,) This gives a map f: 27> T given by <f(@), 0> = fo(m) The measure /(P) defines a certain equi- valence class @(M, T) of measures on T (see Proposition 1.2) The relation with the original problem is as follows Let Mf be a Krieger factor and take 7 = T(M) Then the flow of weights will be a pure point spectrum flow iff the map fis essen- tially injective and the Haar measure on T belongs to @(M, T)
Trang 2uncount-84 T GIORDANO, G SKANDALIS, £.3 WOODS able T(M) (Remark 3.11) Finally, we give an outline of the proof that the same situation holds if 4 is replaced by any countable subgroup of the rationals (Remark 3.10, see also [8])
The best-known fiow is perhaps the Kronecker flow (the flow built over an irrational rotation under a constant ceiling function) In Section 4 we construct a family of ITPFI, factors M-: M(L,, 7,) We prove that if the L, are large enough but not too large, then T(M) is the point spectrum of a Kronecker flow (Proposi- tion 4.1) We give a condition that @(M, T(M)) is the Haar measure class on 7(Al)* However, we are unable to carry out the ergodic decomposition involved in construct- ing the flow of weights from the eigenvalue list for M, which is required to show that fis essentially injective It seems, so far, that only in very special cases one has succeeded in computing an ergodic decomposition However, our investigations did Jead to a number of other interesting results on Krieger factors (sce also [6}, [7]) In Section 2 we use Theorem 1.10 to construct an ITPFI factor M which is not a tensor square (Theorem 2.1) Since every ITPFI of bounded type is an IT PFI, ([6), Theorem 2.1) and hence a tensor square, M is not of bounded type (Coro!- lary 2.3)
The two first-named authors would like to thank E J Woods for his kind invitation to Queen’s University which made this reasearch possible They would also like to express their appreciation to all ihe faculty and staff of the Department of Mathematics and Statistics who contributed to making their stay as pleasant as possible especially Professors P Ribenboim, E Weimar-Woods, and Dr M Khoshkam
We wouid like io thank T Hamachi and M Osikawa for a private communi- cation outlining their results
Notation All the definitions and notation can easily be found in the litera- ture (for instance in [14]) However, let us recall the definitions which are frequentiy used,
DEFINITIONS 1 If g is a normal semi-finite faithful weight on a factor M, then C, denotes the center of the centralizer My == {x © M; of(x)=:x, te R} of — (123,
Chapter 10)
2 A Krieger factor is the crossed product of an abelian von Neumann algebra by an ergodic automorphism
eo
3 A factor M is called an IT PFI if it is of the form M:.: @ (Mi (C), Qi)
&: 1 `
acting on the Hibert spacc 63 (H,, ế,) where Mi, (C) denotes the algebra of i, <1,
k: 1
Trang 3INVARIANTS FOR ITPFI FACTORS 85 4 Tf all the », are bounded by a number nv, M is said to be an ITPFi of bound- ed type
5 Hf all the 7, are equal to 2, M is said to be an ITPFI,
wa 1
6 Let (01, )u>1 be a sequence of states on M,(C) with eigenvalues | Dea › ov, vụ
A ¬
Ta = 7 } ; 0< 4„ < 1 and (L¿)¿>¡ be a sequence oŸ positive integers
Then M(L,, 1,) =: ® (M,C), 2.)°”* denotes the ITPFI, factor correspond-
n>l „
ing to (L,, 4,)n>1-
Finally, O(1) denotes the multiplicative group of complex numbers of modulus i If x is a non-negative real number, [x] stands for its integral part
i THE INVARIANT ¢(M, T)
Let M be a factor, T a subgroup of T(M) While the construction of @(M, T) can be done from the flow of weights (see Remark 1.11), we adopt a somewhat different approach Let @ be a normal semi-finite faithful weight on M If 0 ¢ T(M) there exists a unitary ; oÊ Cự, unique up to a scalar, with of = Adu Let A,(T) be the (abelian) C*-subalgebra (of C,), generated by (uy, 0 €T}
Write A,(T) == C(X,), where X, is the compact space of characters of A,(T).- 1.1 Lemma 1) If x, y¢Xq, then the map f,,,:T -» U(1) defined by f, (0) = z+ Hạ, XS~1Cwạ, y>, is a character of T
2) The map f,: X_ > T given by f(y) = fy Is continuous and injective
3) Uf 0%, == 1, i.e — is periodic, then the image of f, is contained in (T/Z0,)*” ==
= (ET; Cx, 0) = I}
Proof I) Tf AEC, |Al = 1, then (dup, x7 Atty, YY = Hạ, xÈ~1<wạ, yÈ
This shows that f., is well-defined
2) If f(y) = f.2), then (uy, y> = (ug, z> As the up's generate A,(7), the cha- racters vy and z coincide
3) If 66 == 1, then f, (0) = 1
The representation of M in the separable Hilbert space H restricts to a repre- sentation of A,(7) and yields a class 9, of measures on X,
1.2 PRoposiTIon, a) Let & be the equivalence relation on the space of proba-
“a
Trang 486 T GIORDANO, G SKANDALI, E.J WOODS
Then the equivalence class €(T) under Ø of {{j) does not depend on the choices of x in X, and of the probability measure win Fy
b) Let (ue)eer be a choice of unitaries as above satisfying ug 9' -: tytiy’ (0, 8 in T) Let « be a faithful normal state on M Then there exists a probability measure “C= WT, @, a, u)) on ? , whose Fourier-Stieltjes transform is Jñ() == a(u9), and
LET)
Proof a) follows from the equality /2(Z) — f y + £,(2) (the group T is written additively)
b) Note that the choice of an x in X, determines a multiplicative choice of u,, taking (uw, x> =: 1 Also two multiplicative choices of u,’s differ by an element of Ÿ The restriction of the state « to A,(T) determines a probability measure y on X, whose class is in &, If the choice of the #¿'s is given by x, we have
hoy (@)-= V0 dQ) = \ Gụ, về dvQ) = a(ty): a
Xx oO
1.3 Remarks 1 Let @ be a normal semifinite faithful weight on AZ and let « be a normal faithful state Let T be a subgroup of T(M) and let (va)eer be a multi- plicative choice of unitaries as above Let t be an automorphism of MM By [2], Lemma 1.2.10, t-(ug) is a multiplicative choice of unitaries corresponding te pet The equality a(uy) = oo t(t~Xug)) shows that ©,(T) =: ©, AT)
2 If 7 is a positive real number, a7” =: of Therefore @,,(T): - %,(T) 3 Let T’ S T be two subgroups of T(M) and i: T’ + T be the inclusion
Then A,(7’) < A,(T) Therefore we get a surjective map 2: X7 > X23’ Moreover, for every x,y eX, we have fo fl, = f%p,ayy- Hence the class @,(7") is equal to
(€,(7))
Let now M and N be two factors Let g, # be normal semifinite, faithful weights on M and N respectively Let T be a subgroup of T(M) n T(N)
1.4 PROPOSITION If is in ©,(T) and v is in @,(T), then pv is in Cog (T) (and therefore determines this class)
Proof Let (ug)ger, (Ygoer be multiplicative choices of unitaries of C, and C,
satisfying of = Adu, and of == Advy Let œ and 8 be normal faithful states on M and W As ø @ B(ug ® v9) = (up) B(vp) the result follows from Proposition 1.2.b)
Let @ be the weight on P(L*(R)) given by œ(x) = Trace(øx) x e.Z(*(R)).,
Trang 5INVARIANTS FOR ITPFI FACTORS 87
Let w, be the weight on Y(¢*(Z)) given by w(x) = Trace(p-x) xe L(¢*(Z))., where pz is defined by p2(e,) == e7"*e, ((EnJnez denotes the canonical basis of 7°( Z)) If o, w are normal, semi-finite, faithful weights, with of,,¢ = øš„¿ = 1 then there exists ¢€[0, ¢) such that e'’? @ w, and W @ a, are unitarily equivalent (t is such that e? =: (Dự: Dø)s„,¿) (cf [2], Lemma 1.2.5; [3]; [4], Section 5)
1.5 DEFINITION Let M be a factor
a) We denote by @(M, T) the equivalence class @y@.u({T)
b) We denote by @{M, T) the equivalence class €pe0,(T), where 2n/é € T and of,,~¢ = 1
One computes @(M, T) and @(M, T) using Proposition 1.4 and:
1.6 PROPOSITION a) Let h: R > T be given by <h,, 0> =e for teR and 0¢€T Then if mis a probability measure in R equivalent to the Lebesgue measure, h(m)€ @,(7) and therefore determines this class
b) Let Hz: Z T be given by Hn) = hyg Then if mis a probability measure on Z with support Z then Hn) ¢ €, AT) and therefore determines this class
Proof a) Let a be a faithful state on Y(L*(R)) given such that a(g) = = eo dmx) for all g in Z°(R) considered as a multiplication operator Let V, be the multiplication operator by e'™), We have o? = Ad V, If 0 € T, we get
u(V) = \c am = ‹h,, 03 dơn(/) = Ky, 83 đhứm)()
R az
b) is proved similarly ZB
1.7 COROLLARY a) Let o be a faithful, normal state on M Let u = (Up)yer be a multiplicative choice of unitaries of C, with Adu y = of Let f be a strictly positive function on R of Lebesgue integral 1 Then there exists a probability measure u(= UT, @, u, f)) on T whose Fourier-Stieltjes transform is () = @(0,)- Ñ— 0) and pe @(M, T)
b) Let 9 be a faithful, normal state on M with ofn, == 1 Let u = (u)ger be a multiplicative choice of unitaries as above with Usn, = 1 Let f be a strictly positive Junction on Z with sum 1 Then there exists a probability measure u (= pT, @
u, f)) on T whose Fourier-Stieltjes transform is
fi) = glue) and we @{M, T) 2
Trang 688 T GIORDANO, G SKANDALIS, E.J WOODS
Let Ve £(¢*(Z)) be given by Uf(u) = fin + 1) Let ¢ be a normal, semi-finite, faithful weight on M with of.,2 = 1 The flow of weights of M is built over the base transformation S corresponding to the restriction of Ad(1 @ U) to Cyan : and under the constant ceiling function € ({3]; [4], § 4)
1.8 REMARKS a) The equality (1 @ U,) (up @ Vo) (lL @ U,)-!: : e2 @ Vạ) shows that the restriction of the flow F, to 4¿eø(7) is given by translation by h, (Proposition 1.6 a)) In particular, if p¢@(M, T), it is (quasi-invariant and) ergodic under the action of R by addition of A, ({3], II, Theorem 3.1)
b) If g isa periodic weight of period 2z/é, the restriction of the transformation S to Ageo{T) is given by addition of H, (Proposition 1.6 b)) In particular if he @(M, T), it is H.-(quasi-invariant and) ergodic
c) It is useful to get rid of the term fle-**) in Corollary 1.7 b) Let pe @,(7) Let m be a probability measure on Z with support Z By Proposition 1.4, 4 * Hm) € € Cge0AT) = ©(M, T) Moreover p < «+ H{m)
Let us recall that a measure v in 7 (not necessarily H,-quasi-invariant) is
said to be H e-ergodic if for every H;-invariant Borel subset E of f, v(E)::0
or v(7\E) = 0
Let 2’ be the equivalence relation on the set of H.-ergodic probability measures on T given by: 1; A@'py iff there exists ye T such that 6, * fy and py are not mutually singular Note that with the notations of Proposition 1.2, nA’ po iff Hy * H(m)B py, * Hm) We can look at @{M, T) as the equivalence class under #' of w, where pe@,(T) satisfies (0) = p(w)
We now come to the case of ITPFI factors We need the following :
1.9 LEMMA, Let yt be a probability measure on T which is H z-quasi-invariant, (ergodic and) approximately transitive ((4]) Then for all probability measures yw on T with H” %4 wand for all sequences 0, ¢ T with lim e 5Š == | we have
lim (2(6,) — fi'(8,)) = 0 is OO
Proof Let ¢ > 0 As p is approximately transitive, there exist probability measures v, v’ carried by ZH c T and Ho < # Such that | nạ + vì <e/14 and
Ile! — Hạ # v⁄|| < #4
As 9 and ÿ“ are continuous functions on R and periodic of period 2n/é, there exists N such that » > N implies [%(6,) — 1] < Â/4 and jƠ(@,) - 1] < e/4 If
n> N, we have:
IẬ(,) — Â'(6,)| < IÃ(0,) — (uạ + v)^(6,) + lâs(0,)(Ÿ(0,) — 3'(6,))| +
^ ^ &€ é €
Trang 7INVARIANTS FOR ITPFL FACTORS 89
Let M be an ITPFI factor and 7 © T(M) Let ¢ be areal number with 2x/é € T Let pe@(M, 7) By Remark 1.8 b), the transformation (T, Lt H,) is a factor of a base transformation (B, v, S) over which the flow of weights of AZ is constructed under the constant ceiling function € Using then Theorem 8.3, Lemma 2.5 and Remark 2.4 of [4], we get that (T, H, H,) is approximately transitive
1.10 THEOREM Let M be an \TPFI factor and let T be a subgroup of T(M) Let € be areal number with 2n/6¢T Let p and & be two normal, faithful, periodic states on M with period 2n/& and let (Ug)oer, (Voc r be unitaries of M (tg EC, , 0ạc Cụ) with of = Adu, of = Ad vg Then for every sequence (0,)n>1 with 0,¢T and
lim e “Ẽ = 1, we have lim (Jp(uo,)| — |W(va,)|) = 0
Proof We may assume that the choices #g and vg are multiplicative, and s„/; >> == Vaaye = 1 By Proposition 1.2 b), there exist measures pp € @,(T) and ve @,(T) with Â(Ø) = @@¿), 9(0) = W(v,) By Remark 1.8 c), there exist measures pi’, v' € e@(M, T)with < p' and v < ví As p’Bv’, there exists x € T with bye pon~ Therefore 6, * u < v’ As v’ is H.-approximately transitive, we get lim (9,) —
—- (ổy # /)ˆ(0,)) =0 and lim (3/(0,) — 3(6,)) == 0 (Lemma 1.9) The result follows from the equality |(ð; + ¿)^(Ø)| == |0(0)|
L.A RemarK a) The invariants @(M, T) and @.(M, T) can be presented in the following way:
Let (Q, P, F,) be an ergodic flow Let T © R be a subgroup of its L”-point spectrum For all Ø e7, let gạc L°(Q, P), |g9|=1 such that go F, = egy for all t in R Let ~ be a character of the von Neumann algebra L°(Q, P) Put fy == x(g9)7*Z0- We then have for all 0 and @’ in T, fg- fo == fo+o' (cf also [5], Chapter 12) Let now
ƒ:©O — Ï be given by (f(a), 09> = fy(w)
If Af is a factor of type III and (Q, P, F,) is its flow of weights, then the mea- sure f{(P) belongs to the class @(M, T) and therefore determines this class
Assume that F, is constructed over the base transformation (B, v, S) under the constant ceiling function € Let T’ S U(1) be a subgroup of the point spectrum
of S
Let (f)uer be a multiplicative choice of eigenfunctions for S of modulus 1
Let T= {0 ¢ R; ee T’} Define g: B+ T by <g(b), 0> = ƒsoqso(b) Then
the measure g(v) belongs to the class @(M, 7) and therefore determines this class
Trang 890 T GIORDANO, H SKANDALIS, E.J WOODS
2, AN ITPFL FACTOR WHICH IS NOT A TENSOR SQUARE
We use here the results of Section | to construct an ITPFI factor M4 which is not a tensor square As every ITPFI of bounded type is an ITPFI, ((6]) and hence a tensor square, M is not of bounded type (Corollary 2.2)
Let (p,),>1 be a sequence of positive integer multiples of 8, (for instance p, - = 8, k > 1) Let 9, = [] Px-
kal
Forn 2 1, let @, = - Tr(h,-+) be the state on M, ,s„(C)› where h, is diagonal
q
and has coefficients: 1/2 with multiplicity 1 and 27% -* with multiplicity 2 ” 2.1 THEOREM The factor M = @ (MỸ 44, (C); @,) is not a tensor square
n>1 1
Proof Let T:: {0¢R; 2% —1 for some ø > 1} For 0e7 and ø> 1
q +2 n
let „ạ„ be the unitary in Mt AO given by up, = (2° h,)® We have: a,” ^ == Aduy, lf 06T and ø is large enough, uy, — Ì Set ứs = @ „€Äƒ and
n>1
@- © @, We have of = Adu, Hence TS T(A)
z>1
We will show that there exists no probability measure v on ?, such that www€ togs(M, 7); Proposition 1.4 will then imply that M is not a tensor square Let mạ be the probability measure on ?, whose Fourier-Stieltjes coefficients are fi9(0) = (uy) If 1 € Croge(M, T), then by Remark 1.8 c) and by the definition of the equivalence relation # (1.2), there exists ye T such that Họ < ð„* By Theorem 1.10, we get that lim ((2,(0,)' — !f(6,)!) = 0, for every sequence 0, € 7,
n¬oo
with 0, >0 Let 0 — _
qn Log 2 , Je 4% Then @e7 and
Rr nm Ti
ole) = TT oe) =H cos ( i) ke-1 k=1 an
In particular, if
T n rất,
Ủy ==—————, o(u,) = cos (3) = 0
In Log2 °n H 2n
if Ø,— -——— and k <nu—i, 2q, Log2
@,(Ugr,) = Cos Fa > cos lá
Trang 9FNVARIANTS FOR ITPFI FACTORS 9 Hencc 4-00 2 8-3 sz] — _ Fa 2016 n-l T HH 22) > 1—- kesh ” 3 vie so that ] TẾ? 7 (0) >— [1 —————| >——- pứu,) Al sie) 10
If v * ve Croga(M, T), then for n large enough, |5(0,)| < 1/10 and |?(2)J* >
> 6/10 As v is a positive measure, > is positive definite and the matrix
9q) 9(6,) V(r)
^
90) 9(0,—0) — 3()
is hermitian positive But as (1) = ]
đet 4 = 1 + 2 Re(9(0,)9(—6;)®) — 215@,)? — 19@,)? < 0 Z
2.2 COROLLARY The ITPFI factor M is not of bounded type
Proof By Proposition 1.1 of [6], every ITPFI, factor can be written in the form N:+M(Lx, 4,), with Jy 4, < oo, Put L, = l2 | Then we have: M(L;, 4,)®@° = N
k>1
Since every ITPFI of bounded type is an ITPFI, ((6], Theorem 2.1), the ‘result
follows BZ
2.3 REMARK Using exactly the same proof, we can show that for every p 2 2, the ITPFI factor M is not a p™ tensor power It can also be seen that M@ M
is not a p** power, if p > 3 (by the same argument!)
A natural invariant appears to be R(M) = {pe N\ {0}; there exists N, with
N®? = M} If M is an ITPFI, factor, then R(M) = N\{0}
3 AN EXAMPLE OF HAMACHI-OSIKAWA
Trang 1092 T GIORDANO, G SKANDALIS, E.5 WOODS
Let % be a real number in (0,1) In this section we take /, :-: 22k > 0 und consider type If) ITPFI, factors M = @ (M,(C), 9,)°”* =: M(L,, 7,) (cf notation)
k>0
kK
Let 4 -= {(0ER; 2° =: 1, for some ke N} Let 0 € A As ơpt <= | for A large enough, we have 4 & T(M) ([2], Théoréme 1.3.7 (a))
Let (Q, ”) == JE ({0.1, ., £3, w,), where yy, is the measure on {0,1, 4,3
k>0
L,! Ad
J1, — j)! ( +4} ”*
support Z Let # be the equivalence relation on (Q x Z, px B), given by (o, )A(w', m) iff wm, =), for all but finitely many k’s and
given by u,(j) =: Let B be a probability measure on Z, with
3 (@, -~ @,) 2% = m — n
Let (B, v): (Qx Z, x P)/@ (this quotient stands for the ergodic decomposition) Let S be the transformation of (B, v) induced by the addition of 1 (in Z) on (2x2, nxƒ)
The flow of weights of Af is built over the base transformation (B, v, S) under the (constant) ceiling function —LogA ({3], Corollary 11.6.4; cf also Appendix of [6))
Let Wo:Q > Z, be given by (mw) = a((@;)¿>¡) = 3 œ,2* and :B ¬ Z4 be mnduced by the map (œ, øØ) c> Ứog(0) n from @x< Z to Ze
The main feature of this example is coming from:
3.1 THEOREM The map Ứ: ÐB — Z¿, defined above, is essentially injective In particular, the flow of weights of M can be built over the base transformaticir (Zp, Wr), H) under the (constant) ceiling function —Log?, where H denotes the addition of \ in Za
For the proof, we need two straightforward technical lemmas Let p and ,’ be two probability measures on a standard Borel space X As in [10], we set
p(n, w” = | (du(x))!?{d,(x)}M2 =- ( du (x) ) ( đục cy)" dm(x),
2 x dim dm
where m is a measure on_X, with » < mand w’ < m
3.2 LeMMA Let p be a positive real number Let LEN, L 24, ke Z and €eR with 0 < € < (p+ 2)-2 Let pt, yp’ be the measures on Z given by
n_ — EI ẽ/
WD) = Ty a Oe
HG) = wi t+ &)
Trang 11INVARIANTS FOR ITPFI FACTORS 93
Lé
14-€
Proof We may assume k > 0 (interchanging if necessary the roles of ye and yi’)
wk +s BOL -j-—k +i)
3 p
Leto= be the standard deviation of wu If \k| S pø, then p(H, ")>(3/4)e ”
We have = and uG) H jt+i ke j k jk +i k _; ty (ĐÈ Si
tog MELD frog MERI= EHD § Em DOF — hE
HG) ih jt+i S E-G-E+k—ié
{_ a _a-—-b LỆ
(si the inequality Log— > } where E = ~ If j<£ +20 and
b a 1+ é
& < po, then
> @(@-7jq0+ — K-96 i 5 Š (E—=jd+—k 4 E—(2+p)éo “ 4 E—-O+pjø Log MED HG) Hence, _7/2 u(k 4 ) > ex (— _ u()1? 2(E — 2 + p)šø) ke \( „1 Œ£-jq +2) "2 E—(2+p)éo) Note that i - [£+-26]
0=¥Y E-DeV< Y (E—-AuO)
¿0 j~0
Therefore,
pC, tt’) >Š "Hee ; nye + i” HG) > uŒ0.1, , [E +- 2ø]}) “ ie
2(E—(2 p)éo)
By Tchebyshev’s inequality we have g({O,l, , [E + 27ø]}) > 3/4 As
1/8
E::(1 -+ §)ơ3* and as 3 Poe ễÝ2+? < ) by assumption, the above
ol +8 Ve YL
inequality reads p(y, np’) > 3/4 exp, — ————— and as k < po
20°(1 — Vc) (+Ø VL
1
and } — 2 1/2, we get the result Vz | 8 r2 ⁄
3.3 LemMMA Let nạ, tạ be probability measures on a Borel space X with pty fo) = « > 0 Let C,, Cy be Borel subsets of X If u(C)) > 1 — 93/4, j = 1,2,
Trang 1294 T GIORDANO, G SKANDALIS, E.J WOODS
Proof Let m= mị +- Hạ The Cauchy-Schwarz inequality gives
(P6902 69] 4m6) < ulema <3 = 12) dm 2
3
Hence C‡ýU C§ # X @
Proof of Theorem 3.1 Since M is of type IIL and Md, 4,) is type Tao, we have M=M @ Ä(1, 2¿)Ä(L¿ 1,24) Hence we may assume that L,>1 form all & Then for a.e (@, n)<¢Q xX Z, there exists w’eQ such that (w@,n)#(o'.0) Thus it suffices to prove that the map @: 2/R’ > Z, is essentially injective where is the map induced by W, and #’ is the equivalence relation on Ø given by œ.'øŸ iff (@, 0)2(@’, 0) (i.e w, = + @;, for all but finitely many k's and J) (@, — w;,) 2° - 0)
k>0
Let «/, be the o-algebra on Q generated by the m,, K:= 1, .,2—1 and let
wh ~-N ob, Ve ge LQ, Sf, 1), then g-= lim E%*(g) where E“»(g) is the conditional
expectation of g with respect to «/, Let #, denote the ø-algebra on Q generated
—1
by X„, Ss @,2* and w,, k 3 n Let #=:TG, Let 9, denote the o-algebra on
kes n
Q, generated by X, modulo 2”, and let @ =- VQ, Note that > 4,>G,, that if f¢ LQ, &, w) is invariant with respect to #’, then fe L1(Q, 4, «), and that & is the
ø-algebra on Ø generated by the map ¿ Thus the problem is to show that Zand coincide
To prove this, let ¢ > 0 fe LQ, Z, /Ò, Iƒ loa < 1 Then it suffices to show that there exists some m < oo and fy ¢ L(Q, Z,,, «) such that ff — fol < e
Since EZ2(ƒ) is measurable with respect to 4,, there exists a function g on
n~l
loi > rat | such that £Z“»(ƒ):-gø-X„ (In the following we will consi-
der g as a function on Z.)
Choose W < co such that for all? > N JEM f) — Sila < &
.32 n1 Ls
where € oy £ sxp[—- } Since o°(X,) > Yh ——
2 8 ko (Ì + 4#)” - 2**, we have
Lyi
2-#"g3(X„ )> Vo it = ©
% 2502}
By the ratio test there exist infinitely many 7 such that
Trang 13INVARIANTS FOR 1TPFI FACTORS 9S
which gives 22"07(w,) == ara 20°(X,) Let p = 46-12 Choose m2 WN such
*H
that equation (+) is satisfied and 4,, < (p + 2)-7 and L,, > 4
Write E%mn(f) == g0X,4, and E®m(f)=hoX,, Let P, = X,(u) be the
distribution of the random variable X,, and let yx, be the binomial distribution
L,! 3;
————— ————: j=U,l, , ự„
(Œ„ —/)17! +2)”
Half) =
Note that P,., = Đạ# Hy We have:
2e, > ||EZm(ƒ) — E ss(ƒ)||y = \\ Jax) — g(x + 2" /dPy(x) dptg( A):
246,
Let 4 = {x eZ; (ics) — gŒ + 2“/J| dự) <
For xe A4, let Ð, == {7c {0, , F„}; lh(x) -— g(x + 2"7)| <6/8} We have
24a 8 9 32
(Bx) >il— +L — l —- —€@x (- — m,
6 é 64 P
Then P,,(A) > 1 — 6/12
Let x,x’eA with x — x' = k2" where & is an integer < po (> == 4e~1/5
Vind
1 + An
tm(7) = H„(k +7) By Lemma 3.2 we have
Ø = đ(@„) =¬ } Let yj, be the probability measure on Z given by
3 2 3
Ming Hạ ) =: 2——€ TP e~19,
pC m 2 4
As „CB — kh) = H„(B,) > 1— 31/4 and u,(B) > L— 24 we get Bn (By — 1) #O (Lemma 3.3) Hence there exists j such that |A(x) — g(x +- 2”/)| < <ef8 and |ñ(x) g(x’ + 2"%( 7 +4))| < 2/8 Therefore |h(x) — h(x)| <S @/4 Let A’ == An {xe Z;|x — E(X,,)| < 2"~"po} For every class d of inte- gers modulo 2”, such that dn A’ 4O, choose x, in this intersection and put go(d) = == h(x,) If this intersection is empty, put g,(¢@) = 0 For xe Z we put f(x) = = 29(d) where d is the class of x modulo 2”
If xe A’, then |ñ(x) — hạ@)' < &/4,
As o°(X,,) < 2"o?, we get using Tchebyshev’s inequality: P,(A’) 2 1 — — 6/12 —4/p? > 1 — e/3 Hence
2e g it
Trang 1496 T GIORDANO, G SKANDALIS, E.J WOODS
Put fo == Ig- X,, Note that fy is measurable with respect to @,, and
lif /ala<|lfe h* Xnhh + l|hcX„ — hạ s Xnhh < 6 bã
Although the flow of weights of is, for all choices of L,’s, given by rotation in A, T(M) can be larger than A We next give a necessary and sufficient condition to have 7(M) -: A
Note that every x <[0,l] admits a decompositionx = JY} (127% where
0<7/<P
Ciosjcp, is an increasing sequence of nonnegative integers (p,¢NUu { - co}) 3.4 Proposition For n 21, let V,= 3 a nn =: ơ3(2~"X,)
ke:0 TABS
a) Jf liminf V, > 0, then T(M) = A
b) 7 lim inf V, == 0, then T(M) is uncountable
Proof Recall that TS T(M) HĐ Š) L,À@Ì — cos 2*0) < co ([2], Corol-
OE“ k>o
laire 1.3.9)
a) Let x be in [0,1] Write x:+ Yj (—1)/2 7 Note that the closest inte- 0</<P,
aml;
ger to 2’xvis ce”? = (—927 * Put for convenience /_,=0 If §_.<a<J;,
ten
#ø—Ï,—1 n-l
we have 2 4 <!2’x ¢,'<2 4 As for i¢| <1/4, | — cos 2nr>16r*, we get:
Trang 15INVARIANTS FOR ITPEI FACTORS 97
.„ 2X
| —= ——€ †(# ;
By (1), if Logi e 7(M), then yy 4; < oo Therefore Vi, < oo and as
O<j<p, 0<j<p,,
à Đà wo, 2nx
lim inf V, > 0, this sum has to be finite, ie —
n Log 2
b) IÝ lim inf „ = 0, let (/,);>¡ be increasing and Ứì,< 2° ]fx= ` g2 7?
J>1
e, € {0, 1}, then iŸ1;_¡ <Xk< /,, we have: 2*x — [2*x]<2***-4 and 1 — cos 2n(2*x) < ak+1-1.) < 2n?2?“+1~!j, Hence —1 YL — cos2x(24x)) < 8m8 ĐÓ L2 “7< Ke jel kel, /;¿—1 ˆ
<32n? Lib _ 2?Œ-I) 3972 ` tị < co jai kao (1 + A,)? ẤN 2
Therefore for all cholces of ø/s, ie ¢ T(M); hence, T(M) is uncountable og 2
3.5 REMARK IÝ x is a non-dyadic rational number, then there exists « > 0 such that d(2*x,Z) > aforallk > 0 (this follows from the periodicity of the dyadic expansion of x)
Let M= M(L,,4,) be a type IIL factor Then as }) L,4, d(2*x, Z) >
k>0
2m >ơ? Ý L,2, = co, -
& “ Log a
x ¢ T(M) Therefore the dyadic numbers are the onty Logi
rationals in 28" T(M) 2n
By Theorem 3.1 the isomorphism class of the factor M is completely deter- mined by the measure /(y) Our next goal is to find conditions for the measure (y) to be equivalent to the Haar measure For k 2 0 and 6€ 4, let uy, be the unitary
‘1 0
in M(C), wo = , | We have off = Aduy, Put U, = @ (io)? € M
0 2” ne
(note that uw»), = 1 for k large enough) Let ¢ be the state on M, 9 = @ ook, k>0 3.6 REMARK If @¢ A and xe Z,, then 4 has a natural meaning Using
2m Log A
look at the measure /(y) as a measure on 4 We have with the notations of Corol- lary 1.7 b) W(v) = w_ros ;(4, ọ, U, B) Therefore p(y) € @_ Log ,(M, 4) Theorem 3.1 asserts that @_ og ,(M, A) is a complete invariant
this pairing, Z, identifies with the subgroup (4 Z| of 4 We can now
Trang 1698 T GIORDANO, G SKANDALIS, E.J WOODS
By Remark 1.8 c) there exists a probability measure P on 4 (P =: WA, 9, 0, U) cf Proposition 1.2 b)) with P < @) and Ê@) =: @(U,) Note also that P: (jo (with the notations above Theorem 3.1)
j 2
3.7, LEMMA a) For x = J, put 0 = — = get
2 Log 2
fSh Tụ yk roe ae bk
¥Y > cos (202*x)) < — Logie(Uy), < %› ——— ¬¬ cos (2z2*x))
ceo (lL + Aw? kao | —
in 2
b) If x: n3 (—- 12_ ‘i , puto = — 2 Then Log 2 m —Log |p(U)>3 ¥ V1, “=0 ‡ —aiaF “x
Proof a) We have @,(uo,) = it = - As (Ø0 „)| > Re(@,(s„.)): =
*k
+ Rx a—b
= + 2co 22x we get, using the inequality Log — —
1 + Ay b a
ae ke
Logl@(¿ x)| > 4Œ — cos2x2 +) >— _— 4k -(Ï — cos 2n2*x) “a4 2,cos2nx 1— 4,
21
On the other hand, !2;(„;„)|Ê -= 1 — G2 a (1 — cos 272*x), so that
oo Ak
22
Log|y,(u j?€=— gl@x(Ha w)i a +i ( 1 — cos 2z2*x)
b) Let now x = s (—1}⁄ 271, If/_¡ € k </;, we have
j=—=0
9(k—/, kả†,
POWAY £1 — cos 2x2*x<2n?2?6~!), Now, ~— logio(U,)| = — ¥) £, Log|o,(up,)| Therefore,
k>0
mộc TT 2 2(~=l,+1)
—Loglœ(U,) = » > ee ŸẢ Œ_¡ =0)
Trang 17INVARIANTS FOR ITPFI FACTORS 99
ich
” Tư 2 2(k—1)
We put aj = ——— “— 2 +, We know by the proof of Proposition jũ + 4
3.4 that 2 Mi, < ¬ ý a, We hence get
jut
a at
— Log] p(Us)| 24 ¥ 4,23 3) V1, WS
j= 0 j=0
3.8 Proposriion (cf [8], Theorem 2) a) /f lim inf V,,< -+ co, then the measure W(v) is nor equivalent to the Haar measure on Z
b)ÿ en < +00, then the measure w(v) is equivalent to the Haar measure
n>1
of Z¿ and the flow oƒ weights of M has pure point spectrum
Proof Put D = A In By Remark 3.6, Ơc<identiRđes with Z; og
a) Let P be the measure on A with P < /@) and PO) = @(U,), (cf Remark 3.6) Hf w(v) is equivalent to the Haar measure of Z,, then (Ê(0));epe of — 2z2~ >
\ Log a
€C,(D) Therefore lim ÊÍ~ oe n ): =: 0 But by Lemma 3.7 (a),
HAO og iV
>e 44 " Hence lim V, += + co
m7
b) Let F.denote the set of finite subsets of N If A belongs to F, we set mya: 4 A—1 and we denote by lyg<lay < < Fam, the elements of A Put then 0, =: L in (y (— 12 ) For every clement 0 of D, there exist
98 =0
exactly two 4ˆs with Ø„ = 6 (e.g for 0 =0, 4= Ø or {0}; for 0= — — ia Logi A = {1} or {0, 1}) We therefore have
Š IÊ(0#P=—- 9 IÊW,J#< ÿ TỊ e
ged 2 AGF AGFIEGA
(using Lemma 3.7 b)) But by definition of F and by assumption, we have:
YWeow=Yatre™ < +o
AGF IEA n>1
Therefore, P e¢%D) and P is absolutely continuous with respect to the Haar measure of Z, As P < wt) and as both y(v) and the Haar measure are Z-quasi-invariant and ergodic, they are equivalent
Trang 18100 T GIORDANO, G SKANDALIS, E J WOODS
Using Theorem 3.1, we now get that the flow of weights of M has pure
point spectrum A
In particular if lim inf V, < + 00 the factors M([tL,], Avien® are pairwise
+
non isomorphic ([tZ,] is the integral part of tL, cf [7], Corollary 2.6) This is a partial converse to Theorem 3.1 of [6] Note that as condition C ([6], Definition 4.1) is not satisfied and we can not apply Proposition 4.4 of [6]
We now give another partial converse to Theorem 3.1 of [6]:
3.9 Proposition Lei L, and Lj, be two sequences of integers If \im supL,A, <-+ œ and if M =: M(L,,4.) and N= M(Li,, 4,) are isomorphic, then lim (LZ, — Lyd, -= 0
Proof Let f, f’, g and g’ be defined for k > 0 by
2D phy fi= 2D eg > = THẢ and gf =
fe Oe (1 2,2 1-4’, mm
By assumption ƒ, g c/%(N)
[— cosz2~*
Let øze/1{N) be given by 4 = 5 (k >0)
2n2~* Logi
to M and N (cf Lemma 3.7) The algebra “1(N) acts by convolution on the Banach space /*(N) Lemma 3.7 reads:
For 130, let 6, =: — and Up, U5, be the unitaries corresponding
(f # @,-1< — Loglo(Ue | <(g #4),—¡
q) and
Œ ws @)-1 < —LogløC0a DI < (g’ * Q)y—1-
As M and N are isomorphic, we get by Theorem 1.10,
lim (lø(Ua)\ — lø'(U/,)I) = 0
tì S2
As gxa e/%(N), we get lim sup(— Logl@(Ua,)) < +œo Hence lim inf |ø“(U2, =
R
2 HÀ <—Log|p(U; )|, we derive that f’ e/°(N)
(1 + A,)? wa
== lim inf {g(U, )| > 0 As and g’ « £™(N)
Now (g — f)¢C,(N) and (g’ — f.)€C,(N).We derive that (g — f)*ae €C,(N) and (g’ — f’)*aeC,(N) and by (1)
lim ((f * @),-1 + Log ig(Up)|) = 0 and lim ((f’ + a@),-1 + Log |g’(U; i) = 0
Trang 19INVARIANTS FOR ITPFI FACTORS 101 As lim (Log Jø(U,,)ì — Log lø (0,3) = 0, we have (f — f’)*a@eC,(N) Moreover as a@,_, = sin?22-* = 4a,cos?22-*-1 and a, = 1/2, we have
oo go
a 23a, Hence ¥ a,<- ` 3-* = 3/4 We get
8 k=l 2 ro
|œ — ôlh <3/4 < 1, where ð ¡s the unit of (N) and a is invertible in /(N) As €Œg(N)+/(N) c Cạ(N), we have ƒ — ƒ' = (ƒ — ƒ)xa+a~!c Cạ(N) for k 22, a,-, 2 4a, cos?
3.10 REMARK Let now (),),>1 be a sequence of integers p, > 2 Put
n
Qa = |[ Pee An = A”*, Let M = M(L,, A,) be the corresponding factor Let
Keel
also A = {Oe R; 1%? = 1 for some k ¢ N} & T(M) The whole section can be rewritten in this context We obtain:
1) The flow of weights of M is given by the action 4, of R (notations of Remark 1.8 (a)) on A (Theorem 3.1)
n-l r 2
2) With ¥, - "5, —"ee_ 9, we have: T(M) = A iff liminf V, > 0; if
xno (1 + Ay)? đã ,
T(M)# A, then 7(M) is uncountable (Proposition 3.4)
3) If lim inf V,, < 00, the Haar measure on 4 does not belong to @(M, A) If the sequence (V,),>, goes to infinity quickly enough, then the factor M has a pure point spectrum flow of weights (Proposition 3.8)
To have an estimate for the growth of V,, one may use a (weakened) ver- sion of Lemma 3.7:
j j j 2m)
For x =: 0 <j<g,and 2-= h*! #N, and 0 = — “—, we get
I Pa In Logs
n-1 L,A% (1 2 ) Log lø(U,)| n-1 HN q ‘9 )
a (Ì — C0S271đg,Xx)<—LO S ——„- — COS 27,3
» (1 + 4)? I ð129(0a 2» 1— A®% k
2m
and as — Log |(U )}< —Log |p(U,)|, where 0, == — Fleet’ we get — Log |g(U,)|=
Gn Log 4
>8V, Therefore, Š' |ø(U,)J°< W (4 — đ,—;)exp(— 16W,) WẲ this sum is 0<0<1 nel
0ET
finite, then as in Proposition 3.8, the Haar measure on 4 belongs to @(M, A) In particular if 0, is a real number and D isa subgroup of Q, then by (It)
and (3), the factors, whose flow of weights is pure point spectrum with point spectrum 0)D, are ITPFT, (cf [8])
4) Iflim sup = < oo and if M(L,, 4,) and M(L;, 4,) are isomorphic, k Prt
Trang 20102 T GIORDANO, G SKANDALIS, E.J WOODS
3.11 Remark The occurrence of uncountable 7(M) (for factors M acting on a separable Hilbert space) was somewhat unexpected (cf [13]) Proposition 3.4 and Remark 3.10.2 state that 7(M) is uncountable iff the multiplicities are not too large This suggests the following interpretation of:this phenomenon Among the type III factors, the type III, factors are ‘“‘closer’’ to the semifinite factors Similarly, among the type If[, factors, those with uncountable | T (M) are “closet” to the semifinite factors
4 ANOTHER EXAMPLE
Let 5 be an irrational number, 0 < 6 < I, and let (p,/9,),>, be the sequence of its best rational approximations (see [9], Chapter 10) Let 4 be a real number with 0 < 2 < 1 and (Z£,).>1 be a sequence of positive integers Let M == M(L,., zy be the associated ITPFI, factor
We first focus on Connes’ invariant 7(M) We know ([3], Theorem 1.3.7(b))
Qn
that for all sequences L,, - Te (M) Log?
2nb vy La
4.1 PROPOSITION a) We have a" eT™M) iff ¥ oo hn < + 00
Log 4 k>i đấy
b) Uf lim inf £,2°* “4 > 0, then TIM) © _ -(Z @ b2)
k Vics Log ¿
24
2nb
Proof a) By corollaire 1.3.9 of [2] we have Ty T(M) iff & L2 “(1 —
Og 4
1 1
— cos 2nbq,) < +00 But for all k 2 1, -—— <|bg, — p,} <— - Since for
Qr+1 W441
2 2r°
|fi < 1/2, 8?? < l — cos2mxí < 2m3, we get -—— <1 — cos 2nbq, — and
Weta W414 59k + L2 — cos2nbg,) < + 00 iff YA k>1 k>1 LH 2 2n0 Ty
b) Assume that lim inf [Ni fk > 0 if e T(M), then » K 4
Gir Log 2 Kol đấ+1
Hea (1 cos 2n0q,) < + co Therefore y- đến (I — cos 2x0qg,)< co In par-
Gi kết để
ticular, lim —— Gis (1 — cos 2x6q,) == 0 Let r, be the closest integer to 0g, (k > 1)
đ
We get lim Te (0g, — #,)% -= 0, ie lim Mee (0q, — r„) == 0, hence there exists
I Ut
K such that for k > K, Ut |Ủ@y — rạL < 1/4 Write đý¿y == 4/4, E đy—y (cf [9))
Trang 21INVARIANTS FOR ITPFI FACTORS 103
I€k2K + 1, we have O(a.q% + G1) — ple — Pa- ACOGg — 84) + OGn a — Peas %+! ang _
I đ.—:
and as a, < > 1, we have
Wang, + I-12) — Ogle — My-al < tery (Og, — el + l@yT~„ — rx-¡| <— — 1
Crs qị~1 2
Therefore, ay, +r,—-, is the closest integer to O0q,41, Le Pyar = đẹy -E Pgằ—Ð fork>K + 1 As the matrix | Px 4K is invertible in 14,(Z) (cf [9], Chapter
Pray Wat
r Pr q
10) there exist two integers m and n with| K | = | x x II then
Fư+i Pray Uqiih”
has by induction r, == mp, + nq,, for all k= K Taking the limit of wk, we get ứ:
0 := mồ +1-n Y
From now on we will assume that the two conditions of Proposition 4.1
1A? đ 2m
are satisfied, namel —z— < +: œand lim inf y2 %% Mk >0 Let 7=———'
7 Lf Gert Gv Log /
(Z ® bZ) (= T(M))
We want now tofind conditions for the Haar measure on T to be in @(M, T)
(Definition 1.5 a))
1 0 Hd ASk eM
For k>0, put Uy == Cy lo se, where Cụ = Ta ey Let Pr be
the state ø@ a,- We have
a
1 Vk 2mibg, | ' Ak anibq, 3%
gy) == A *S 1 + 4% „ Re + ¿re 1 + A% 51 — —44 1+2 _ 1 ~ cos 22bq,)
Lute @L
s ———— (1 — cos2nbq,) < 00, we know (ef [1]) that U = LÔ, (uy, k) k>1 1+”
makes sense in M = @ (MC), 9,)°** Moreover, ؇„„ro„; = AGU (as for all k,
k>1
Os Lona == Ad u, (ef [2], Lemma 1.3.8)) Also ofjjroga =: 1
By Remark 1.8.c), there exists a probability measure yt on T B= wT, o, @, VU) (cf Proposition 1.2 b)) in @,(7), with fu ( Ti (n+ øm)) = @(U”") The support
og 2
Trang 22104 T GIORDANO, G, SKANDAL1S, E, J WOODS
L2 đ
4.2 LEMMA For ke N put vo, = -
(+2*}* đu Me have
y lọ(U*jð<— ®Đ(—20) —_ n= [He Jaa [E]}e Ị exp ( nh) — _ 30
Proof We have '@(U")| < |øo,()l”* and
-} 1 2h inbg, 3%), loud) = T+ 2% = (: — mm pts (1 ++ 2%)? "_ 1/2 (1 — cos 2rnba.)) Hence 1 ( 22%
Log |g,(uZ)i 6 i0kW)¡ < — —- {| —————- (I — cos 2anb )} 2 La +“? ( %‹
n 7 c
AsS ———— </|bq,— p,| < ——— we get for * <n<—tetL,
24,41 Wk +1 mn > 2 4n* 2 2nq, 2 đị++ đị+a đị +: 1 — cos2mnbq¿—Ì — cos [ Say | 2 +œ 1 —2
Y woes SY exp (- An, ¬" ‹— PPC- 2 2
Ass fk +1 ne ¬ +1 q‹ 1 — exp| — k
° ° Vr
LA tk, 0, then the Haar measure m of
G+ 2) diay
G = {xeT; <x, 2n/Log2> = 1} belongs to © 15,2 (M, T)
4.3 PROPOSITION 7ƒ lim inf
k
Proof By Lemma 4.2, for k large enough,
A ( 2nb ) ? 2exp (—cq,)
H|n-———|I S——————
Log A}! 1 — exp(—2c)
Trang 23INVARIANTS FOR ITPEL FACTORS 105
where c is a positive constant As g, 2 k for all k,
a(n 22.) ifn 22
P“ ”Log2 x mỉ
Therefore » < m By Remark I.8 c), zøz belongs to @_1,, ,(M, T) ZB
2 2 =142Y5 n>1 < œo "e7
We now consider two factors M and N corresponding to the sequences (Ƒ„,2”*} and (L;, 2%*) We have:
_t -< + coand that q,4, >
4.4 PROPOSITION Assume that limsup L,A%
k đị+:
2
2 3q, for all k If M and N are isomorphic, then lim (L, — Lae = 0
K-00 đr+1
Proof Let 9 = @ (ø ®) and g’ = @ (o®9 be periodic states on M and N
k>1i A4 k k>1 ak
and let U and U’ be the corresponding unitaries of M and N (cf notations of Lemma 4.2) As in Lemma 3.7, one checks easily:
Ey (1 — cos 2nbq,q,) < — Log |p(U™)| < VV L2" (1 — cos2xb4,4„)
& (1 Ame _Ằ ee sil — 2% mm
and
Ly Atk q LA
— #——(l!— <— (Or <V 1 —cos 2nbq,q,)
a (+ wep (1 — cos 27bq,9,) < — Log |ø'(U”»)| đi 1 cos 2zb/,4,)
For all & > 1, put 0y = 2m(Ùđ, — p,„) We have
“- |, Ẻ
3@k++ đr+a đu+a đụ+1
Let f, f', g and g’ be defined for k 2 1 by
12” 5 LA‘
A= aay (1 — cos2zbqi), g, = TT (1 — cos 2nbqj),
Lo a — 4
, = na my (1 — cos 2xbq?Z), #øx = TT Ly At , LA (1 — cos2zögp)
: : + 1 — cos 2nbq¡q›
Let A be the infinite matrix (a;,);, where a,, =- If j <k, we 1 — cos 2nbq;
have 4; , = L= 00s 45 and ik | = 08 40% AS đ;+yi > 3g; and [0,1 < , 1 — cosg,0, CN: 1 — cosđ;+¡Ú,
< 2 < a < in we get ik <1-ecos(3) al, Hence 2;y < 4i-k WF
Trang 24106 T GIORDANO, G SKANDALIS, E J WOODS
— ; ‘sae Ì —cosg,0, 10 441
Jj 2k, we have a;, == 1 = 608 460; and “the S221, We have|-“Đ| <
, 1 — cosq,0, aj x | — cos q,0; 10;
2 ; 2m 2m đ;+1
<i) and as 1g.) < 14x01 < < OM we get YH <
Gj+o Án 2 đr+i 3 địa
bà ] — cos-,- < ————z—=:- Hence for j > k, đ,, < 3*”1, 1 — cos “= 3 s
AT OME ang Lath | rai
L—2' 2 12% yas
we get that the sequence - Log |o(U**)| < Ja; g, is bounded By Theorem 1.10, j>1
As fir S & IN - 1S bounded by assumption,
we derive that lim (—Log |@(U%)| +- Log |@’(U"*))) == 0, hence fi < Yajafi <
k-00 del
| f +
-=Log |@'(U “*) | is bounded and as g, < Oa hs we get that /”, ø' ce/(N) For an infinite matrix D = (2đ; ¿), put [LĐ|I=:sup 3,12; ¿| Let of = {D;|'D) <
j>L kè1
< -+-0o} Then (.#, || -|Ì) is a Banach algebra and acts naturally on /%(N) Let AES be the closed subalgebra % =- {De ;DheC,(N) if he C(N)}, Le
B= [Dew lim $5 Idjal — 0 for all nen}
IPS fed
Then 4:: (2;„)c 2 Moreover for all j > 1
Yaa +L l4 Ki,
k#j k::7/+1 6
Therefore [|,A — 1}; < 5/6 <1 and A is invertible in 2
As g ~- fe C,(N), we get that 4g — AfeC,(N) Therefore Af he C,(N), where fy, =: ~-Log |g@(U%)| In the same way, Af —h’eC,(N), where Ii =
:— Log;@(U'®), By Theorem 1.10, we get that h—jh’e C,{(N) Hence, A(f — f')€ C,(N) As A is invertible in #, f — f’ € C,(N) A 4.5 Remark We have actually just proved the following somehow stronger statement than Proposition 4.4:
7% 2 dk
If lim sup L,/ < + ©œ, g ¡>34, and if Ó_,„„;(M, 7) :6_„y;(N, T), diss
then lim(L, — Lik qe
Trang 25TNVARIANTS FOR ITPFf FACTORS 107
In particular, in that case
© tog i(M, 7) # Ÿ_voya(M @ M, T) =: © tog (M, T) * © 155 ÚM, T), and the Haar measure m of G does not belong to ©_,,, ,(M, T) (cf Proposition 4.3)
Acknowledgement The first author was financially supported in part by NSERC {Canada ) and the Swiss National Fund for Scientific Research; the secend author by NSERC (Canada) They wish to thank these institutions
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T GIORDANO, G SKANDALIS, E J WOODS
Department of Mathematics and Statistics, Queen's University,
Kingston, Ontario K7L 3N6, Canada