Báo cáo toán học: "On the computation of invariants for ITPFI factors " potx

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

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)

tn [8], Hamachi and Osikawa consider the ITPFI, factors M(L,, 22, O<^2<l, and prove that for „ sufficiently large, the ñow of weights is pure point spectrum

In Section 3 we study this family of factors We compute for all sequences LZ, the flow of weights by showing that the map f indicated above is essentially injective (Theorem 3.1) We give a condition on the ZL, that @(M, T) contains the Haar measure and hence the flow is pure point spectrum (Proposition 3.8) Proposition 3.4 gives a precise condition for T(M) to be either 4 = {OE R; jit*o - 1 for some

k € N} or uncountable This result gives some insight into the occurrence of

uncount-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

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

INVARIANTS FOR ITPFI FACTORS 85

4 Tf all the », are bounded by a number nv, M is said to be an ITPFi of bound-

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

bility measures on T given by: uAv iff there exists y¢T such that wis equivalent to

5, * v (where 5, is the Dirac measure)

86 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

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

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

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)).,

where p is defined by pf(t) = e'f(t); fe L°(R) Then the (dominant) weight ¢ @ @

on M@L(L(R)) does not depend within unitary equivalence on the normal semi- -finite faithful weight y on M (cf [2], Lemma 1.2.5; [3], Theorem II.1.1; [4], Section 4).

INVARIANTS 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)()

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

Let U, ¢ £(L°(R)) be given by Ứ,ƒ(s) = ƒ(s + #) The flow of weights F, of M

is given by the restriction of Ad(1 © U,) to Cggw ((3); [4], § 4)

88 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

INVARIANTS 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

b) Such a construction can be done for an ergodic action of any locally com- pact abelian group

90 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,

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

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,

FNVARIANTS FOR ITPFI FACTORS 9

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

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

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

In [8], Hamachi and Osikawa consider the ITPFI, factors M = M(L,, 2,

9 < À < | They prove that for L, large enough, the flow of weights of M has pure point spectrum We study here this family of factors We compute for all sequences L,, the flow of weights of M We then give estimates on the growth of the L,’s for this flow to have pure point spectrum.

92 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)

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),

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

WD) = Ty a Oe

HG) = wi t+ &)

(uf) = 0 if 7 <O0or j > L),

INVARIANTS FOR ITPFI FACTORS 93

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, then Cạn C; # Ø.

94 T GIORDANO, G SKANDALIS, E.J WOODS

Proof Let m= mị +- Hạ The Cauchy-Schwarz inequality gives

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

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 < &

where € oy £ sxp[—- } Since o°(X,) > Yh ——

Lyi 2-#"g3(X„ )> Vo it = ©

By the ratio test there exist infinitely many 7 such that