Quasiregular representations of the infinite-dimensional Borel group

Journal of Functional Analysis 218 (2005) 445–474

Quasiregular representations of the infinite-dimensional Borel group

Sergio Albeverio

and Alexandre Kosyak

aInstitut fu¨r Angewandte Mathematik, Universita¨t Bonn, Wegelerstr. 6, D-53115 Bonn, Germany

bSFB 256, Bonn, BiBOS, Bielefeld–Bonn, Germany

cCERFIM, Locarno and Acc. Arch., USI, Switzerland

dIZKS, Bonn, Germany

eInstitute of Mathematics, Ukrainian National Academy of Sciences, 3 Tereshchenkivs’ka, Kyiv 01601, Ukraine

Received 10 February 2004; accepted 20 March 2004 Communicated by Paul Malliavin

Abstract

The notion of quasiregular (Representation of Lie groups, Nauka, Moscow, 1983) or geometric (Grundlehren der Mathematischen Wissenschaften, Band 220, Springer, Berlin, New York, 1976; Encyclopaedia of Mathematical Science, Vol. 22, Springer, Berlin, 1994, pp.

1–156) representation is well known for locally compact groups. In the present work an analog of the quasiregular representation for the solvable infinite-dimensional Borel groupG¼Bor^N₀ is constructed and a criterion of irreducibility of the constructed representations is presented.

This construction usesG-quasi-invariant Gaussian measures on someG-spacesXand extends the method used in Kosyak (Funktsional. Anal. i Prilozˇhen 37 (2003) 78–81) for the construction of the quasiregular representations as applied to the nilpotent infinite- dimensional groupB^N₀:

r2004 Elsevier Inc. All rights reserved.

MSC:22E65; 28C20; 43A80; 58D20

Keywords:Infinite-dimensional groups; Borel group; Regular representations; Quasiregular (geometric) representations; Irreducibility; Quasi-invariant measures; Ergodic measures; Ismagilov conjecture;

Solvable group

Corresponding author. Fax: 38044-235-2010.

E-mail addresses:kosyak@imath.kiev.ua, kosyak01@yahoo.com (A. Kosyak).

0022-1236/$ - see front matterr2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.jfa.2004.03.009

1. Introduction

1.1. The setting and the main results

With any action a:G-AutðXÞ (AutðÞ denoting the group of all measurable automorphisms) of a groupGon aG-spaceX (i.e. a space on whichGacts) andG- quasi-invariant measure m on X one can associate a unitary representation p^a;m;X:G-UðL²ðX;mÞÞ;of the groupGby the formulaðp^a;m;X_t f ÞðxÞ ¼ ðdmða_t¹ðxÞÞ=

dmðxÞÞ¹⁼²fðat¹ðxÞÞ;fAL²ðX;mÞ: Let us setaðGÞ ¼ fatAAutðXÞjtAGg: Let aðGÞ⁰ be centralizer of the subgroup aðGÞ in AutðXÞ: aðGÞ⁰¼ fgAAutðXÞjfg;a_tg ¼ ga_tg¹a¹_t ¼e8tAGg:

Conjecture 1 (Kosyak[27,28]). The representationp^a;m;X:G-UðL²ðX;mÞÞis irreducible if and only if

(1) m^g>m8gAaðGÞ⁰\feg;(where >stands for singular), (2) the measuremis G-ergodic.

We say that a measuremisG-ergodic iffðatðxÞÞ ¼fðxÞ8tAGimpliesfðxÞ ¼const for all functionsfAL¹ðX;mÞ:

In this paper we shall prove Conjecture 1 in the case where G is the infinite- dimensional group, namely the Borel groupG¼Bor^N₀;the spaceX¼X^m being the set of left cosets Gm\Bor^N; Gm suitable subgroups of the group Bor^N and m any Gaussian product-measures onX^m:See below for explanation of the concepts used here.

1.2. Regular and quasiregular representations of locally compact groups

Let G be a locally compact group. The right r (respectively left l) regular representationof the groupGis a particular case of the representationp^a;m;Xwith the spaceX¼G;the actiona being the right actiona¼R(respectively the left action a¼L), and the measurem being the right invariant Haar measure on the groupG (see for example[9,21,22,45]).

Aquasiregular representationof a locally compact groupGis also a particular case of the representationp^a;m;X (see for example[45, p. 27]) with the spaceX ¼H\G;the actiona being the right action of the group G on the space X and the measurem being some quasi-invariant measure on the spaceX (this measure is unique up to a scalar multiple). In[21,22]this representation is also calledgeometric representation.

1.3. An analog of the regular and quasiregular representations of infinite-dimensional groups and the Ismagilov conjecture

The work of Gel’fand played a decisive role in representation theory in general and in representation theory of infinite-dimensional groups, in particular see[11–13].

Using his orbit method, developed in [19], Kirillov described in [20] all unitary irreducible representations of the completion in strong operator topology of the groupUðNÞ ¼lim

-nUðnÞ(where lim

- stands for inductive limit).

This approach was generalized by Ol’shanskiı˘ for the inductive limits of other classical groups KðNÞ ¼lim

-nKðnÞ where K is U;O or Sp: In [38] the complete classification of the so-called ‘‘tame’’ representations of the group KðNÞ was obtained. The book[37] deals with the representation theory of the automorphism groups of infinite-dimensional Riemannian symmetric spaces.

The book of Ismagilov [16] is devoted to the representations of two classes of infinite-dimensional Lie groups: that of current groups and that of diffeomorphism group and some of their semidirect product.

The book of Neretin[34]is devoted to the representations theory of the following infinite-dimensional groups: groups of diffeomorphisms of manifolds, groups associated to Virasoro or Kac–Moody algebras, infinite groups of permutations SN;groups of operators in Hilbert spaces, groups of currents, and finally groups of automorphisms of measure spaces.

The book of Albeverio and coauthors[2]is devoted to representation theory of gauge groups and related topics.

LetSN¼S

nX1 Snbe the group of finite permutation of the natural numbers. All indecomposable central positive definite functions onSN;which are related to factor representations of II₁;were given by Toma[42].

Later Vershik and Kerov obtained the same result by a different method in[43]

and gave a realization of the representation of type II1 in [44].

In [35,36] Obata construct and classifies a family U^y;w of uncountably many irreducible representations of the group SN: This family consists of induced representations.

In[18]generalized regular representationsfTz:zACgof the groupSNSNwere studied. These representations are deformations of the biregular representation of SN in l²ðSNÞ: A two-parameter family of the generalized regular representations Tz;z⁰ of the group SN was mentioned also in[18]. In [8]the corresponding spectral measure P_z;z⁰ was studied. The correlation functions are of a determinantal form similar to those studied in random matrix theory.

In[7]the asymptotics of the Plancherel measuresMnfor the symmetric groupsSn

is studied. It is shown thatMnconverge to the delta measure supported on a certain subset O of R² closely connected to Wigner’s semicircle law for distribution of eigenvalues of random matrices. In particular this gives a positive answer to a conjecture of Baik et al.[5].

In the present article we will consider only the approach which deals with the analog for infinite-dimensional groups of the regular and quasiregular representa- tions of finite-dimensional groups. Let G be an infinite-dimensional topological group. To define an analog of the regular representation, let us consider some topological groupG;˜ containing the initial groupG as a dense subgroupG%¼G˜ (G% being the closure ofG). Suppose we have some quasi-invariant measuremonX¼G˜

with respect to the right action of the groupG;i.e.a¼R;R_tðxÞ ¼xt¹:In this case we shall call the representationp^a;m;G˜ananalog of the regular representation. We shall denote this representation byT^R;m;and the Conjecture 1 is reduced to the following Ismagilov conjecture.

Conjecture 2. The right regular representation T^R;m:G-UðL²ðG;˜ mÞÞis irreducible if and only if

(1) m^Lt>m8tAG\feg;

(2) the measuremis G-ergodic.

Remark 3. In the case of the right regular representation, the group aðGÞ⁰¼ RðGÞ⁰CAutðGÞ˜ obviously contains the groupLðGÞ;the image of the groupG with respect to the left action.

The work[14]initiated the study of representations of current groups, i.e. groups CðX;UÞ of continuous mappings X/U; where X is a finite-dimensional Riemannian manifold andU is a finite-dimensional Lie group.

The regular representation of infinite-dimensional groups, in the case of current groups, was studied firstly in [1,3,4,15] (see also the book [2]). An analog of the regular representation for an arbitrary infinite-dimensional group G; using a G- quasi-invariant measure on some completionG˜of such a group, is defined in[23,25].

For X ¼S¹; U a compact or non-compact connected Lie group, a Wiener measures on the loop groups G˜¼CðX;UÞ were constructed and their quasi- invariance were proved in[31–33].

Conjecture 2 was formulated by Ismagilov for the group G¼B^N₀ and any Gaussian product measure on the groupG˜¼B^N and was proved for this case in [23,24]. Here G¼B^N₀ is the nilpotent group of finite upper triangular matrices of infinite order with unities on the principal diagonal andB^Nis the group of all upper triangular matrices of infinite order with unities on the principal diagonal.

For any product measure on the groupB^NConjecture 2 was proved in[26]under some technical assumptions on the measure.

In [25] the criterion was proved for groups of the interval and circle diffeomorphisms. For the group of the interval diffeomorphisms the Shavgulidze measure[40] was used, the image of the classical Wiener measure with respect to some bijection. For the group of circle diffeomorphisms the Malliavin measure[32]

was used.

Whether the Conjecture 2 holds in the general case is an open problem.

Let us consider the special case ofG-spaces, namely the homogeneous spaceX¼ H\G;˜ whereHis a subgroup of the groupG˜and the measuremis some quasi-invariant measure on X (if it exists) with respect to the right action of the group G on the homogeneous spaceH\G:˜ In this case we call the corresponding representationp^a;m;H\G˜ ananalog of the quasiregular or geometric representationof the groupG(see[27]).

In[27,28]the Conjecture 1 was proved for thenilpotent group G¼B^N₀ and some G-spacesX^m;mAN;being the set of left cosetsGm\B^N;whereGmare some subgroups of the groupB^N:Heremis an arbitrary Gaussian product-measure onX^m:In[29]it was shown that Conjecture 1 holds for the inductive limit G¼SL₀ð2N;RÞ ¼ lim-nSLð2n1;RÞ;of the special linear groups (simple groups) acting on a strip of lengthmANin the space of real matrices infinite in both directions, and the measure mbeing the product Gaussian measure.

In the present article we prove Conjecture 1 for thesolvable infinite-dimensional Borel group G¼Bor^N₀ acting on G-spaces X^m;mAN; where X^m is the set of left cosetsG_m\Bor^N;andG_mare some subgroups of the groupBor^N:The measuremcan be any Gaussian product-measure onX^m:

2. The infinite-dimensional Borel groupBor^N₀

Let Ekn be infinite-dimensional matrix units k;nAN: We define the infinite- dimensional group of upper triangular matrices

G˜ ¼Bor^N¼ x¼ X

1pkpn

xknEknjxknAR;xkka0;k;nAN

( )

and the subgroup

G¼Bor^N₀ ¼ fxABor^NjxI is finiteg;

whereI¼P

kANE_kk is a neutral element in the groupBor^N:

We callBor^N₀ theinfinite-dimensional Borel group. ObviouslyBor^N₀ is the inductive limitBor^N₀ ¼lim

-mBorðm;RÞ; of the finite dimensional (classical) Borel group Borðm;RÞ ¼ x¼ X

1pkpnpm

xknEknjxknAR;xkka0;1pkpnpm

( )

;

with respect to the imbedding Borðm;RÞ{x/xþEmþ1;mþ1ABorðmþ1;RÞ: For mANwe also define the subgroupsGm resp.G^m of the groupBor^N as follows:

G_m¼ xABor^Njx¼ X

mokpn

x_knE_knþX^m

k¼1

E_kk

( )

;

G^m¼ xABor^Njx¼ X

1pkpm;kpn

xknEknþ X^N

k¼mþ1

Ekk

( )

:

Since Bor^N¼G_mG^m the space X^m of left cosets X^m¼G_m\Bor^N is isomorphic to the group G^m: By construction, the right action R of the group G is well defined on the space X^m: More precisely if we define the decomposition x¼xmx^m:

Bor^N{x/x_mx^mAG_mG^m; the right actionRt will be defined as follows:

R_tðx^mÞ ¼ ðx^mt¹Þ^m; x^mAG^m;tABor^N₀:

Define the measurem^m:¼m^m_ðb;aÞ on the spaceX^mCG^m by the formula dm^m_ðb;aÞðxÞ ¼

#

1pkpm;kpn dm_ðbkn;aknÞðxknÞ;

whereb¼ ðbknÞ_kpn;a¼ ðaknÞ_kpn;b_kn40;a_knAR and the one-dimensional Gaussian measurem_ðb;aÞ is defined as follows:

dm_ðb;aÞðtÞ ¼ ðb=pÞ¹⁼²expðbðtaÞ²Þdt; b40; aAR:

Lemma 4. We haveðm^m_ðb;aÞÞ^RtBm^m_ðb;aÞ for all tABor^N₀;where Bmeans equivalence.

Proof. Let us fix some tABor^N₀: Since the group Bor^N₀ is the inductive limit so tABorðp;RÞ for some pAN: Hence we are in the case of the right action of some locally compact groupGon some finite-dimensional homogeneous spaceX¼H\G with some quasi-invariant measure.

Let us suppose thatp4m;ifppmthe proof will be even simpler. We define two subgroupG^mðpÞandG_mðpÞof the groupBorðp;RÞ

G^mðpÞ ¼G^m-Borðp;RÞ; G_mðpÞ ¼G_m-Borðp;RÞ:

Then Borðp;RÞ ¼GmðpÞ G^mðpÞ for mop: Since for tABorðp;RÞ the right action x/R_tðxÞchanges only a finite number of coordinates of the pointxAX^m;so we are in some finite dimensional subgroupX^mðpÞ ¼G_mðpÞ\Borðp;RÞCG^mðpÞof the group X^mCG^m: The measure m^m_ðb;aÞ is a product Gaussian measure, hence the projection m^m_ðb;aÞðpÞof this measure onto this subgroupG^mðpÞ

dm^m_ðb;aÞðpÞðxÞ ¼

#

1pkpm;kpnppdm_ðbkn;aknÞðxknÞ

is equivalent with the corresponding Haar measure on this subgroup. We note that the Haar measuredhðxÞon the groupBorðp;RÞis equal

dhðxÞ ¼ 1 jdetðxÞj

Y

1pkpnpp

dxkn¼ Y

1pkpp

jxkkj

!1

Y

1pkpnpp

dxkn;

where dx is the Lebesgue measure on the real line R: Hence the Haar measure dh^mðpÞðxÞon the group G^mðpÞis equal

dh^mðpÞðxÞ ¼ 1 jdetðxÞj

Y

1pkpm;kpnpp

dxkn¼ Y

1pkpm

jxkkj

!1

Y

1pkpm;kpnpp

dxkn:

So fortAG^mðpÞCBorðp;RÞthe Haar measuredh^mðpÞis right invariant by definition of the Haar measure. It is easy to verify that for anothertAG_mðpÞCBorðp;RÞ it is quasi-invariant. &

Let us define the representation

T^R;m;m:Bor^N₀/UðHm¼L²ðX^m;m^m_ðb;aÞÞÞ by the following formula:

ðT_t^R;m;mfÞðxÞ ¼ ðdm^m_b;aðR¹_t ðxÞÞ=dm^m_b;aðxÞÞ¹⁼²fðR¹_t ðxÞÞ:

It is natural to call this representation an analog of the quasiregular or geometric representation.

Let us set fortAR;k;nAN;1pkpnpm;

S_kn^LðmÞ ¼X^N

m¼n

b_km 2

1 2bnm

þa²_nm

;kon; S_nn^LðmÞ ¼2 X^N

m¼n

b_nma²_nm;

S^L;_kn ðm;tÞ ¼t² 4

X^N

m¼n

b_km bnm

þX^N

m¼n

b_km

2 ð2akmþtanmÞ²: Let alsoP_k¼Pm

n¼1E_nn2E_kk;1pkpm:In the casem¼2 we have P1 ¼ 1 0

0 1

; P2¼ 1 0

0 1

;

expðtE12Þ ¼ 1 t 0 1

and expðtE12ÞP1¼ 1 t

0 1

:

Theorem 5. Four following conditions(i)–(iv)are equivalent for the measurem¼m^m_ðb;aÞ: (i) the representation T^R;m;m is irreducible;

(ii) m^Lt>m for all tABorðm;RÞ\feg;where LtðxÞ ¼tx;xAX^m; (iii)

ðaÞ m^LPk>m; 1pkpm;

ðbÞ m^LexpðtEknÞ>m8AR\f0g; 1pkonpm;

ðcÞ m^{LexpðtEknÞPk}>m8tAR; 1pkonpm;

8>

<

>:

(iv)

ðaÞ S^L_kkðmÞ ¼N; 1pkpm;

ðbÞ S_kn^LðmÞ ¼N; 1pkonpm;

ðcÞ S^L;_kn ðm;tÞ ¼N8tAR; 1pkonpm:

8>

<

>:

MoreoverðiiiÞðaÞ3ðivÞðaÞ;ðiiiÞðbÞ3ðivÞðbÞandðiiiÞðcÞ3ðivÞðcÞ:

Remark 6. We note that the measure m^m_ðb;aÞ on the space X^m is Bor^N₀-right-ergodic since it is a product measure.

We note also that conditions (iii)(a) for 1pkom are the particular cases of conditions (iii)(c) for t¼0 and 1pkon¼m: Indeed expðtEkmÞPkj_t¼0¼ P_k;1pkom:

Proof of Theorem 5. It is sufficient to prove the following implications:

ðiÞ ) ðiiÞ )

ðiiiÞðaÞ ) ðivÞðaÞ ðiiiÞðbÞ ) ðivÞðbÞ ðiiiÞðcÞ ) ðivÞðcÞ 8>

<

>:

9>

=

>;) ðiÞ:

PartsðiÞ ) ðiiÞ ) ðiiiÞare obvious. To proveðiiiÞ ) ðivÞit is sufficient to consider the elementsP_k¼Pm

n¼1E_nn2E_kk;1pkpmin the groupBorðm;RÞ(see Lemma 9), the one parameter subgroups expðtEknÞ ¼IþtEkn;1pkonpm;tAR\f0g of the group Borðm;RÞ (see Lemma 10) and the following images of these subgroups:

expðtEknÞPk;tAR(see Lemma 12).

Remark 7. In[27,28]it was proved that in the case of the connected nilpotent group

G¼Bðm;RÞ ¼ Iþx¼Iþ X

1pkonpm

xknEknjxknAR

( )

it is sufficient, forðm^m_ðb;aÞÞ^Lt>m^m_ðb;aÞ8tABðm;RÞ\feg to verify only for one-parameter subgroups expðtEknÞ ¼IþtE_kn;1pkonpm;tAR\f0g (which generateG).

However in the case of the solvable (non-connected) classical Borel group G¼Borðm;RÞ it is not sufficient to verify the conditions ðm^m_ðb;aÞÞ^Lt>m^m_ðb;aÞ8tA Borðm;RÞ\feg for one-parameter subgroups expðtEknÞ;1pkpnpm;tAR\f0g:Even if we add all the elementsPk;1pkpmit will still not be sufficient, in general. This can be seen by the following.

Counterexample. Let m¼2 and let the measure m²_ðb;aÞ be defined by taking a_1n¼ a2nþ1¼b_1n¼1;b2nþ1¼ ðnþ1Þ²;nAN: In this case S₁₁^LðmÞ ¼S^L₂₂ðmÞ ¼S^L₁₂ðmÞ ¼N:

By Lemmas 9 and 10, we conclude that ðm²_ðb;aÞÞ^LtEkk>m²_ðb;aÞ8tAR\f0g;k¼1;2 and ðm²_ðb;aÞÞ^LexpðtE12Þ>m²_ðb;aÞ;8tAR\f0g: But S^L;₁₂ ðm;2Þ ¼²₄²PN

n¼2b1n

b2nþPN n¼2b1n

2ð2a1nþ 2a2nÞ²¼PN

n¼2 1

n²oN:Hence, by Lemma 12 we conclude thatðm²_ðb;aÞÞ^{Lexpð2E12ÞP1}Bm²_ðb;aÞ: The idea of the proof of irreducibility(i.e. partðivÞ ) ðiÞ): Let us denote byA^mthe von Neumann algebra generated by the representationT^R;m;m

A^m¼ ðT_t^R;m;mjtAGÞ⁰⁰:

We show that conditions (iv) implyðA^mÞ⁰CL^NðX^m;m^mÞ:Using the ergodicity of the measure m^m¼m^m_ðb;aÞ this proves the irreducibility. Indeed in this case an operator AAðA^mÞ⁰ should be the operator of multiplication by some essentially bounded function aAL^NðX^m;m^mÞ: The commutation relation ½A;T_t^R;m;m ¼0 8tABor^N₀ impliesaðxtÞ ¼aðxÞðmodm^mÞ 8tABor^N₀;so by ergodicity of the measure m^m on the space X^m we conclude that A¼a¼const: This then proves part ðivÞ ) ðiÞ of Theorem 5.

The inclusion ðA^mÞ⁰CL^NðX^m;m^mÞ is based on the fact that the operators of multiplication by independent variablesx_kn;1pkpm;kpn;may be approximated (if conditions (iv) are valid) in the strong resolvent sense, by some polynomials in the generatorsA^R;m_kn ¼_dt^dT_IþtE^R;m;m_knj_t¼0;k;nAN;kpni.e. that the operatorx_kn are affiliated to the von Neumann algebraA^m:

Definition. Recall (see[10]) that a non necessarily bounded operatorAin a Hilbert spaceH is said to be affiliated to a von Neumann algebraM of operators in this Hilbert spaceH;which is denoted byAZM;if expðitAÞAM for alltAR:

Remark 8. We will prove the approximation of xkn firstly for one vector 1AL²ðX^m;m^m_ðb;aÞÞ: Secondly, the approximation also holds for some dense (in the spaceL²ðX^m;m^m_ðb;aÞÞ) setDof analytic vectors for the corresponding operators

D¼l:s:ðX^a¼ Y

1pkpm;kpn

x^a_kn^knjaALÞ;

where L¼ fa¼ ðaknÞ_1pkpm;kpng are the set of finite (i.e. akn¼0 for a large n ) multi-indicesa_kn¼0;1;y: andl:s:ðf_nÞmeans linear space generated by the set of vectorsðf_nÞ:So using the[39, Theorem VIII, 25]we conclude that the convergence holds in the strong resolvent sense. The proof is the same as the proof of[24, Lemma 2.2, p. 250]. Since the generatorsA^R;m_kn are affiliated to the von Neumann algebraA^m so the limitxkn is also affiliated.

We have for the generators the following expressions:

A^R;m_kn ¼X^m

r¼1

xrkDrn; mokpn; A^R;m_kn ¼X^k

r¼1

xrkDrn; 1pkpm;kpn;

whereDkn¼@=@xknbknðxknaknÞ:

The approximation uses conditions (iv) and is based on the following estimation (see for example[6, Chapter I, Section 52])

xARminⁿ Xⁿ

k¼1

a_kx²_kXⁿ

k¼1

x_k¼1

!

¼ Xⁿ

k¼1

1 a_k

!1

: ð1Þ

We will also use the same estimation in a slightly different form

xARminⁿ Xⁿ

k¼1

a_kx²_kXⁿ

k¼1

x_kb_k¼1

!

¼ Xⁿ

k¼1

b²_k a_k

!1

: ð2Þ

The extremum in (2) is obtained forx_k¼^b_a^k_k Pn k¼1

b²_k ak

1

: In what follows, we will say that two series P

kANak andP

kANbk with positive elementsa_k;b_k40are equivalentand will use the notationP

kANa_kBP

kANb_kif they are convergent or divergent simultaneously. It is easy to see that for positive ak;bk40;kAN the following asymptotic holds:

X

kAN

ak

a_kþb_kBX

kAN

ak

b_k: ð3Þ

Here and in the following we shall use for two sequencesðanÞ_nAN andðbnÞ_nAN the notation anBbn as n-N; if for some const C1;C240 and a large numbers NANholds

aNpC1bNpC2aN:

In this case we say that the sequences are ‘‘equivalent at infinity’’.

Proof of partðiiiÞ ) ðivÞ: This follows from Lemmas 9–12. Let us denote bym_kthe measure

m_k¼

#

n¼k

m_ðbkn;aknÞ:

Then obviously m^m_ðb;aÞ¼#^m_k¼1m_k: We will prove ðiiiÞ ) ðivÞ only for m¼2: For another m42 the proof is similar. To prove ðiiiÞðaÞ3ðivÞðaÞ it is sufficient to considerm¼1 and the measurem¹¼m¹_ðb;aÞ:In this caseP1¼ E11:We prove a little more, namely

Lemma 9. The following three conditions are equivalent:

(1) ðm¹_ðb;aÞÞ^Lt>m¹_ðb;aÞ8tABorð1;RÞ\feg ¼GLð1;RÞ\feg;

(2) ðm¹_ðb;aÞÞ^LE11ðxÞ ¼m¹_ðb;aÞðxÞ>m¹_ðb;aÞðxÞ;

(3) S₁₁^LðmÞ ¼2P

nANb_1na²_1n¼N:

Proof. Let us consider t:¼tE11ABorð1;RÞ: We have LtðxÞ ¼tx¼P

nANtx1nE1n; so

dðm¹_ðb;aÞÞ^LtðxÞ ¼

#

nAN

dm^L_ðb^t

1n;a1nÞðx1nÞ ¼

#

nAN

ffiffiffiffiffiffi b_1n p r

expðb1nðtx1na1nÞ²Þdtx1n

¼

#

nAN

ffiffiffiffiffiffiffiffiffiffi t²b1n

p s

expðt²b_1nðx1na_1n=tÞ²Þdx_1n¼

#

nAN

dm_ðt2b1n;a1n=tÞ

¼dm¹_ðt2b;a=tÞðxÞ:

It is known (see [17,41]) that two Gaussian product measures m¹_ðb;aÞ and m¹_ðb0;a⁰Þ on X¹DR^N¼RR? are equivalent if and only if (1) m¹_ðb;0ÞBm¹_ðb0;0Þ and (2) m¹_ðb;aÞBm¹_ðb;a0Þ: Otherwise they are orthogonal. Condition (1) is equivalent with Q

nAN 4b1nb1n0

ðb1nþb1n0Þ²40 and condition (2) is equivalent with P

nANb1nða1na1n0Þ²oN: Obviously m¹_ðt2b;0Þ>m¹_ðb;0Þ since Q

nAN 4t²b1nb1n

ðt²b1nþb1nÞ²¼Q

nAN 4t²

ðt²þ1Þ²¼0 if jtja1; so ðm¹_ðb;aÞÞ^Lt ¼m¹_ðt2b;a=tÞ>m¹_ðb;aÞ8tAR;jtja1;ta0: In the case t¼ 1 we have ðm¹_ðb;aÞÞ^LE11 ¼m¹_ðb;aÞ so ðm¹_ðb;aÞÞ^LE11>m¹_ðb;aÞ if and only if P

nANb_1nða1nþa_1nÞ²¼4P

nANb_1na²_1n¼N: &

EquivalenceðiiiÞðbÞ3ðivÞðbÞfollows from the following

Lemma 10. For the measurem²_ðb;aÞ¼m₁#m2 one has ðm₁#m2Þ^LexpðtE12ÞBðm1#m2Þ8tA R\03

S^L₁₂ðmÞ ¼1 4

X^N

n¼2

b_1n b2n

þ1 2

X^N

n¼2

b_1na²_2noN: ð4Þ

Proof. We will use two obvious formulas (the second formula follows from the first one)

1ffiffiffip p

Z

R

expðbx²þcxÞdx¼ 1 ffiffiffib p exp c²

4b

; ð5Þ

ffiffiffib p r Z

R

expðb=2½ðxþsÞ²þx²Þdx¼exp bs² 4

: ð6Þ

Since

expðtE12Þ ¼ 1 t 0 1

and expðtE12Þ x_1n x2n

¼ 1 t 0 1

x_1n x2n

¼ x_1nþtx_2n x2n

;

we have for the Hellinger integralHðm;nÞ(see[30]):

Hððm₁#m2Þ^LexpðtE12Þ;m₁#m2Þ

¼Y^N

n¼2

Hððmðb_1n;a1nÞ#m_ðb2n;a2nÞÞ^LexpðtE12Þ;m_ðb1n;a1nÞ#m_ðb2n;a2nÞÞ

¼Y^N

n¼2

Z

R²

ffiffiffiffiffiffiffiffiffiffiffiffi b_1nb_2n p² r

exp b_1n

2 ½ðx1nþtx_2na_1nÞ² þðx1na_1nÞ² b_2nðx2na_2nÞ²

dx_1ndx_2n

ð6Þ¼Y^N

n¼2

Z

R¹

ffiffiffiffiffiffi b_2n p r

exp t²b_1nx²_2n

4 b_2nðx2na_2nÞ²

dx_2n

¼Y^N

n¼2

Z

R¹

ffiffiffiffiffiffi b_2n p r

exp x²_2n b2nþt²b_1n 4

þ2b2na2nx2nb2na²_2n

dx2n

ð5Þ¼Y^N

n¼2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2n

b_2nþt²b1n

4 vu

uu

t exp b²_2na²_2n

b2nþ^t2b₄¹ⁿb2na²_2n

!

¼Y^N

n¼2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1þ^t_4b^2b_2n¹ⁿ

q exp t²b1na²_2n 4 1þ^t_4b^2b1n

2n

0

@

1 A40

if and only if

S₁₂^LðmÞ ¼1 4

X^N

n¼2

b_1n b_2nþ1

2 X^N

n¼2

b_1na²_2noN: &

Remark 11. To obtain the same conditions we may use the generator corresponding to the left action of the group expðtE12Þon the spaceX²:

Indeed if ðm²_ðb;aÞÞ^LexpðtE12ÞBm²_ðb;aÞ we can define a one-parameter unitary group T^L;m

2 ðb;aÞ

expðtE12Þ as follows:

ðT^L;m

2 ðb;aÞ

expðtE12Þf ÞðxÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdm²_ðb;aÞÞ^{L1expðtE12Þ}ðxÞ

dm²_ðb;aÞðxÞ vu

ut fðL¹_expðtE12ÞðxÞÞ:

A direct calculation gives us the generator A^L;m

2 ðb;aÞ

12 ¼d

dtT^L;m

2 ðb;aÞ

IþtE12j_t¼0¼ X^N

n¼2

x_2nD_1n;

whereD1n ¼@=@xknb1nðx1na1nÞ: Finally we get

jjA^L;m

2 ðb;aÞ

12 1jj² ¼ X^N

n¼2

x2nD1n1

2

¼X^N

n¼2

jjx2nb_1nðx1na_1nÞjj² ¼X^N

n¼2

b1n

2 1 2b_2nþa²_2n

¼S^L₁₂ðmÞ:

To proveðiiiÞðcÞ3ðivÞðcÞwe use

Lemma 12. For the measure m²_ðb;aÞ we haveðm²_ðb;aÞÞ^{LexpðtE12ÞP1}Bm²_ðb;aÞ8tAR3

S₁₂^L;ðm;tÞ ¼t² 4

X^N

n¼2

b1n

b_2nþ1 2

X^N

n¼2

b1nð2a1nþta2nÞ²oN: ð7Þ

Proof. Since

expðtE12ÞP1¼ 1 t 0 1

1 0

0 1

¼ 1 t

0 1

and

expðtE12ÞP1

x_1n x2n

¼ 1 t

0 1

x_1n x2n

¼ x1nþtx_2n x2n

;

we have for the Hellinger integralHðm;nÞ(see[30]):

Hððm²_ðb;aÞÞ^{LexpðtE12ÞP1};m²_ðb;aÞÞ

¼Y^N

n¼2

Hðm_ðb1n;a1nÞ#mðb2n;a2nÞÞ^{LexpðtE12ÞP1};ðm_ðb1n;a1nÞ#mðb2n;a2nÞÞ

¼Y^N

n¼2

Z

R²

ffiffiffiffiffiffiffiffiffiffiffiffi b1nb2n

p² r

exp b1n

2 ½ðx1nþtx2na1nÞ² þ ðx1na1nÞ² b2nðx2na2nÞ²

dx1ndx2n

¼ð6Þ

ðsince x1nþtx2na1n¼ ðx1na1nÞ þ ð2a1nþtx2nÞÞ

¼Y^N

n¼2

Z

R¹

ffiffiffiffiffiffi b2n

p r

exp b1nð2a1nþtx2nÞ²

4 b2nðx2na2nÞ²

! dx2n

¼Y^N

n¼2

Z

R¹

ffiffiffiffiffiffi b_2n p r

exp x²_2n b2nþt²b_1n 4

þx2nð2b2na2nþtb1na1nÞ b1na²_1nb2na²_2n"

dx2n

¼ð5ÞY^N

n¼2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b_2n b2nþt²b_1n

4 vu

uu

t exp ð2b2na_2nþtb_1na_1nÞ² 4b2nþt²b1n

ðb1na²_1nþb_2na²_2nÞ

!

¼Y^N

n¼2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1þt²b_1n

4b2n

vu uu

t exp b_1nð2a1nþta_2nÞ² 4 1þ^t_4b^2b1n

2n

0

@

1 A40

if and only if

S₁₂^L;ðm;tÞ ¼t² 4

X^N

n¼2

b1n

b_2nþ1 2

X^N

n¼2

b_1nð2a1nþta_2nÞ²oN: &

This finishes theproof of Theorem5ðiiiÞ ) ðivÞ:

Proof of Theorem 5 ðivÞ ) ðiÞ: Let us denote by /fnjnANS the closure of the linear space generated by the set of vectorsðf_nÞ_nAN in a Hilbert space H: For the measure dm_ðb;aÞðxÞ ¼ ðb=pÞ¹⁼²expðbðxaÞ²Þdx on R we shall consider the

following expectations, using the notationMf forfAL¹ðR;m_ðb;aÞÞ; with Mf :¼

Z

R

fðxÞdm_ðb;aÞðxÞ;

Mx¼a; Mx²¼ ð2bÞ¹þa²¼:c; Mx³¼3ð2bÞ¹aþa³Bac; ð8Þ

Mx⁴¼3ð2bÞ²þ6ð2bÞ¹a²þa⁴Bc²; ð9Þ

Mjx²Mx²j²¼Mx⁴ ðMx²Þ² ¼2ð2bÞ²þ4ð2bÞ¹a²Bð2bÞ¹c: ð10Þ IfD¼d=dxbðxaÞandDkn¼@=@xknbknðxknaknÞwe have

MD²1¼ b=2; MjD1j²¼b=2; MjD²1j²¼3ðb=2Þ²; ð11Þ

MjðD² ðMD²1Þ1j²¼2ðb=2Þ²; ðDkn1;1Þ ¼0; ð12Þ

ðDkn1;D_rs1Þ ¼0; ððD²_knMD²1Þ1;D_knD_rs1Þ ¼0 for ðknÞaðrsÞ: ð13Þ Form¼1 we have

A^R;1_1n ¼x11D1n;1pn; A^R;1_kn ¼x1kD1n;2pkpn:

Lemma 13. We have for nAN

x11x1nA/A^R;1_1kA^R;1_nk1¼x11x1nD²_1k1jkAN;nokS: Moreover x11x1nZA¹:

Proof. We have A^R;1_1kA^R;1_nk1¼x₁₁x_1nD²_1k1: Since b_k¼MD²_1k1¼ ^b₂^1k and P

kt_kMD²_1k1¼1 hence X^N

k¼1

t_kA^R;1_1kA^R;1_nk x₁₁x_1n

" #

1

2

¼ jjx11x_1njj² X^N

k¼1

t²_k D²_1kþb_1k 2

1

2

:

Using (2) andak¼c11c1njjðD²_1kþ^b₂^1kÞ1jj²¼c11c1n2ð^b₂^1kÞ²B2ð^b₂^1kÞ² we have

minftkg

X^N

k¼1

tkA^R;1_1kA^R;1_nk x11x1n

" #

1

2 X^N

k¼1

tkMD²_1k1¼1 8<

:

9=

;

¼ X^N

k¼1

b²_k a_k

!1

%%%!

N-N

03N¼X

kAN

b²_k a_k¼X

kAN

ðb1k=2Þ²

2ðb1k=2Þ²¼1=2 X

kAN

1: &

Lemma 14. One has

x₁₁A/x₁₁x_1kjkANS3S₁₁^LðmÞ ¼N: Moreover x11ZA¹:

Proof. Sincebk¼Mx1k¼a1kandP

ktkMx1k¼1;hence X

k

tkx11x1kx11

" # 1

2

¼ jjx11jj² X

k

tkx1k1

" # 1

2

¼c₁₁ X

k

t_kðx1ka_1kÞ

2

¼c₁₁ X

k

t²_kjjx1ka_1kjj²

¼c₁₁ X

k

t²_k 1 2b1kBX

k

t²_k 1 2b1k

:

Using (2) we have

minftkg

X^N2

k¼N1

tkx11x1kx11

" # 1

2 X^N2

k¼N1

tkMx1k¼1 8<

:

9=

;

¼ X^N2

k¼N1

b²_k a_k

!1

%%%!

N2-N

03N¼X

kAN

b²_k a_kBX

kAN

2b1ka²_1k¼S₁₁^LðmÞ: &

So x11 and x11x1k are affiliated to the von Neumann algebra A¹ and hence x1kZA¹;kAN: This completes the proof of the Theorem 5ðivÞ ) ðiÞform¼1:

Form¼2 we have

A^R;2_1n ¼x11D1n;1pn; A^R;2_kn ¼x1kD1nþx2kD2n;2pkpn:

We have 4 conditions:

S^L₁₁ðmÞ ¼S^L₁₂ðmÞ ¼S^L₂₂ðmÞ ¼N; S^L;₁₂ ðm;tÞ ¼N8tAR:

Consider the two cases:

(a)PN

m¼2b_1m=b_2m¼N; (b)PN

m¼2b1m=b2moN:

We use the expressionA^R;2_1n A^R;2_kn ¼x11x1kD²_1nþx11x2kD1nD2n

Lemma 15. We get for nAN

x11x1nA/A^R;2_1kA^R;2_nk1jkAN;nokS3X^N

n¼2

b1k=b2k¼N:

Proof. We have

A^R;2_1kA^R;2_nk ¼x11D1kðx1nD1kþx2nD2kÞ ¼x11x1nD²_1kþx11x2nD1kD2k: Sinceb_k¼MD²_1k1¼ ^b1k₂ andP

kt_kMD²_1k1¼1 hence X^N

k¼2

tkA^R;2_1kA^R;2_nk þx11x1n

" #

1

2

¼ jjx11jj² X^N

k¼2

t²_k x1n D²_1kþb1k

2

þx2nD1kD2k

& '

1

2

:

Using (8), (11) and (12) we have ak¼c11 x1n D²_1kþb1k

2

þx2nD1kD2k

& '

1

2

¼c11 c1n2 b1k

2 2

þc2n

b1k

2 b2k

2

!

Bðb²_1kþb1kb2kÞso, using (2) we get

minftkg

X^N

k¼2

tkA^R;1_1kA^R;1_nk þx11x1n

" #

1

2 X^N

k¼2

tkMD²_1k1¼1 8<

:

9=

;

¼ X^N

k¼2

b²_k a_k

!1

%%%!

N-N

03N¼X^N

k¼2

b²_k a_kBX^N

k¼2

b²_1k b²_1kþb1kb2k

B^ð3ÞX^N

k¼2

b1k

b_2k: &

So, in case (a) we have x11x1kZA²;kAN: By Lemma 14 we have x11A/x11x1kj1okS3S^L₁₁ðmÞ ¼N: Since x11;x11x1k and x11D1k are affiliated to

the algebra A² we conclude thatx_1k andD_1k are also affiliated to the algebraA²: Hence A^R;2_nk x1nD1k¼x2nD2kZA²;2pnpk and x2nx2pD²_2kZA²;2pnpppk: By analogy with Lemma 13 we conclude that x2nx2p;2pnpp are affiliated to the algebraA²:

Lemma 16. We have for nX2

x_2nA/x_2nx_2kjkAN;nokS3S^L₂₂ðmÞ ¼N:

Proof. The proof is similar to the one of Lemma 14. &

Sox_2kZA²;kX2:This proves the irreducibility form¼2 in case (a).

Now we consider case (b). For 2pppnokwe use the expression A^R;2_pkA^R;2_nk ¼ ðx1pD_1kþx_2pD_2kÞðx1nD_1kþx_2nD_2kÞ

¼x1px1nD²_1kþ ðx1px2nþx2px1nÞD1kD2kþx2px2nD²_2k:

Lemma 17. We get for p;nAN;2pppn

x2px2nþbðp;nÞx1px1nA/A^R;2_pkA^R;2_nk 1jkAN;nokS if S1¼ X^N

k¼nþ1

b²_2k=ðb1kþb2kÞ²¼N

and when the limit existslimmb_mðp;nÞ ¼bðp;nÞAR;where

b_mðp;nÞ ¼ X^m

k¼nþ1

b1kb2k

a_kðp;nÞ X^m

k¼nþ1

b²_2k a_kðp;nÞ

!1

ð14Þ

and

akðp;nÞ

¼c1pc1n2 b_1k 2

2

þ(c1pc2nþc2pc1nþ2a1pa2na2pa1n"b_1k 2

b_2k

2 þc2pc2n2 b_2k 2

2

: