L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Éric RICARD

The von Neumann algebras generated byt-gaussians Tome 56, n^o2 (2006), p. 475-498.

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THE VON NEUMANN ALGEBRAS GENERATED BY t-GAUSSIANS

by Éric RICARD (*)

Abstract. — We study thet-deformation of gaussian von Neumann algebras.

They appear as example in the theories of Interacting Fock spaces and conditionally free products. When the number of generators is fixed, it is proved that if tis sufficiently close to1, then these algebras do not depend ont. In the same way, the notion of conditionally free von Neumann algebras often coincides with freeness.

Résumé. — Dans la théorie des probabilités non commutative, beaucoup de déformations ou généralisations de la notion de produit libre sont apparues, comme les concepts de probabilités libres conditionnelles et d’espaces de Fock interactifs.

L’un des premiers exemples d’algèbres ainsi obtenu est l’objet de cet article : les algèbres de von Neumann engendrées par un nombre finind’opérateurst-gaussiens.

Il s’avère qu’à n fixé, si t est suffisamment proche de 1, alors ces algèbres ne dépendent pas de t. Plus généralement, on donne une condition qui assure un isomorphisme entre un produit libre conditionnel et un produit libre réduit usuel.

1. Introduction

The notions of free and non commutative probabilities originally ap- peared in the works of Voiculescu in the 80’s (see [16] for instance) to study von Neumann algebras, in particular the von Neumann algebraL(Fn)associated to the free group withn generators. Since then, this domain has expanded rapidly, and is now considered as a subject in itself. It has con- nexions with combinatorics, classical probabilities, mathematical physics and of course operator algebras. Naturally, people become interested in finding generalizations or deformations of the free probability or the free product constructions to obtain new non commutative probability spaces,

Keywords:Conditionnal free product, interacting Fock space.

Math. classification:46L54, 46L10.

(*) The author would like to thank the Wrocław department of mathematics for introducing him to the subject and also for their kind hospitality.

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especially from the combinatoric point of view. One of the first successful attempt was made by Bożejko and Speicher ([7], see also [5]) in introducing the so calledq-deformation of the free factor. These von Neumann algebras were studied in the last years, at the moment it is known that they share some properties with the free group algebras: they are factor [14], non injec- tive [13], and solid [15] (for some values of the parameterq). Nevertheless, it is still unknown if they really differ from the free group algebras.

With the same spirit, Bożejko, Leinert and Speicher introduced in [6]

the concept of conditional freeness. It is closely related to the construction of completely positive maps on free product made by Boca in [2, 3]. They were able to describe the combinatoric underlying this notion in a way similar to that of the freeness in terms of Cauchy Transforms and their reciprocal. The school of Accardi also developed its own deformation by considering the algebra generated by position operators on an interacting Fock space (see [1]). Unfortunately, very few papers on these two topics were concerned with the study of the resulting von Neumann object. One of the motivation for the present work is to start such a study like in theq-case.

We will mainly focus on specific examples of von Neumann algebras, the t-gaussian von Neumann algebras. They have the advantage to be both a model for conditional freeness and an algebra on an interacting Fock space.

They even appear as a limit object for the theory of thet-convolution of measures of Bożejko and Wysoczański in [8]. So they seem to be quite central in all those deformations of the free probability.

As an illustration, we recall the definition of the conditional freeness, and explain how thet-gaussian algebra appears. Let (Ai, φi, ψi)be ∗-algebras equipped with a pair of states. Assume that allAi’s lie in a bigger algebra Awith a stateφ. Then we say that the algebrasAi’s are conditionally free forφprovided that whenever

a_j ∈A_i_j, i₁6=i₂6=...6=i_n, ψ_i_j(a_j) = 0 we have

φ(ai₁...ai_n) =φi₁(ai₁)...φi_n(ai_n).

It is clear that for any given (Ai, φi, ψi) it is possible to construct φ on the free productA=∗ⁿ_i=1A_i so that theA_i’s are conditionally free. In the situation whereφi =ψi, one recovers the classical notion of freeness, that isφis the free product of theψ_i. We letψbe the free product of theψ_i on A.

Given some probability measuresµ_i, ν_i(with compact support), one can consider them as states on the space of polynomials in one variableC[Xi]

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in a natural manner by the formula µi(X_i^k) = Z

x^kdµi(x).

The free product ofnalgebras of polynomials in one variable is exactly the space of polynomials innnon commuting variablesChXiii6n. So, using the above construction, from the measuresµi, νi one can build two new states φ and ψ on ChX_ii, so that the algebras (C[X_i], µ_i, ν_i) are conditionally free in(ChXii, φ, ψ). The distribution ofX1+X2 with respect toφis the compactly supported measureµso that

φ((X₁+X₂)^k) = Z

x^kdµ(x).

The c-free convolution of(µ1, ν1)and(µ2, ν2)is then defined as (µ1, ν1)^c(µ2, ν2) = (µ, ν1ν2)

wheremeans the usual free convolution (i.e.ν1ν2is the distribution of X1+X2 with respect to ψ). This operation^c on couples of measures is commutative and associative. Limits theorems were obtained in this frame- work. For instance, a central limit theorem consists in finding the possible limit distributions ofX₁+...+X_n/√

n, where theX_i are identically dis- tributed (and centered) and are conditionally free. The measures (µ, ν) appearing at the limit, are parameterised by the second moments ofµand ν. So up to normalizationµt(X²) = 1andνt(X²) =t. The von Neumann algebra arising from the GNS construction of the first state on the conditional free product∗ⁿ_i=1(C[X_i], µ_t, ν_t), isΓ_t,nthet-gaussian algebra withn generators. Whent= 1, this is just the classical object of free probabilities, that isΓ_1,n=L(Fn).

The main result of this paper states that forn>2 Γt,n=

( Γ1,n if t∈h

n n+√

n,_n−ⁿ^√_ni Γ1,n⊕B(`2) otherwise ,

where as usual B(`₂) stands for the bounded operators on a separable Hilbert space. Consequently, this result is a bit disappointing for operator algebras, as the deformations may not produce new objects.

In the next section, we give a precise construction ofΓ_t,narising from the theory of one mode interacting Fock space and very basic results about it.

The third section is devoted to the proof of the main result. In the last section, we give some conditions that ensure the equality between the reduced free product of commutative von Neumann algebras(A_i, ψ_i)and their conditional reduced free product (Ai, φi, ψi). These conditions are satisfied

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for the t-gaussian algebras (in the first case), however, we prefer to give slightly different proofs for these examples as one can extract information from them to get some partial results on interacting Fock space.

2. Basics

In the whole paper, twill be a positive real number t >0. We will also use standard notations,B(H)andK(H)will denote the bounded and the compact operators on the Hilbert spaceH.

For a given Hilbert space H, with real partH_R, equipped with a scalar producth., .i, we denote byF1, the Fock space built onH:

F1=CΩ⊕k>1H^⊗k. More generally, thet-deformed Fock space is given by

Ft=CΩ⊕k>1t^k−1H^⊗k.

Here,t^k−1H^⊗k simply means the space H^⊗k where the scalar product is multiplied byt^k−1.

In the most part of the paper, we will assume thatHis finite dimensional, saydimH=n. Let(e_i)_i=1...n be a real basis of H_R. Then one can define a canonical basis forFt. Its elements ei are indexed by wordsi in the n letters1, ..., n, and

ei= 1

√t^k−1

ei₁⊗...⊗ei_k for i=i1...ik.

As usual, fore∈ H_R, the creation operator associated toeis defined on Ft by

lt(e)Ω =e

lt(e)(h1⊗...⊗hk) =e⊗h1⊗...⊗hk.

It is well known thatlt(e)extends to a bounded operator onFt. Its adjoint is given by

l_t(e)^∗Ω = 0 lt(e)^∗h=hh, eiΩ

lt(e)^∗(h1⊗...⊗hk) =t < h1, e > h2⊗...⊗hk. Thet-gaussian associated toeis the operators^t(e) =lt(e) +lt(e)^∗.

We are interested in the von Neumann generated by thet-gaussians. Let s^t_i=lt(ei) +lt(ei)^∗ fori= 1, ...n.Ct,n andΓt,nrefer to theC^∗-algebra and the von Neumann algebra generated by thes^t_i’s, i= 1, ..., n:

Γt,n ={lt(e) +lt(e)^∗;e∈ H_R}⁰⁰⊂B(Ft).

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This deformation of the Fock Hilbert space is a particular case of inter- active Fock spaces. This type of objects was introduced in [1]. The basic idea is to modify the scalar product at each levelH^⊗k by a positive scalar λk.

The vector stateh.Ω,ΩionB(Ft)is called the vacuum state and will be denoted byφ. In the caset= 1,φis a trace on Γ1,n, so we will prefer the notationτ.

In the following, we sum up all basic results about the von Neumann algebra generated by a singlet-gaussians^twhenH=C.

Proposition 2.1. — Γt,1 is in GNS position with respect toφ, which is faithful on it.

The distribution ofs^twith respect toφis given by 1

2π

√ 4t−x²

1−(1−t)x²1_[−2^√_t,2^√_t]dx if t> ¹2

1 2π

√4t−x²

1−(1−t)x²1_[−2^√_t,2^√_t]dx+ 1−2t 2−2t(δ 1

−√

1−t

+δ√1 1−t

) if t < ¹₂.

The map ρ : Γ_t,1 → Γ_1,1 given by ρ(s^t) = √

ts¹ extends to a normal representation.

Moreover, given i 6 n, there are ∗-isometric normal representations πi : Γ_t,1→Γ_t,n given byπ(s^t) =s^t_i.

Proof. — The computations of the distribution of s^t can be found in [6, 8, 17]. TheG-transform of the distribution ofs^tforφis given by

G_st(z) = (¹₂−t)z+¹₂√ z²−4t

z²(1−t)−1 = 1

z− 1

z− t

z− t z− t

. ..

It is obvious thatΩis cyclic forΓ_t,1so is also separating since the algebra is commutative andφis faithful. In particular, the spectral measure of√

ts¹ is absolutely continuous with respect to that of s^t, this implies that the mapρ: Γt,1→Γ1,1 is well defined and normal.

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Note that the matrix ofs^t in the natural orthonormal basis is given by





 0 1

1 0 √

√ t

t 0 √

t . .. . .. . ..

√t 0 √ t . .. . .. . ..





 .

So at theC^∗-level,ρis just the restriction of the Calkin map (ρ:B(F1)→ B(F₁)/K(F₁), whereKstands for the compact operators) sinces^t−√

ts¹ is of rank 2 and Γ1,1 does not meet the compact operators (no atoms in the measure).

In Γt,n, s^t_i is unitarily equivalent to s^t⊕(√

ts¹)∞ corresponding to its decomposition in reducing subspaces (they are indexed by reduced word

starting by a letter which is not ai).

Remark 2.2. — There are natural isometries between allFt by identi- fying the canonical basis. This allows to consider all Γt,n as subalgebras ofB(F₁). When we will talk about the Calkin map, we mean the classical quotient mapρ:B(F1)→B(F1)/K(F1). Using this identification, we have ρ(C_t,n) =ρ(C_1,n)for any0< tand16n <∞.

Remark 2.3. — From the densities, it is clear that as von Neumann algebras, we have

Γt,1=

( Γ1,1 ⊕ C² if t < ¹₂ Γ1,1 if t> ¹2.

Let A_i ≈Γ_t,1 be the algebra generated by s^t_i in Γ_i,n. By the previous proposition, there are two given states onAi. One is coming from the vacuum state (denoted byφ, no confusion since it coincides with the vacuum on Γt,n) and another one coming from the vacuum state on Γ1,1 that is τ ρπ_i⁻¹ (call itψeven if it depends oni).

The following is well known

Proposition 2.4. — The algebras(A_i, φ, ψ)are conditionally free with respect to the vacuum state, that is

φ(a1...ap) =φ(a1)...φ(ap)wheneveraj∈Ai_j, i16=i26=...6=ipandψ(aj) = 0.

Moreover(Γt,n, φ,Ω)is in GNS position.

We postpone the proof of this fact to the next section.

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Let U be an orthogonal transformation on H_R and still denote its ten- sorization withId_C byU. The first quantization Γ(U)ofU is the unitary onF_tgiven by

Γ(U) = Id_CΩ⊕k>1U^⊗^k.

Then,Γ_t,n is stable by conjugation byΓ(U)and for anye∈ H_R, Γ(U)s^t(e)Γ(U)^∗=s^t(U e).

3. The von Neumann algebra Γ_t,n 3.1. Factoriality

The following is well known to specialists:

Proposition 3.1. — LetM ⊂B(H)be a von Neumann algebra, if M contains a non zero compact operator, then eitherM =B(K)⊗Idd with H = K^d and d is the smallest rank of a non zero compact projection in M orM is not a factor and has a direct summand isomorphic toB(K)for someK.

Proof. — We know thatM contains a finite rank projection. So letpbe a finite rank projection of minimum rankd>1. Let(pi)i∈I be a maximal family of mutually orthogonal projections equivalent top(so allpi’s have the same finite rank). Letq=Pp_i, we show thatqis central.

Ifqis not central then there is somex∈M so that(1−q)xq6= 0. Hence we can assume thatRanxis orthogonal toRanqandxq6= 0. There must be an i so that xpi 6= 0. Consider xpix^∗, it is non zero and of rank less or equal tod. By definition of d, p0 = xpix^∗ must have at most one non zero eigenvalue, and has at least one asp06= 0. Sop0 is a multiple (say 1) of a projection of rankd. Moreover,pix^∗xpi is also a non zero projection hence it must be equal top_i. Sop₀ is equivalent top. But the range of p₀ is orthogonal toqso p0q=qp0= 0, this contradicts the maximality of I.

Soqis central.

Let ui,j be the partial isometries between pi and pj i, j ∈ I. It is not hard to see thatui,j is a system of matrix units that generates qM. So if q= 1thenM is a factor, elseM is not a factor and has a direct summand

isomorphic toB(`2(I)).

In the cased= 1, the argument is much simpler: letξbe in the range ofp as above. ThenM contains all projections onK=M.ξ(using conjugation of p). So if K 6= H, then the projection q onto K belongs to both M⁰

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(by definition) and M (because q = Pp_i with p_i projection onto lines corresponding to an onb in K), so M is not a factor and has a direct summand isomorphic toB(K). IfK=H the same kind of arguments gives thatM =B(K).

3.2. Orthogonal polynomials

LetU_n be the Tchebychef polynomial of the second kind of degreen.

Proposition 3.2. — The orthonormal polynomials fors^t with respect toφare given by:

v₀(X) = 1 v₁(X) =X =√

tU₁ X 2√ t

v_n(X) =√ t

U_n X

2√ t

−1 t −1

U_n−2 X 2√ t

forn>2.

The orthonormal polynomials for√

ts¹are given byun(X) =Un

X 2√ t

. Proof. — According to the continued fraction decomposition of the G- transform of the measure, the unital orthogonal polynomials must satisfy the relations

XP_n(X) =P_n+1(X) +tP_n−1(X) forn>2 XP1(X) =P2(X) +P0(X), P0(X) = 1, P1(X) =X.

Then one deduces that this relation is satisfied withP_n(X) = 2ⁿ√

tⁿ⁻¹v_n(X).

See also [4, 17] for detailed proofs.

The polynomialsunandvnare related one to each other, lettingα= ¹_t−1

(R)

vn=√ t

un−αu_n−2 u_2n=αⁿv₀+ 1

√t

n

X

k=1

α^n−kv_2k

u_2n+1= 1

√t

n

X

k=0

α^n−kv_2k+1.

Lemma 3.3. — At the algebraic level, we have for i1 6= ... 6= il and α_j >1, withi=i^α₁¹...i^α_l^l:

uα₁(s^t_i₁)...uα_l−1(s^t_i_l−1)vα_l(s^t_i

l)Ω =ei.

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Proof. — We havev_α_l(s^t_i

l)Ω =e_iαl l

as thev_j are the orthonormal polynomials fors^t_i

l. Now,s^t_i_l−1 acts on tensors starting with a letteril exactly as√

ts¹_i_l−1 so uαl−1(s^t_i_l−1)vα_l(s^t_i_l)Ω =e_iαl−1

l−1 i^αl_l , and so on.

Proof of Proposition 2.4. — Using multilinearity, we need to show that fori₁6=...6=i_landα_j>1, we have

φ u_α₁(s^t_i

1)...u_α_l(s^t_i

l)

=φ(u_α₁(s^t_i

1))...φ(u_α_l(s^t_i

l)).

From the relations(R), we have φ(u_α_k(s^t_i

k)) =

α^α^k^/2 ifαk is even 0 otherwise.

But by the previous lemma

0 = 1

√tφ uα₁(s^t_i₁)...vα_l(s^t_i

l)

=φ uα₁(s^t_i₁)...uα_l(s^t_i

l)

−αφ uα₁(s^t_i₁)...uα_l−2(s^t_i_l) . Then we conclude by induction onPαk.

The statement about the GNS position can then be easily deduced from

the conditional freeness (see Lemma 4.1).

Let

η= Ω−√ tα

n

X

k=1

ekk. Define a functional onB(F_t)by:

ψ(x) =˜ hxΩ, ηi.

Lemma 3.4. — The functionalψ˜coincides with τ ρonCt,n.

Proof. — The set of polynomials in the letters s^t_i is dense isC_t,n. As a consequence, the linear span of productsud₁(s^t_i₁)...ud_m(s^t_i_m), with dj >1, m>0 and i₁ 6=i₂ 6=...6= i_m is dense inC_t,n. So we only need to prove that as soon asm>1 (using freeness):

0 = hud1(s^t_i₁)...udm(s^t_i_m)Ω, ηi

= D

ud₁(s^t_i₁)...ud_m(s^t_i_m)Ω, 1−√

tα

n

X

k=1

v2(s^t_k) ΩE

= φ

1−√ tα

n

X

k=1

v2(s^t_k)

ud₁(s^t_i₁)...ud_m(s^t_i_m)

! . So we need to evaluate expressions of the form

φ(ab1....bl)

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with a ∈ A_j₀, b_u ∈ A_j_u with j₀ 6= j₁ 6= ... 6= j_l+1 and ψ(b_i) = 0 (with possiblyl= 0). Using conditional freeness, and denoting byb=b1...bl and x=φ(b) =φ(b₁)...φ(b_l)

φ(ab) = φ (a−ψ(a))b

+ψ(a)φ(b)

= φ(a)φ(b).

Using this equality, we get asφ(v2(s^t_i)) = 0 hud1(s^t_i

1)...u_d_m(s^t_i

m)Ω, ηi = φ 1−√

tαv₂(s^t_i

1) u_d₁(s^t_i

1)...u_d_m(s^t_i

m) . In this scalar product, there will be a factor

φ 1−√

tαv2(s^t_i₁)

ud₁(s^t_i₁) .

From the formula above, ifd₁is odd thenu_d₁ is in the span of{v_2k+1;k>

0}, so this quantity is 0. Ifd1= 2p>2then φ

1−√ tαv₂(s^t_i

1) u_d₁(s^t_i

1)

= φ

1−√ tαv₂(s^t_i

1)

α^p+ 1

√t

p

X

j=1

α^p−jv2j(s^t_i₁)

= α^p−α.α^p−1= 0.

Corollary 3.5. — The mapρextends to a normal (surjective) repre- sentationΓt,n →Γ1,n.

Proof. — From the above lemma, it follows that ψ˜ is positive on Ct,n. As it is normal, it extends to a normal state onΓ_t,n (that we will denote by ψ). Now, the GNS representation ofψgives the normal representation.

Corollary 3.6. — Γ_t,n has a direct summand isomorphic toΓ_1,n. Remark 3.7. — If the mapρis not isomorphic, thenΓ_t,nis not a factor.

Otherwise it is a typeII1factor.

Remark 3.8. — Denoting byc^t= (s^t₁)²+...+ (s^t_n)², we have η =α

(n+ 1 α)−c^t

Ω.

We will study this operator in the next sections.

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3.3. Caset < n/(n+√

n)and t > n/(n−√ n) 3.3.1. Caset <1/2

In this section, we focus on the case t <1/2. It is a particular case of the next one but the arguments are simpler.

Whent <1/2, the measure ofs^tcontains one atom at the point ^√¹

1−t>

√2t. Since in the Calkin algebraρ(s^t) =ρ(√

ts¹)≈√

ts¹, we deduce that Ct,1contains a compact operator, corresponding to the point projection on

√1

1−t (denoted by P) computed on s^t. Since Ωis cyclic, this projection is one dimensional. From the decompositions^t_i ≈s^t⊕(√

ts)_∞, we also get thatp=P(s^t_i)is a one dimensional projection. The range ofpis the linear span of

ξ_i= Ω + 1

√1−t

∞

X

k=1

α^1−k² e_ik. This vector makes sense as0< α <1 when0< t <1/2.

According to the Remark 3.7 and Proposition 3.1, on one hand Γ_t,n (∞> n>2) is eitherB(`2(I))or not a factor. And on the other hand it is not a factor or is typeII1.

Corollary 3.9. — Fort <1/2, the von Neumann algebra Γt,n is not a factor for26n <∞. Moreover it has a direct summand isomorphic to B(`₂)and another one toΓ_1,n and the stateφis not faithful.

Lemma 3.10. — The vectorξ1is not cyclic forΓt,n.

Proof. — It suffices to show that η is orthogonal to Γ_t,nξ₁. But ξ₁ = P(s^t₁)Ω, for P a multiple of the Dirac function at ^√¹

1−t. For anyx∈Γt,n, we have

hxξ1, ηi=hxP(s^t₁)Ω, ηi=τ(ρ(xP(s^t₁))).

But we know thatρ(P(s^t₁)) = 0sinceP(s^t₁)is compact.

Proof. — The projectionP(s^t₁)is of course minimal, so asΓt,nξ1is infinite dimensional (forn>2), for theB(`₂(I))summand provided by Propo- sition 3.1,Iis infinite. Then, the stateφcan not be faithful because of the

B(`₂)summand.

3.3.2. Caset < n/(n+√

n)andt > n/(n−√ n)

Lemma 3.11. — Fort /∈h

n n+√

n, ⁿ

n−√ n

i

, theC^∗-algebraC_t,ncontains a compact operator.

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First proof. — It is possible to compute explicitly the distribution of c^t= (s^t₁)²+...+ (s^t_n)² forφusing the R-transforms machinery developed in [6]. We denote by γi =s^t_i², these variables are conditionally free with respect to the distribution ofts¹². From Theorem 5.2 in [6], we know the following relations

Rc^t(z) =nRγ_i(z) G_ct(z) = 1

z−R_ct(G_tc1(z)).

TheR andG-transforms oftc¹ are obtained as usual using freeness and a change of variable. The computation gives

G_γ_i(z) =1 2

1−2t+p

1−4t/z z(1−t)−1 R_γ_i(z) = 1

1−tz Rc^t(z) = n

1−tz R_tc1(z) = nt

1−tz

G_tc1 =(1−n)t+z−p

(n−1)²t²−2(n+ 1)tz+z² 2tz

G_ct =(2t−1)z+ (1−n)t−p

(n−1)²t²−2(n+ 1)tz+z² 2z

(t−1)z+n+t(1−n) . We can recover the distribution of c^t with respect to φ from the G- transform. It turns out that it has atoms at ^n+t(1−n)_1−t = n+ ¹_α provided thatt /∈h

n n+√

n, ⁿ

n−√ n

i .

Actually an eigenvector for the eigenvaluen+_α¹ is given by ζ=√

tΩ +X

k>1

1 (nα)^k

X

|i|=2k i_2j+1=i_2j+2

ei.

Second proof . — We use the Khinchine inequalities for free products see [11, 9].

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Consider the operator

Tk = X

p>1 k1+...+kp=2k

kieven i1∈{1,...,n}

i₁6=i26=..6=ip

uk1(s^t_i₁)uk2(s^t_i₂)...ukp(s^t_i_p).

In the Calkin algebra, we can identify ρ(Tk) = X

p>1 k1+...+kp=2k

kieven i₁∈{1,2}

i16=i26=..6=ip

uk₁(√

ts¹_i₁)uk₂(√

ts¹_i₂)...uk_p(√ ts¹_i_p).

Thus, we have

ρ(T_k)Ω = X

|i|=2k i2j+1=i2j+2

e_i.

Andkρ(T_k)Ωk₂=n^k/2.

Asρ(Tk)Ωis homogeneous of degree2k, from the Khinchine inequalities kρ(T_k)k6(2k+ 1)kρ(T_k)Ωk₂.

But expanding the polynomials using conditional freeness, it comes that φ(Tk) =n^kα^k.

So if n|α| > √

n then ρ is not isometric and hence there is a compact

operator inCt,n.

Remark 3.12. — Let f(x) = 1

2xπ q

x−t(1−√ n)²

t(1 +√

n)²−x

(t−1)x+n+t(1−n) 1_[t(1−^√_n)2,t(1+√ n)²]. The distribution ofc^twith respect toφis

f(x)dx ift∈h

n n+√

n,_n−ⁿ^√_ni f(x)dx+

(n−1) t−_n+ⁿ^√_n

t−_n−ⁿ^√_n n(1−t)²+t(1−t) δ_n+1

α ift /∈h

n n+√

n, ⁿ

n−√ n

i.

Lemma 3.13. — We haveker c^t−(n+_α¹)

=Cζ.

Proof. — We will prove it in several steps. We already know that ζ is one eigenvector forc^t. By valuation of a vector inFt, we mean the index of its first non zero component according to the natural filtration ofF_t. Soζ has valuation 0. In the following, we letξ be one eigenvector (if it exists!)

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not inCζ. We can assume that the valuation ofξis bigger than 1 and that ξis real.

From the computation of theG-transforms above, the spectrum oftc¹ is exactly the interval[t(1−√

n)², t(1 +√

n)²](as τ is faithful).

First step: The valuation ofξis 1.

First notice than on tensors of length bigger than 3,c^tacts exactly astc¹. Sincektc¹k< n+_α¹, there is no eigenvector with valuation bigger than 3.

Now we focus on the valuation 2. OnH ⊗ H. We consider the basis given byf1=e11+...+enn, f2=e11−e22, ..., fn =e11−enn, and the vectors f_ij =e_ij withi6=j.

First the component ofξonf1 is 0. Indeed, it is clear thathc^tf1,Ωi=n and for other basis vectorsf (and vectors of valuation bigger than 3), we havehc^tf,Ωi= 0. So ifξhas valuation 2, necessarily

0 = (n+ 1

α)hξ,Ωi=hc^tξ,Ωi=√

nhξ, f1i.

On the other hand, as above, on vectors of valuation 2 with no component onf1,c^tacts astc¹. So there is no such vectors.

Consequently, dim kerc^t−(n+ _α¹) 6 n+ 1. Since c^t is invariant by conjugation with unitaries coming from the first quantization. If ξ has valuation 1, we can assume that the component of degree1 ofξise1.

Second step: ξ∈Span{(c^t)^ke1;k>1}.

Let f be a continuous function on Rvanishing on the spectrum of tc¹ strictly smaller than 1 except thatf(n+_α¹) = 1. Asρ(c^t) =tc¹, it follows thatf(c^t)is self-adjoint compact. Modifyingf, we can assume thatf(c^t) is exactly the projection ontokerc^t−(n+_α¹).

Then, f(c^t)e1 is non zero as

hf(c^t)e1, ξi=he1, f(c^t)ξi=he1, ξi= 1.

So f(c^t)e₁ is one eigenvector. It is collinear with ξ for its component of degre 1 is collinear toe1and is non vanishing as there is no eigenvector of valuation greater than 2.

Third step: ξdoesn’t exist.

It is clear by induction that Span{(c^t)^ke1;k > 0} = Span{fk, k > 0}

with

fk = X

|i|=2k i_2j+1=i_2j+2

ei1.

We letξ=P

k>0xkfk. we have

c^tf_k=ntf_k−1+ (n+ 1)tf_k+f_k+1 k>1 c^tf0= (nt+ 1)f0+tf1.

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Consequently the sequencex_i has to satisfy the recursion formula:

(n+_α¹)x₀= (nt+ 1)x₀+ntx₁

(n+_α¹)xk=ntxk+1+ (n+ 1)txk+tx_k−1 k>1.

Hence,xk=a(nα)^−k+bα^k, with b6= 0as soon as n6= 1.

But then with this values the series Px_kf_k is not convergent ! So ξ

doesn’t exists

Theorem 3.14. — Forn>2 andt /∈h

n n+√

n,_n−ⁿ^√_ni

, as von Neumann algebras, we have

Γt,n =B(`2)⊕Γ1,n. Moreover the stateφis not faithful onΓt,n.

Proof. — Let 1−q be the central support of the representation ρ. As above, we know thatq6= 0. We have to show thatqΓ_t,n=B(`₂).

Since Ωis cyclic forΓt,n,qΩis cyclic inqFt forqΓt,n (hence non zero).

Letpbe the projection ontoCζ. We have that forx∈Γ_t,n 0 =τ(ρ(qx)) =hqxΩ, ηi=αhxΩ, c^t−(n+ 1

α) qΩi.

So(c^t−(n+_α¹))qΩ = 0, andqΩ =λζ for someλ6= 0. Sop6q, andqΓt,n

contains a one dimensional projection on a cyclic vector so is isomorphic to B(`2) (as Γt,nζ is infinite dimensional). The state can not be faithful

because of theB(`2)summand.

3.4. Case n/(n+√

n)< t < n/(n−√ n)

At the algebraic level, we have already seen that for i1 6=... 6= il and α_j >1, withi=i^α₁¹...i^α_l^l:

uα₁(s^t_i₁)...uαl−1(s^t_i_l−1)vα_l(s^t_i_l)Ω =ei. We define an antilinear mapS onSpanei by

S(ei) =vα_l(s^t_i

l)uα_l−1(s^t_i_l−1)...uα₁(s^t_i₁)Ω.

Lemma 3.15. — The map S satisfies that for any i₁, ..., i_l (not neces- sarily withi16=...6=il) and any polynomialsP1, ..., Pl:

S(P1(s^t_i₁)...Pl(s^t_i_l)Ω) =Pl(s^t_i_l)...P1(s^t_i₁)Ω.

Proof. — Clear by induction onn.

Lemma 3.16. — For _n+ⁿ^√_n < t < _n−ⁿ^√_n,S extends to a bounded oper- ator.

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Proof. — We decompose S. Let i₁ 6= ... 6= i_l and α_j > 1 and define antilinear operators by:

A(e_i) =u_α_l(s^t_i

l)u_α_l−1(s^t_i_l−1)...u_α₁(s^t_i

1)Ω B(ei) =uα_l−2(s^t_i_l)uα_l−1(s^t_i_l−1)...uα₁(s^t_i₁)Ω.

with the convention thatu_k= 0 ifk <0. ThenS=√

t(A−αB).

From the relations between v and u, it follows that A sends tensors of lengthl to a sum of tensors of lengthl,l−2, .... Fixk>0, and denote by Ak the component ofAthat sends tensors of lengthl to tensors of length l−2k. Let J be the antiunitary ofFt that reverses the order of tensors.

PutC=Pn

s=1l1(es)²J, then it is not hard to see that for|i| − |j|= 2k:

hAke_i, e_ji=α^kf(i, j)hC^ke_i, e_ji.

wheref(i, j)can take the value1 or ^√¹

t.

Since the coefficients of C are all positive, we get that kA_kk6 |α|^k

√t kC^kk.

And asC^∗C=nId, we get that kAk6 1

√t

∞

X

k=0

(√ n|α|)^k which is convergent provided that _n+ⁿ^√_n < t < _n−ⁿ^√_n.

The same kind of arguments shows thatB is bounded.

Now, we assume that _n+ⁿ^√_n < t < _n−ⁿ^√_n.

Lemma 3.17. — One has that for anyi16=... 6=il andk6nand any polynomialsP₁, ..., P_l:

h

SP1(s^t_i₁)...Pl(s^t_i_l)S, s^t_ki

= 0.

Proof. — Just a writing game, and the boundedness ofS.

Corollary 3.18. — For _n+ⁿ^√_n < t < _n−ⁿ^√_n, the state φis faithful on Γt,n.

Proof. — Ω is cyclic for Γ⁰_t,n as by the previous lemma SΓt,nS ⊂Γ⁰_t,n

andS is invertible (becauseS²= Id).

Corollary 3.19. — For _n+ⁿ^√_n < t < ⁿ

n−√

n,Γ_t,ndoes not contain any compact operator. Moreover theC^∗-algebrasC_t,nandC_1,nare isomorphic.

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Proof. — If Γ_t,n contains a compact operator, then as above there is a direct summand ofΓt,n isomorphic to B(`2(I)). But as Ωis separating, I must consist of one point, which is impossible ass^t₁ as no eigenvector.

The second assertion follows from the first one since the Calkin map is

then an isomorphism in both case.

Theorem 3.20. — For _n+ⁿ^√_n < t < _n−ⁿ^√_n, the von Neumann algebra Γt,n is isomorphic toΓ1,n.

Proof. — One just needs to show that the mapρis faithful. To do so it suffices to show thatψis faithful.

Letb=S(1−√ tαPn

i=1v2(s^t_i))S∈Γ⁰_t,n. We have forx∈Γt,n

ψ(x) =hb^∗xΩ,Ωi.

Applying it tox=y^∗y and asΩis cyclic it follows thatbis positive. Say b=a², then

ψ(x^∗x) =kxaΩk².

Moreover the distribution ofb forφ is absolutely continuous with respect to the Lebesgue measure (see Remark 3.12), it follows that there is a net of elements(ci)in the von Neumann generated by bso thatciaΩ→Ω. Now, if forx∈Γt,n, ψ(x^∗x) = 0then

0 = limcixaΩ = limxciaΩ =xΩ.

Sox= 0as Ωis cyclic.

Remark 3.21. — The Tomita-Takesaki operator S is bounded onΓ_t,n. Remark 3.22. — It is possible to reverse all the arguments above. To do so, define a normal linear formφ˜onΓ1,n by

φ(x) =˜ D xΩ,X

k>0

α^k X

|i|=2k i2j+1=i2j+2

e_iE .

It is well defined because the vector on the right side makes sense as

√n|α|<1.

It is clear that on the set of polynomials in s¹_i, it coincides with φ. By continuity, we get that onΓt,n, we have

φ= ˜φρ.

Then the GNS construction ofΓ_1,n forφ˜givesρ⁻¹.

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3.5. Case t=n/(n±√ n)

In this situation, we will adopt the strategy of the last remark.

Let P be the set of noncommutative polynomials in the letters X1, ..., Xn. It has a natural∗-algebra structure.

For0< r <1, we define a linear functional φ_r onP: for i₁ 6=...6=i_l and αj >1

φ_r

u_α₁(X_i₁)...u_α_l(X_i_l)

= (

(r²α)P_α

i/2 if allα_i are even

0 otherwise.

Formally, we can identifyP with the∗-algebra generated by√

ts¹_i inC1,n

(sayπ(Xi) =√

ts¹_i) and to∗-algebra generated bys^t_i inCt,n (sayσ(X1) = s^t_i).

Let Tr be the second quantization associated to rId on Γ1,n, it is unital completely positive and letπ(P)invariant.

The functionalφr is made so that φr(P) =φ

σ(π⁻¹(Tr(π(P)))) . Indeed, fori16=...6=il andαj>1

π⁻¹(T_r(π(u_α₁(X_i₁)...u_α_l(X_i_l)))) = π⁻¹(T_r(u_α₁(√ ts¹_i

1)...u_α_l(√ ts¹_i

l)))

= π⁻¹(rP_α

iuα₁(√

ts¹_i₁)...uα_l(√ ts¹_i_l))

= rP_α

iuα₁(Xi₁)...uα_l(Xi_l).

and by conditional freeness:

φ

σ(uα₁(Xi₁)...uα_l(Xi_l))

= φ

uα₁(s^t_i₁)...uα_l(s^t_i_l)

= (

αP

αi/2 if allα_i are even

0 otherwise.

Lemma 3.23. — The functionalφr is positive onP.

Proof. — The ∗-algebra Ai generated by Xi in P has two particular functionals on it. The first one is the restriction ofφr(still denoted byφr) and the second one isψπ. From the definition ofφrthe algebra(Ai, φr, ψπ) are conditionally free. Hence the result will follow from [6] Theorem 2.2 (or Proposition 3.2 in [2]) provided that we can show the positivity ofφ_r and ψπ on each Ai. For ψπ this is obvious asψ is a state. For φr, it suffices to notice that the distribution ofX_i forφ_rthe distribution ofrs^t_i forφ, so

comes from a probability measure.

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Corollary 3.24. — The functional φ_rπ⁻¹ extends to a normal state onΓ1,n(denoted byφr).

Proof. — By the preceding Lemma, it is positive onπ(P)which is weak-∗

dense inΓ1,n. The result follows from the representation φr(π⁻¹(x)) =D

xΩ,X

k>0

(r²α)^k X

|i|=2k i_2j+1=i_2j+2

ei

E .

It is well defined because the vector on the right side makes sense as r²√

n|α|=r²<1.

Theorem 3.25. — TheC^∗-algebrasC_n/(n±^√_n),nand C1,n are isomor- phic and the stateφis faithful onC_n/(n±^√_n),n.

Proof. — There is a state onC1,ndefined byφ˜= lim_r→1,Uφr, where the limit is taken along some non trivial ultrafilter. It is clear from the formulas onP, that forx∈Ct,n, we have

φ(x) = ˜φ(ρ(x)).

We consider the GNS construction ofφ˜forC1,n. The underlining Hilbert space is exactlyF_n/n±^√_n from the above identities, so this GNS construction is the inverse of the Calkin mapρ.

Assume x∈C_n/(n±⁺ ^√_n),n is such thatφ(x²) = 0, thenxΩ = 0. It follows that ψ(ρ(x)) = hxΩ, ηi = 0. So ρ(x) = 0 as ψ is faithful on Γ_1,n, so

x= 0.

The analogue result for the von Neumann algebras is a consequence of the next section. In particular it follows that the mapφ˜is actually normal.

3.6. Case n=∞

This case was already treated in [18] and was a starting point to this work.

Ift6= 1then it can be checked that 1

k

X

i=1

(s^t_i)² −→

k→∞(1−t)P+tId.

in the weak-* topology, whereP is the orthogonal projection ontoΩ. AsΩ is cyclic, Proposition 3.1 ensures that

Γ_t,∞=B(F_t).

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4. Generalizations

In this section, we explain how to extend the previous results.

Let I_i be bounded measurable subsets of R,A_i=L_∞(I_i, µ_i)be ncom- mutative von Neumann algebras. The integration with respect toµi will be calledφ_iands_iis the identity function onI_i. Assume that we are given ψi, distinguished normal states on Ai. As a consequence,ψi has a density fi with respect toφi, we putf˚i =fi−1. We will denote byvⁱ_k anduⁱ_k the orthogonal polynomials forφi andψi.

We consider the algebraic conditional free product A˜=∗ⁿ_i=1(Ai, φi, ψi).

The stateφis the conditional free product state andψis the free product of theψi. Then we can talk about the von Neumann reduced conditional free product, corresponding to GNS construction ofφ, we denote it byA

A⊂B(F) φ=h.Ω,Ωi.

F is the conditional Fock space and Ωthe vacuum state, we refer to [12]

Section 6 for more details about this construction. Actually this is also a particular case of the construction of completely positive maps on full free products by Boca in [2, 3] when the amalgamation is over the com- plex field. One of the main trouble comes from the fact that in general the stateφis not faithful onA(fort-gaussians for instance). Nevertheless, the injections i_i : A_i → A are normal and isometric and preserve the states (i.e.i^∗_i(φ) =φi, so we will drop the indexes) (this corresponds to Propo- sition 2.1). Unfortunately, it seems that in general there is no conditional expectation fromA toAi which is state preserving.

In this situation, the Lemmas 3.3 and 3.4 remain true. Fori₁6=... 6=i_l andαj >1, leti=i^α₁¹...i^α_l^l and define:

e_i=uⁱ_α¹

1(s_i₁)...uⁱ_α^l−1

l−1(s_i_l−1)vⁱ_α^l

l(s_i_l)Ω.

Lemma 4.1. — The family(e_i)i∈{1,...,n}^<∞is an orthonormal basis ofF. If the densitiesfi are bounded (or inL²(µi)), the stateψon the algebraic free product extends to a normal state ofA.

Proof. — To prove the first assertion we will need to evaluate expressions of the formφ(ab1...bld), wherea∈Ai₀,bk∈Ai_k,ψi_k(bk) = 0andd∈Ai_l+1.