COMPLETELY POSITIVE MAPS ON COXETER GROUPS AND THE ULTRACONTRACTIVITY OF THE

INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1998

COMPLETELY POSITIVE MAPS ON COXETER GROUPS AND THE ULTRACONTRACTIVITY OF THE

q -ORNSTEIN–UHLENBECK SEMIGROUP

M A R E K B O ˙ Z E J K O

Institute of Mathematics, University of Wroc law pl. Grunwaldzki 2/4, 50-384 Wroc law, Poland

E-mail: bozejko@math.uni.wroc.pl

1. Coxeter groups. In this note we give an application of the following result on the symmetric group S _n :

Theorem 1. For fixed n ∈ N let us consider the permutation group S n and denote by π i ∈ S n (i = 1, ..., n − 1) the transposition between i and i + 1. Furthermore, let operators T _i ∈ B(H) (i = 1, ...n − 1) on some Hilbert space H be given, with the properties:

(i ) T _i ^∗ = T i for all i = 1, ..., n − 1;

(ii ) kT _i k ≤ 1 for all i = 1, ..., n − 1;

(iii ) The T i satisfy the braid relations:

T i T i+1 T i = T i+1 T i T i+1 for all i = 1, ..., n − 2, T i T j = T j T i for all i, j = 1, ..., n − 1 with |i − j| ≥ 2.

Now let us define a function

ϕ : S n −→ B(H) by quasi-multiplicative extension of

ϕ(e) = 1, ϕ(π i ) = T i ,

i.e. for a reduced word S n 3 σ = π _i(1) . . . π _i(k) we put ϕ(σ) = T _i(1) . . . T _i(k) . Then ϕ is a completely positive map, i.e. for all l ∈ N, f _i ∈ CS n , x _i ∈ H (i = 1, . . . , l) we have

D X ^l

i,j=1

ϕ(f _j ^∗ f i )x i , x j

E ≥ 0.

1991 Mathematics Subject Classification: Primary 51F15, 20F55.

Research partially supported by KBN grant 2P03A05108.

The paper is in final form and no version of it will be published elsewhere.

[87]

By our previous result from [BSp1], Theorem 1 is equivalent to the following:

Theorem 2. Under the assumptions of Theorem 1 the operator P ⁽ⁿ⁾ = P _T ⁽ⁿ⁾ = X

σ∈S

ϕ(σ) =

= (1 + T 1 )(1 + T 2 + T 2 T 1 ) . . . (1 + T n−1 + T n−1 T n−2 + · · · + T n−1 . . . T 1 ) satisfies

P ⁽ⁿ⁾ ≥

n

Y

k=2

c _k (q) > 0, where

c k (q) = (1 − q ² ) ⁻¹

k

Y

l=1

(1 − q ^l )(1 + q ^l ) ⁻¹ . Moreover , by Gauss formula

c k (q) ≥ c(q) = (1 − q) ⁻¹

∞

Y

l=1

(1 − q ^l )(1 + q ^l ) ⁻¹ = (1 − q) ⁻¹

+∞

X

l=−∞

(−1) ^l q ^l

. In the proof we need the following lemma:

Lemma 3. If T ⁱ ∈ B(H) satisfy the braid relations of Theorem 1 , then for 1 ≤ r <

k < n − 1, we have

(T n−1 T n−2 . . . T k )(T n−1 T n−2 . . . T r ) = T n−1 (T n−1 T n−2 . . . T r )(T n−1 . . . T k+1 ).

P r o o f. The proof of the Lemma follows by induction on k:

Let k = n − 2. Then by the braid relations we get (T n−1 T n−2 )(T n−1 T n−2

| {z }

T n−3 . . . T r ) =

= T n−1 T n−1 T n−2 T n−1

| {z }

T n−3 . . . T r =

= T n−1 (T n−1 T n−2 T n−3 . . . T r )T n−1 . The next step looks as follows:

(T n−1 T n−2 T n−3 )(T n−1 T n−2 T n−3 . . . T r ) =

= (T n−1 T n−2 )(T n−1 T n−3 T n−2 T n−3

| {z }

. . . T r ) =

= (T _n−1 T _n−2 )(T _n−1 T _n−2

| {z }

T _n−3 T _n−2 T _n−4 . . . T _r ) =

= T n−1 T n−1 T n−2 T n−1 T n−3 T n−2 (T n−4 . . . T r ) =

= T n−1 T n−1 T n−2 T n−3 (T n−1 T n−2 )(T n−4 . . . T r ) =

= T n−1 (T n−1 T n−2 T n−3 . . . T r )(T n−1 T n−2 ).

Next we need the following important lemma:

Lemma 4. Let T i ∈ B(H) and

R k (T 1 , . . . , T k−1 ) = R k = 1 + T k−1 + T k−1 T k−2 + . . . + T k−1 T k−2 . . . T 1 ,

where k = 2, 3, . . . , n. Then

(a) R k (1 − T k−1 T k−2 . . . T 1 ) =

= (1 − T _k−1 ² T _k−2 . . . T ₁ )(1 + T _k−1 + T _k−1 T _k−2 + . . . + T _k−1 T _k−2 . . . T ₂ ) =

= (1 − T _k−1 ² T _k−2 . . . T ₁ )R _k−1 (T ₂ , T ₃ , . . . , T _k−1 ),

(b) R n (1 − T n−1 T n−2 . . . T 2 T 1 )(1 − T n−1 T n−2 . . . T 2 ) . . . (1 − T n−1 ) =

= (1 − T _n−1 ² T _n−2 . . . T ₂ T ₁ )(1 − T _n−1 ² T _n−2 . . . T ₂ ) . . . (1 − T _n−1 ² T _n−2 )(1 + T _n−1 ).

P r o o f. Let us start with the case k = 3. Since R 3 = 1 + T 2 + T 2 T 1 , we have R 3 (1 − T 2 T 1 ) = 1 + T 2 − T ₂ ² T 1 − T 2 T 1 T 2 T 1 =

= 1 + T 2 − T ₂ ² T 1 − T ₂ ² T 1 T 2 =

= (1 − T ₂ ² T 1 )(1 + T 2 ).

Now we consider the case k = 4. By natural calculations using Lemma 3 we get R 4 (1 − T 3 T 2 T 1 ) = (1 + T 3 + T 3 T 2 ) − (T ₃ ² T 2 T 1 )(1 + T 3 + T 3 T 2 ) =

= (1 − T ₃ ² T 2 T 1 )(1 + T 3 + T 3 T 2 ).

Therefore, using the case k = 3, we have

R ₄ (1 − T ₃ T ₂ T ₁ )(1 − T ₃ T ₂ ) = (1 − T ₃ ² T ₂ T ₁ )(1 − T ₃ ² T ₂ )(1 + T ₂ ).

Repeating this process we get the proof of the Lemma.

This implies the next lemma.

Lemma 5. If

P ⁽ⁿ⁾ = X

σ∈S

ϕ T (σ) = P ⁽ⁿ⁻¹⁾ (1 + T n−1 + . . . + T n−1 . . . T 1 ) =

= P ⁽ⁿ⁻¹⁾ R n = R 2 R 3 . . . R n , and kT _i k ≤ q < 1, then

kR _n ⁻¹ k ≤ (1 − q) ⁻¹

n−1

Y

k=1

(1 + q ^k )

n

Y

k=3

(1 − q ^k ) ⁻¹ . (∗∗)

P r o o f. By Lemma 4 we have R n =

n−2

Y

k=1

(1 − T _n−1 ² T n−2 . . . T k )(1 + T n−1 )

1

Y

l=n−1

(1 − T n−1 . . . T l ) ⁻¹ . But, since kT _i k < q < 1, therefore

k(1 − T n−1 . . . T _(n−1)−k ) ⁻¹ k ≤ (1 − q ^k ) ⁻¹ and we infer the estimation of Lemma 5.

Now we can state Theorem 2 in a stronger version.

Theorem 6. If kT ⁱ k ≤ q < 1 and the assumptions of Theorem 1 are satisfied , then

(i) P ⁽ⁿ⁾ ≥ ω(q)(P ⁽ⁿ⁻¹⁾ ⊗ 1),

where

ω(q) ² = (1 − q ² ) ⁻¹

∞

Y

k=1

(1 − q ^k )(1 + q ^k ) ⁻¹ .

(ii ) P ⁽ⁿ⁾ ≤ 1

1 − q (P ⁽ⁿ⁻¹⁾ ⊗ 1).

P r o o f. The proof follows from the following considerations:

(a) We know from the results of [BSp1] that P ⁽ⁿ⁾ ≥ 0.

Since, by Lemma 5, kR ⁻¹ _n k ≤ ¹ _c for some c > 0, therefore k(R ⁻¹ _n ) ^∗ R ⁻¹ _n k ≤ 1

c ² , and this implies

R n R ^∗ _n ≥ c ² . But, because

P ⁽ⁿ⁾ = (P ⁽ⁿ⁻¹⁾ ⊗ 1)R n , and P ⁽ⁿ⁾ = P ^(n)∗ , we obtain

[P ⁽ⁿ⁾ ] ² = P ⁽ⁿ⁻¹⁾ R _n R ^∗ _n P ⁽ⁿ⁻¹⁾ ≥ c ² [P ⁽ⁿ⁻¹⁾ ] ² and hence

P ⁽ⁿ⁾ ≥ c(P ⁽ⁿ⁻¹⁾ ⊗ 1), where c = ω(q).

(b) The statement (ii) of Theorem 2 follows from the two facts:

P ⁽ⁿ⁾ = P ⁽ⁿ⁻¹⁾ R n

and

R n = 1 + T n−1 + T n−1 T n−2 + . . . + T n−1 . . . T 1 . Therefore kR n k < _1−q ¹ and again as before we have

P ⁽ⁿ⁾ ≥ 1

1 − q (P ⁽ⁿ⁻¹⁾ ⊗ 1).

So, the proof of Theorem 6 is complete.

This theorem is also valid for all finite and affine Coxeter groups (for more details see [BSp4]). Theorem 1 comes from investigations in harmonic analysis on groups (see [B1], [BSz]) and on perturbed cannonical commutation relations. In the paper with R.

Speicher ([BSp1]) we considered the following relations c i c ^∗ _j − qc ^∗ _j c i = δ ij 1

for a real q with |q| ≤ 1, and we needed essentially the fact that the function ϕ : S _n −→ C, π 7−→ q ^|π|

is a positive definite function for all n, where |π| denotes the number of inversions of π.

For other proofs of that result see [BKS, BSp1, BSp2, BSp4, BSz, Spe, Z].

R. Speicher in [Spe] considered more general commutations relations d _i d ^∗ _j − q ij d ^∗ _j d _i = δ _ij 1

for

−1 ≤ q _ij = q _ji ≤ 1,

and he founded the existence of a Fock representation by central limit arguments. Our construction of the q _ij relations depends on some operator T which is a self-adjoint contraction on a Hilbert space H and satisfies the braid or Yang-Baxter relations of the following form:

T ₁ T ₂ T ₁ = T ₂ T ₁ T ₂ ,

where T 1 = T ⊗ 1 and T 2 = 1 ⊗ T on H ⊗ H ⊗ H are the natural amplifications of T to H ⊗ H ⊗ H.

From Theorem 1 we get more general construction of deformed commutation relations of the Wick form:

d i d ^∗ _j − X

r,s

t ^ir _js d ^∗ _r d s = δ ij 1

(see also Jorgensen et al. [JSW] and [BSp4] for similar considerations).

2. Applications. Next we examine the deformed commutation relations from an operator spaces’ point of view. If we assume that kT k = q ≤ 1 and if we take G i = d i +d ^∗ _i , then we prove that the operator space generated by the G _i is completely isomorphic to the canonical operator Hilbert space R ∩ C, which means

N

X

i=1

a _i ⊗ G i

≈ max 

N

X

i=1

a _i a ^∗ _i

1/2

,

N

X

i=1

a ^∗ _i a _i

1/2 

for all bounded operators a ₁ , ..., a _N on some Hilbert space. This generalizes the Theorem of Haagerup and Pisier [HP], who obtained that result for free creation and annihilation operators, (see also [VDN] and [Buch]). As another application of our construction we have obtained a large class of non-injective von Neumann algebras, when considering the von Neumann algebra V N (G ₁ , ..., G _N ) generated by G ₁ , ..., G _N . For more details see [BSp4, BKS].

3. The ultracontractivity of the q-second quantization functor Γ _q . Let T : H −→ K be a contraction beetween real Hilbert spaces. Then the linear map defined on elementary tensors by

F _q (T )(f ₁ ⊗ . . . ⊗ f n ) = T f ₁ ⊗ . . . ⊗ T f n

extends to a contraction from q-Fock spaces F q (H) to F q (K). Here F q (H) is the completion of the full Fock space L ∞

n=0 H ^⊗n with respect to the new scalar product hf 1 ⊗ . . . ⊗ f n , g 1 ⊗ . . . ⊗ g n i _q = δ n,m

X

σ∈S

q ^inv(σ) hf _σ(1) , g 1 i . . . hf _σ(n) , g n i.

The creation operators are defined as:

c ^∗ (f 0 )(f 1 ⊗ . . . ⊗ f n ) = f 0 ⊗ f 1 ⊗ . . . ⊗ f n , f j ∈ H

and c(f ) = [c ^∗ (f )] ^∗ .

Let G(f ) = c(f ) + c ^∗ (f ) for f ∈ H. Let Γ q (H) be the von Neumann algebra generated by G(f ), f ∈ H, and

τ q (S) = hSΩ, Ωi _q , S ∈ Γ q (H).

One can show that τ _q is a trace on Γ _q (H).

If dim H = ∞, then we showed that Γ q (H) is a factor.

If e ₁ , e ₂ , . . . , e _N is an orthonormal basis of H, then we put G _i = G(e _i ), (i = 1, . . . , N , N = ∞, 1, 2, . . .). In this setting the following theorem holds:

Theorem 7 ([BKS], Theorem 2.1.1). Let T be as above, then there exists a unique map Γ _q (T ) : Γ _q (H) −→ Γ _q (K) such that Γ _q (T )(X)Ω = F _q (T )(XΩ) for every X ∈ Γ _q (H).

The map Γ q (T ) is bounded , normal , unital , completely positive and trace preserving.

We note that Γ _q is a functor, namely if S : H −→ K and T : K −→ J are contractions, then Γ q (ST ) = Γ q (S)Γ q (T ).

If H is a real Hilbert space and T t = e ^−t I for t ≥ 0, then the completely positive maps P _t ^q = Γ q (T t ), t ≥ 0, on Γ q (H), form a semigroup, called the q-Ornstein-Uhlenbeck semigroup. The q-Ornstein-Uhlenbeck semigroup extends to a semigroup of contractions of the non-commutative L ^p spaces, which are symmetric on L ² . Its infinitesimal generator on L ² is the number operator N ^q , i.e. P t = exp(−tN ^q ), where N ^q is the unbounded self-adjoint operator defined as N ^q Ω = 0 and

N ^q f 1 ⊗ . . . ⊗ f n = nf 1 ⊗ . . . ⊗ f n , f 1 , . . . , f n ∈ H.

Ph. Biane [Bia] proved Nelson’s hypercontractivity of the q-Ornstein-Uhlenbeck semi- group P _t , extending the results of Nelson and Gross. In that paper Ph. Biane also showed ultracontractivity for q = 0 using some results of the author (see [B2]). Now we prove the ultracontractivity of that semigroup for all q ∈ [−1, 1].

Theorem 8. Let X be in the eigenspace of N ^q , with eigenvalue n. Then (i ) kXk _L

≤ C(q)(n + 1)kXk ² _L ;

(ii ) For t ≥ 0, P t maps L ² into L ^∞ = V N q (G 1 . . . G N ) and for t ≤ 1 kP _t ^q k _L

→L

≤ c q t ^−3/2 .

(iii ) (Poincar´ e-Sobolev inequality). If Q _q (X) = hXN ^q XΩ, Ωi is a non-commutative complete Dirichlet form (on an appropriate domain) on L ² (Γ q (H), τ q ), in the sense of [DL] , then there exists a constant c _q ≥ 0 such that for all X in the domain of Q _q we have

kXk ² _L

≤ c q (|τ q (X)| ² + Q q (X)).

For the details of the proof of this theorem see [B3] and [Bia].

References

[B] D. B a k r y, L’hypercontractivit´ e et son utilisation en th´ eorie des semigroupes, Lectures on Probability Theory, Lecture Notes in Mathematics, vol.1581, Springer, Berlin Hei- delberg New York, (1994), 1-114.

[Bia] Ph. B i a n e, Free Hypercontractivity, Prepublication N 350, Paris VI, (1996), 1–19.

[B1] M. B o ˙z e j k o, Positive definite kernels, length functions on groups and a noncommu- tative von Neumann inequality , Studia Math. 95 (1989), 107–118.

[B2] M. B o ˙z e j k o, A q-deformed probability , Nelson’s inequality and central limit theorems, Non-linear fields, classical, random, semiclassical (P. Garbaczewski and Z. Popowicz, eds.), World Scientific, Singapore, (1991), 312–335.

[B3] M. B o ˙z e j k o, Ultracontractivity and strong Sobolev inequality for q-Ornstein-Uhlenbeck semigroup (to appear).

[BSp1] M. B o ˙z e j k o and R. S p e i c h e r, An Example of a Generalized Brownian Motion, Comm. Math. Phys. 137 (1991), 519–531.

[BSp2] M. B o ˙z e j k o and R. S p e i c h e r, An Example of a generalized Brownian motion II . In : Quantum Probability and Related Topics VII (ed. L. Accardi), Singapore: World Scientific (1992), 219–236.

[BSp3] M. B o ˙z e j k o and R. S p e i c h e r, Interpolation between bosonic and fermionic relations given by generalized Brownian motions, Math. Zeit. 222 (1996), 135–160.

[BSp4] M. B o ˙z e j k o and R. S p e i c h e r, Completely positive maps on Coxeter Groups, deformed commutation relations, and operator spaces, Math. Annalen 300 (1994), 97–120.

[BKS] M. B o ˙z e j k o, B. K ¨ u m m e r e r and R. S p e i c h e r, q-Gaussian Processes: Non-commu- tative and Classical Aspects, Comm. Math. Phys. 185 (1997),129–154.

[BSz] M. B o ˙z e j k o and R. S z w a r c, Algebraic length and Poincare series on reflection groups with applications to representations theory , University of Wroc law, preprint (1996), 1–25.

[Buch] A. B u c h h o l z, Norm of convolution by operator-valued functions on free groups, Proc.

Amer. Math. Soc., (1997).

[CL] E. A. C a r l e n and E. H. L i e b, Optimal hypercontractivity for Fermi fields and related non-commutative integration inequalities, Comm. Math. Phys. 155 (1993), 27–46.

[CSV] T. C o u l h o n, L. S a l o f f - C o s t e, and N. Th. V a r o p o u l o s, Analysis and Geometry on Groups, Cambgidge Tracts in Mathematics, vol.100, Cambridge University Press, 1992.

[DL] E. B. D a v i e s and J. M. L i n d s a y, Non-commutative symmetric Markov semigroups, Math. Zeit. 210 (1992), 379–411.

[G] L. G r o s s, Hypercontractivity and logarithmic Sobolev inequalities for the Clifford- Dirichlet form, Duke Math. J. 42 (1975), 383–396.

[H] U. H a a g e r u p, An example of a non nuclear C

- algebra which has the metric approx- imation property , Invent. Math. 50 (1979), 279–293.

[HP] U. H a a g e r u p and G. P i s i e r, Bounded linear operators between C

-algebras, Duke Math. J. 71 (1993), 889–925.

[JSW] P. E. T. J o e r g e n s e n, L. M. S c h m i t t and R. F. W e r n e r, Positive representations of general commutation relations allowing Wick ordering , J. Func. Anal., 134 (1995),3–99.

[N] E. N e l s o n, The free Markoff field , J. Func. Anal. 12 (1973), 211–227.

[P] G. P i s i e r, The Operator Hilbert Space OH , Complex Interpolation and Tensor Norms, Mem. Amer. Math.Soc. 1996.

[Spe] R. S p e i c h e r, A non commutative central limit theorem, Math. Zeit. 209 (1992), 55–66.

[VDN] D. V. V o i c u l e s c u, K. D y k e m a and A. N i c a, Free random variables, CRM Mono- graph Series No.1, Amer. Math. Soc., Providence, RI, 1992.

[Z] D. Z a g i e r, Realizability of a model in infinity statistics, Comm. Math. Phys. 147

(1992), 199–210.