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

(1)

INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1998

L

∞

-KHINTCHINE–BONAMI INEQUALITY IN FREE PROBABILITY

A R T U R B U C H H O L Z

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

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

Abstract. We prove the norm estimates for operator-valued functions on free groups sup-

ported on the words with fixed length (f = P

|w|=l

a

w

⊗λ(w)). Next, we replace the translations by the free generators with a free family of operators and prove inequalities of the same type.

1. Introduction. The constants in the classical Khintchine inequality and in its extensions (for the products of Rademacher functions R

i

) are unbounded at infinity (see [Bn], see also [H1] where the best constant for k = 1 was found):

X

i₁...i_k

a

i₁,...,i_k

R

i₁

. . . R

i_k

_L

2p

≤ (2p − 1)

^k²

X

i₁...i_k

a

i₁,...,i_k

R

i₁

. . . R

i_k

_L

2

.

In the free case (for free generators, or more generally for Leinert sets) the constants are bounded at infinity. It was shown by M. Leinert in [M] that the square summable functions supported on Leinert’s subsets of discrete groups are convolvers (the symbols of the convolution operators):

kf k

_{V N (G)}

≤ √

5 kf k

_l2(G)

.

In the paper [Bo1] by Bo˙zejko the same result (with the best constant) was obtained as the limit version of Khintchine inequality:

kf k

L_2p

≤ (C

p

)

^2p¹

kf k

L₂

where kf k

^2p_L

2p

= (f ∗ ˜ f )

^∗p

(e) , ˜ f (g) = f (g

⁻¹

) and C

_p

=

^2p_p

₁

p+1

are the Catalan numbers.

The same kind of norm equivalence (with bounded constant) holds for the functions supported on words with fixed length (on free groups). The following estimate was proved

1991 Mathematics Subject Classification: Primary 47A30; Secondary 47A80, 60E15.

Research supported by KBN grant 2P03A05108.

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

[105]

(2)

by U. Haagerup in [H2]:

kf k

_{V N (F)}

≤ (l + 1) kf k

l²(F)

where supp f ⊂ {w ∈ F : |w| = l} and F is a free group.

In the paper [HP] some estimate for operator-valued functions was shown. For a function f : F → B(H) supported on the words of length one with the values in the algebra of bounded operators on a Hilbert space H Haagerup and Pisier obtained:

(1) kf k

B(H)⊗V N (F)

≤ 2 max n

X

g∈supp f

f (g)

^∗

f (g)

1/2

,

X

g∈supp f

f (g) f (g)

^∗

1/2

o .

In the same paper the above estimate was extended to the case of a family of operators G

i

which is free in the Voiculescu’s sense (for the notion of freeness see [V] or [VDN]):

(2)

X

i

a

_i

⊗ G

_i

≤ 2 max n

X

i

a

^∗_i

a

_i

1/2

,

X

i

a

_i

a

^∗_i

1/2

o

where a

i

are elements of B(H) and G

i

form the free family. See also [BSp] for another generalization of (2).

The two above estimates are extended in this note. In Section 2 we show the following inequality for operator-valued functions supported on the words of fixed length:

X

|w|=l

a

w

⊗ λ(w)

≤ (l + 1) max {k (a

pq

) k

X_k

: k ∈ {0, . . . , l}} ,

where p, q are the words of the length l − k and k respectively, a

_i

∈ B(H), (a

_pq

) :

⊕

{q∈F:|q|=l−k}

H → ⊕

{p∈F:|p|=k}

H, and a

pq

= 0 if |pq| 6= l.

In the last section the same inequality was shown for the products of operators G

i

which form a free family with respect to the state φ(·) = h·ξ|ξi:

(3)

X

|i|=l

a

i

⊗ G

i

≤ ∆

^max{0,l−1}

(l + 1) max{k(a

pq

)k

X_k

: k ∈ {0, . . . , l}},

where ∆ = max

i

{kG

i

ξk, kG

^∗_i

ξk} , G

i

= G

i₁

. . . G

i_l

, a

i

∈ B(H) and a

pq

= 0 if p

k

= q

1

. In the scalar case the inequality (3) was obtained by M. Bo˙zejko in [Bo2].

2. The free groups’ case. In this section E

_l

(G) denotes the subset of words of length l of a free group G (considered with a fixed family of free generators g

1

, . . . , g

N

).

For w ∈ E

_l

(G) we denote by w

_i

the i-th letter in the reduced word w, i.e.

w = w

1

. . . w

l

, where w

i

∈ {g

1

, . . . , g

N

, g

₁⁻¹

, . . . , g

_N⁻¹

},

and by

_k|

w (w

_k|

) we denote the product of the first k-letters (last l − k-letters):

k|

w =







w for k > l, Q

k

i=1

w

_i

for l ≥ k > 0,

e for k ≤ 0,

w

_k|

=







w for k < 0, Q

l

i=k+1

w

i

for 0 ≤ k < l,

e for k ≥ l.

First we recall a decomposition of the left translation operator by a free generator g (see the proof of Proposition 1.1 in [HP]):

(4) λ(g) = P

_g

λ(g) + λ(g)P

_g−1

.

(3)

Here and in what follows P

w

denotes the orthogonal projection of l

²

(G) onto the closed span of {δ

_wg

∈ l

²

(G) : |wg| = |w| + |g|} (P

_e

= Id

_l2(G)

). In an obvious way (4) gives a decomposition for any word:

(5) λ(w) =

|w|

X

k=0

P

_k|w

λ(w)P

_w⁻¹

k|

. From now on we write T

k

(w) instead of P

_k|w

λ(w)P

_w⁻¹

k|

. For w, v ∈ E

1

(G) it is easy to verify that:

T

0

(w)T

0

(v)

^∗

= δ

w,v

(Id − P

w

) , T

1

(w)

^∗

T

1

(v) = δ

w,v

(Id − P

_w−1

) . The above observation in an elementary way implies the following lemma:

Lemma 1. Let |w| = |v| then:

i ) T

_k

(w)T

_k

(v)

^∗

= δ

_w_k|_,v_k|

T

_k

(

_k|

w)(Id − P

_w_k+1

)T

_k

(

_k|

v)

^∗

, ii ) T

k

(w)

^∗

T

k

(v) = δ

_k|w,_k|v

T

0

(w

k|

)

^∗

(Id − P

_w⁻¹

k

)T

0

(v

k|

).

In the case when |w| = |v| and w

_k|

6= v

_k|

(resp.

_k|

w 6=

_k|

v), the operators T

k

(w) and T

_k

(v) have orthogonal domains (resp. images), thus we get:

(6) X

w∈E_l(G)

T

_k

(w) = (X

_l

(pq)T

_k

(pq))

_(p,q)∈E

k(G)×E_l−k(G)

, where X

l

is the characteristic function of the set E

l

(G).

Our aim is to estimate the norm of the following operator:

X

w∈E_l(G)

a

_w

⊗ λ(w),

where a

w

∈ B(H) (H - a Hilbert space). For the upper estimate it is enough to compute the norm of the operator:

(a

pq

⊗ T

k

(pq))

_(p,q)∈E

k(G)×El−k(G)

. Here, of course, a

pq

= 0 if |pq| < l. By Lemma 1 we have:

k (a

pq

⊗ T

k

(pq)) × (a

_pq

⊗ T

k

(pq))

^∗

k =

= k (a

pq

⊗ T

k

(p)(Id − P

_q₁

)) × (a

_pq

⊗ T

k

(p)(Id − P

_q₁

))

^∗

k

= k (a

_pq

⊗ T

_k

(p)(Id − P

_q₁

))

^∗

× (a

_pq

⊗ T

_k

(p)(Id − P

_q₁

)) k

= k (a

_pq

⊗ (Id − P

_p_k

)(Id − P

_q₁

)) k

²

≤ k (a

_pq

) k

²

.

In order to prove the opposite inequality we observe that for any vector of the form

⊕

_q∈E_l−k_(G)

v

q

we have:

(7) k ⊕ v

q

k

⊕_{q∈El−k(G)}H

= k ⊕ (v

q

⊗ δ

_q⁻¹

)k

_⊕_{q∈El−k(G)}H⊗₂l²(G)

, k (a

pq

) ⊕ v

q

k = k (a

pq

⊗ T

k

(pq)) ⊕ (v

q

⊗ δ

q⁻¹

)k.

The above considerations prove the second statement of the following theorem:

Theorem 2. Let G be a free group generated by the set {g

1

, . . . , g

N

} of free generators,

and a

w

(w ∈ E

l

(G)) be a family of bounded operators on a Hilbert space. Then the

following statements hold :

(4)

i ) k P

w∈El(G)

a

w

⊗ λ(w)k ≥ max {k (a

pq

) k

X_k

: k ∈ {0, . . . , l}} , ii ) k P

w∈E_l(G)

a

_w

⊗ λ(w)k ≤ (l + 1) max {k (a

pq

) k

_X_k

: k ∈ {0, . . . , l}} , where (a

pq

) : ⊕

_q∈E_l−k_(G)

H → ⊕

_p∈E_k_(G)

H and a

pq

= 0 if pq 6∈ E

l

(G).

The statement (i) can be proved similarly as in (7) but instead of ⊕(v

_q

⊗ δ

_q−1

) we take v = P v

q

⊗ δ

_q⁻¹

and observe that a

w

⊗ T

m

(w)v ⊥ a

w

⊗ T

n

(w)v if m 6= n.

3. The case of the reduced free product of B(H)’s. We use the following no- tations throughout this section. (H

i

, ξ

i

) denotes a Hilbert space H

i

with a distinguished normalized vector ξ

i

. For any space H

i

we consider a bounded operator g

i

∈ B(H

i

) with the property: hg

_i

ξ

_i

|ξ

i

i = 0 and we denote by G

i

the extension of g

_i

onto the free product

∗(H

i

, ξ

i

). By i we denote the sequence (i

1

, . . . , i

n

) with the property i

1

6= i

2

6= . . . 6= i

n

and by

_k|

i, i

_k|

the restrictions of i to the (i

₁

, . . . , i

_k

) and (i

_k+1

, . . . , i

_n

) respectively.

As in the previous section we first recall a decomposition of the operators G

i

(see proof of Proposition 4.9 in [HP]):

(8) G

_i

= e

_i

G

_i

(1 − e

_i

) + G

_i

e

_i

,

where e

i

denotes the orthogonal projection of ∗(H

i

, ξ

i

) onto the space Cξ ⊕ ⊕

^∞n=1

⊕

_i=i₁_6=i₂_6=...6=i_n

H

_i^◦

1

⊗

₂

. . . ⊗

₂

H

_i^◦

n

.

This gives a decomposition for the product of G

i

(under the assumption that i

1

6= i

2

6=

. . . 6= i

n

):

(9)) G

i₁

G

i₂

. . . G

i_n

=

n

X

k=0 k

Y

p=1

G

i_p

(1 − e

i_p

)

n

Y

q=k+1

G

i_q

e

i_q

.

From now on the operators which arise under the sum on the right hand side we denote by S

k

(i).

The S

k

’s have properties analogous to (i) and (ii) from Lemma 1:

Lemma 3. Let i and j have the same length. Then the following statements hold : i ) S

k

(i)S

k

(j)

^∗

= δ

i_k|,j

k|

S

k

(

_k+1|

i) Q

n

p=k+2

hG

i_p

G

^∗_i_p

ξ|ξi

S

k

(

_k+1|

j)

^∗

, ii ) S

_k

(i)

^∗

S

_k

(j) = δ

_k|_i,_k|_j

S

_k

(i

_k|

)

^∗

Q

k

p=1

hG

^∗_i_p

G

_i_p

ξ|ξi S

_k

(j

k|

).

Hence S

k

’s have the same orthogonality properties as T

k

’s. This gives the equality:

(10) X

|i|=n

S

_k

(i) = (1 − δ

_p_k_,q₁

)S

_k

(pq)

(p,q)

,

where (1 − δ

p_k,q₁

)S

k

(pq) : ⊕

_|q|=n−k

∗ (H

i

, ξ

i

) → ⊕

_|p|=k

∗ (H

i

, ξ

i

). In the same way as in the previous section we obtain:

k(a

pq

⊗ S

k

(pq))k = k(Id ⊗ E

_p,p

) × (a

_pq

⊗ Id) × (Id ⊗ F

q,q

)k,

where the first and last matrices on the right hand side are diagonal of order |p| = k and

|q| = n − k respectively and such that:

E

p,p

= (1 − e

p_k

)

k

Y

m=1

kG

p_m

ξk, F

q,q

= S

0

(q

1

)

l−k

Y

m=2

kG

^∗_q

m

ξk.

(5)

Finally we obtain the inequality:

k(a

pq

⊗ S

k

(pq))k ≤ max

p

n Y

^k

m=1

kG

p_m

ξk o max

q

n kG

q₁

k

l−k

Y

m=2

kG

^∗_q_m

ξk o

k(a

pq

)k.

Now the second statement of the following theorem is clear:

Theorem 4. Let G

ⁱ

(i ∈ {1, . . . , m}) be as above. Then under the assumption that G

i

are contractions we have:

i ) k P

|i|=n

a

i

⊗ G

i

k ≥ δ

ⁿ

max{k(a

pq

)k

X_k

: k ∈ {0, . . . , n}}, ii ) k P

|i|=n

a

_i

⊗ G

i

k ≤ ∆

^max{0,n−1}

(n + 1) max{k(a

_pq

)k

_X_k

: k ∈ {0, . . . , n}}, where δ = min

i

{kG

i

ξk, kG

^∗_i

ξk}, ∆ = max

i

{kG

i

ξk, kG

^∗_i

ξk}, G

i

= G

i₁

. . . G

i_n

and a

pq

= 0 if p

_k

= q

₁

.

The first statement can be verified by substitutions of vectors (as in the proof of Theorem 2 (i)).

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^p

(G), Ann. Inst.

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Paris 286 (1978), A259–A262.

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^∗

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

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^∗

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