Positivity DOI 10.1007/s11117-015-0332-x

Positivity (r,s)

The noncommutative H p spaces

(A; ∞ ) and H p (A; 1 )

Turdebek N. Bekjan · Kanat Tulenov · Dostilek Dauitbek

Received: 15 October 2014 / Accepted: 27 February 2015 © Springer Basel 2015 (r,s)

Abstract In this paper we introduce the noncommutative H p (A; ∞ ) and H p (A; 1 ) spaces, and prove the contractivity of the underlying conditional expectation Φ on these spaces. We also give results on duality and complex interpolation. Keywords von Neumann algebra · Subdiagonal algebras · Vector valued noncommutative Hardy spaces · Riesz factorization Mathematics Subject Classification

46L51 · 46L52

1 Introduction Let M be a finite von Neumann algebra equipped with a normal faithful tracial state τ . Let D be a von Neumann subalgebra of M, and let Φ : M → D be the unique normal faithful conditional expectation such that τ ◦ Φ = τ . A finite subdiagonal algebra of M with respect to Φ is a w∗ -closed subalgebra A of M satisfying the following conditions: (i) A + J (A) is w ∗ -dense in M; (ii) Φ is multiplicative on A, i.e., Φ(ab) = Φ(a)Φ(b) for all a, b ∈ A;

T. N. Bekjan (B) L.N. Gumilyov Eurasian National University, Astana 010008, Kazakhstan e-mail: [email protected] K. Tulenov · D. Dauitbek Al-Farabi Kazakh National University, Almaty 050038, Kazakhstan e-mail: [email protected] D. Dauitbek e-mail: [email protected]

T. N. Bekjan et al.

(iii) A ∩ J (A) = D, where J (A) is the family of all adjoint elements of the element of A, i.e., J (A) = {a ∗ : a ∈ A}. The algebra D is called the diagonal of A. It’s proved by Exel [10] that a finite subdiagonal algebra A is automatically maximal in the sense that if B is another subdiagonal algebra with respect to Φ containing A, then B = A. This maximality yields the following useful characterization of A, where A0 = A ∩ ker Φ (see [1]): A = {x ∈ M : τ (xa) = 0, ∀a ∈ A0 }.

(1.1)

Given 0 < p ≤ ∞ we denote by L p (M) the usual noncommutatve L p -spaces associated with (M, τ ). Recall that L ∞ (M) = M, equipped with the operator norm (see [18]). The norm of L p (M) will be denoted by · p . For p < ∞ we define H p (A) to be closure of A in L p (M), and for p = ∞ we simply set H∞ (A) = A for convenience. These are so called Hardy spaces associated with A. They are noncommutative extensions of the classical Hardy space on the torus T. This noncommutative Hardy spaces have received a lot of attention since Arveson’s pioneer work. For references see [1,5–7,14,18], whereas more references on previous works can be found in the survey paper [18]. The theory of vector-valued noncommutative L p -spaces are introduced by Pisier in [16] for the case M is hyperfinite. Junge introduced these spaces for general setting in [12] (see also [9,13]). Let 0 < p, r, s ≤ ∞ such that 1/ p = 1/r + 1/s. Define the space L (r,s) p (M, ∞ ) of all sequences x = (xn ) of operators in L p (M) for which there is a bounded sequence y = (yn ) in M and operators a ∈ L r (M) and b ∈ L s (M) such that xn = ayn b, ∀ n ≥ 1. Put

(xn ) p;(r,s) = inf a r sup yn ∞ b s , n

where the infumum runs over all possible decompositions of x = (xn ) as above. Then (r,s) L p (M, ∞ ) is a Banach space whenever r, s ≥ 2 (see [9]). Similar way one can prove L (r,s) p (M, ∞ ) is a quasi-Banach space whenever r, s > 0. We put right

Lp

(∞, p)

(M, ∞ ) = L p

( p,∞)

le f t

(M, ∞ ),

L p (M, ∞ ) = L p

and (2 p,2 p)

L p (M, ∞ ) = L p

(M, ∞ ).

(M, ∞ )

(r,s)

The noncommutative H p

(A; ∞ ) and H p (A; 1 ) spaces

Given 0 < p ≤ ∞, a sequences x = (xn ) belongs to L p (M; 1 ) if there are (u kn )k,n≥1 and (vnk )n,k≥1 in L 2 p (M) such that xn =

∞ 

u kn vnk

k=1

for all n and ∞ 

u kn u ∗kn ∈ L p (M)

k,n=1

∞ 

∗ vnk vnk ∈ L p (M).

n,k=1

Here all series are required to be convergent in L p (M) (relative to the w*-topology in the case of p = ∞). L p (M; 1 ) is a quasi-Banach space (Banach space whenever p ≥ 1) when equipped with the norm

x L p (M;1 )

1/2   1/2      ∞ ∗   ∞ ∗    , = inf  u kn u kn   vnk vnk   p

k,n=1

n,k=1

p

where the infimum runs over all decompositions of (xn ) as above. (r,s) We now define the spaces H p (A; ∞ ) and H p (A; 1 ) by a similar way. Definition 1 Let 0 < p, r, s ≤ ∞ such that 1/ p = 1/r + 1/s. (r,s)

(i) We define H p (A, ∞ ) as the space of all sequences x = (xn )n≥1 in H p (A) which admit a factorization of the following form: there are a ∈ Hr (A), b ∈ Hs (A) and a bounded sequence y = (yn ) ⊂ A such that xn = ayn b, ∀n ≥ 1. Put

x H (r,s) (A, p

∞)

= inf{ a r sup yn ∞ b s }, n

where the infimum runs over all factorizations of (xn ) as above. The spaces right

Hp

(∞, p)

(A; ∞ ) = H p

(A; ∞ )

and le f t

Hp

( p,∞)

(A; ∞ ) = H p

(A; ∞ )

of all sequences (xn ) which allow uniform factorizations xn = yn b and xn = ayn with a, b ∈ H p (A) and a bounded sequence (yn ) ⊂ A, respectively. Moreover, in the symmetric case put (2 p,2 p)

H p (A; ∞ ) = H p

(A; ∞ ).

T. N. Bekjan et al.

(ii) Let 0 < p ≤ ∞. We define H p (A; 1 ) as the space of all sequences x = (xn )n≥1 in H p (A) which can be decomposed as xn =

∞ 

u kn vnk , ∀n ≥ 1

k=1

for two families (u kn )k,n≥1 and (vnk )n,k≥1 in H2 p (A) such that ∞ 

u kn u ∗kn ∈ L p (M) and

k,n=1

∞ 

∗ vnk vnk ∈ L p (M).

n,k=1

Here all series are required to be convergent in L p (M) (relative to the w*topology in the case of p = ∞). It is equipped with the norm 1/2  ∞ 1/2       ∗   ∞ ∗    , u u v v

x H p (A;1 ) = inf  kn kn  nk nk    p

k,n=1

n,k=1

p

where the infimum runs over all decompositions of x as above. (r,s)

In Sect. 2 we proved basic properties of H p (A, ∞ ) and H p (A; 1 ). In Sect. 3, for 0 < p, r, s ≤ ∞, we extended the conditional expectation Φ to a contractive pro(r,s) (r,s) jection from H p (A; ∞ ) onto L p (D; ∞ ) and from H p (A; 1 ) onto L p (D; 1 ), respectively. Section 4 is devoted to dual of H p (A; 1 ), predual of H p (A; ∞ ). Interpolation properties of H p (A; 1 ) and H p (A; ∞ ) proved in Sect. 5. We will use this ∞ - and 1 -valued version of the noncommutative Hardy space H p (A) to study noncommutative version of Hardy martingales, operator space analytic UMD property and operator space analytic convexity. For references see [11,16,20]. (r,s)

2 Basic properties of H p

(A; ∞ ) and H p (A; 1 )

Lemma 1 Let 0 < p, r, s ≤ ∞ such that 1/ p = 1/r + 1/s. (r,s)

(i) If (xn ) ∈ L p (M; ∞ ), then there exist h ∈ Hr (A), g ∈ Hs (A) and (z n ) ⊂ M such that h −1 , g −1 ∈ A, and for all n, xn = hz n g, and supn z n ∞ ≤ 1. Moreover,  = inf h r sup z n ∞ g s , 

(xn ) p;(r,s)

n

where the infimum runs over all factorizations of (xn ) as above. le f t right (M; ∞ ). (ii) L (r,s) p (M; ∞ ) = L r (M; ∞ ) · L s

(r,s)

The noncommutative H p

(A; ∞ ) and H p (A; 1 ) spaces

(r,s)

Proof (i) If x ∈ L p (M; ∞ ), then for any ε > 0 there is a bounded sequence y = (yn ) in M and operators a ∈ L r (M) and b ∈ L s (M) such that for all n xn = ayn b, yn ≤ 1, and a r b s < x p;(r,s) +ε. Let a ∗ = u|a ∗ | and b = v|b| the polar decompositions 1

1

of a ∗ and b, respectively. Put c = (|a ∗ |2 +ε) 2 and d = (|b|2 +ε) 2 . Clearly, |a ∗ |2 ≤ c2 and |b|2 ≤ d 2 . Then by Remark 2.3 in [9] there exist contractions ω, θ ∈ M such that |a ∗ | = ωc, |b| = θ d. Since c ∈ L r (M), d ∈ L s (M) and c−1 , d −1 ∈ M, by Theorem 3.1 in [3] there exist h ∈ Hr (A), g ∈ Hs (A) and unitary operators ν, w ∈ M such that c = hν, d = wg, and h −1 , g −1 ∈ A. Obviously, xn = h[νω∗ u ∗ yn vθ w]g. Put z n = νω∗ u ∗ yn vθ w, then it is clear that supn z n ∞ ≤ 1. The norm estimate is clear. (1) (2) (ii) Let x ∈ L (r,s) p (M; ∞ ), then x n = ayn b. Choosing x n = a and x n = yn b for all n, we see that xn = xn(1) xn(2) , ∀n, (1)

le f t

and (xn ) ∈ L r

(2)

right

(M; ∞ ), (xn ) ∈ L s

(M; ∞ ).

Proposition 1 Let 0 < p, r, s ≤ ∞ such that 1/ p = 1/r + 1/s. Then H p(r,s) (A; ∞ ) = {(xn ) ∈ L (r,s) p (M; ∞ ) : (x n ) ⊂ H p (A)}.

(2.1)

(xn ) p;(r,s) = (xn ) H (r,s) (A, ) , ∀ (xn ) ∈ H p(r,s) (A; ∞ ).

(2.2)

Moreover, p

(r,s)

∞

(r,s)

Proof The inclusion H p (A; ∞ ) ⊂ {(xn ) ∈ L p (M; ∞ ) : (xn ) ⊂ H p (A)} is clearly. Let (yn ) ∈ {(xn ) ∈ L (r,s) p (M; ∞ ) : (x n ) ⊂ H p (A)}. Then by (i) of Lemma 1 there exist a ∈ H2 p (A), b ∈ H2 p (A) and z n ∈ M such that yn = az n b, ∀n, and a −1 , b−1 ∈ A, and supn z n ∞ ≤ 1. By Proposition 3.3 in [3], we have that z n = a −1 yn b−1 ∈ H p (A) ∩ M = A, ∀n. (r,s)

Hence (yn ) ∈ H p

(A; ∞ ). So (2.1) holds. Using (i) of Lemma 1 we get (2.2).

T. N. Bekjan et al.

Theorem 1 Let 0 < p, r, s ≤ ∞ such that quasi-Banach (or Banach for r, s ≥ 2) space.

1 p

=

1 r

(r,s)

+ 1s . Then H p

(A, ∞ ) is a

Proof By (2.2), it is suffices to show H p(r,s) (A, ∞ ) is a closed linear subspace of (r,s) (1) (2) (r,s) (1) L p (M, ∞ ). Let (xn ), (xn ) ∈ H p (A, ∞ ) and α, β ∈ C. Then (αxn + (1) (2) ∈ H p (A). By Proposition βxn(2) ) ∈ L (r,s) p (M, ∞ ) and for all n, αx n + βx n (1) (2) (r,s) (r,s) 1, we have that (αxn + βxn ) ∈ H p (A, ∞ ), i.e., H p (A, ∞ ) is a linear ( j) (r,s) (r,s) subspace of L p (M, ∞ ). Next to prove H p (A, ∞ ) is closed. Let (xn ) ∈ (r,s) (r,s) H p (A, ∞ ) ( j = 1, 2, . . .) and (xn ) ∈ L p (M, ∞ ) such that ( j)

lim (xn ) − (xn ) p;(r,s) = 0.

j→∞

Since ( j)

( j)

xn − xn p ≤ (xn ) − (xn ) p;(r,s) , ∀n ∈ N, ( j)

it follows that lim j→∞ xn − xn p = 0, so xn ∈ H p (A). Using Proposition 1 we obtain (xn ) ∈ H p(r,s) (A, ∞ ), i.e., H p(r,s) (A, ∞ ) is closed. Corollary 1 Let 0 < p, r, s ≤ ∞ such that 1/ p = 1/r + 1/s. Then H p(r,s) (A; ∞ ) = Hr

le f t

right

(A; ∞ ) · Hs

(A; ∞ ).

Lemma 2 Let 0 < p ≤ ∞. If x ∈ L p (A; 1 ), then for each n there exist (akn )k≥1 ⊂ H2 p (A), (bnk )k≥1 ⊂ H2 p (A) and (ynk )k≥1 ⊂ M such that xn =

∞ 

akn ynk bnk ,

k=1 −1 −1 )k≥1 , (bnk )k≥1 ⊂ A, and supn ynk ∞ ≤ 1 for all n and k. Moreover, where (akn

1/2  ∞ 1/2       ∗   ∞ ∗    ,

x L p (M;1 ) = inf  a a sup

y

b b kn kn  nk ∞  nk nk   n k,n=1

p

n,k=1

p

where the infimum runs over all decompositions of x as above. Proof Let (xn ) ∈ L p (M; 1 ). Then for ε > 0 thereare two families (u kn ), (vnk ) ∈  ∗ v , ∗ u v ∈ L (M), v u u L 2 p (M) such that xn = ∞ p kn kn ∈ L p (M) k=1 kn nk nk nk and   1/2   1/2  ∞   ∞ ∗  ∗     < x L (M; ) + ε. u u v v kn kn  nk nk  p 1   k,n=1

p

n,k=1

p

(r,s)

The noncommutative H p

(A; ∞ ) and H p (A; 1 ) spaces

Let u ∗kn = ϑkn |u ∗kn | and vnk = νnk |vnk | be the polar decompositions of u ∗kn and vnk ,

ε ε for all n and k, respectively. Put ckn = (|u ∗kn |2 + 2k+n ) 2 and dnk := (|vnk |2 + 2k+n )2 . ∗ 2 2 2 2 It is clear that |u kn | ≤ ckn and |vnk | ≤ dnk . By Remark 2.3 in [9], there exist contractions ωkn , θnk ∈ M such that |u ∗kn | = ωkn ckn , |vnk | = θnk dnk . Notice that −1 −1 ckn ∈ L r (M), dnk ∈ L s (M) and ckn , dnk ∈ M. Hence, by Theorem 3.1. in [3], there exist unitary operators μkn , wnk ∈ M and h kn ∈ H2 p (A), gnk ∈ H2 p (A) such −1 that ckn = h kn μkn and dnk = wnk gnk , and h −1 kn , gnk ∈ A. Clearly, 1

xn =

∞ 

1

∗ ∗ h kn [μkn ωkn ϑkn νnk θnk wnk ]gnk .

k=1

Set ∗ ∗ ynk = μkn ωkn ϑkn νnk θnk wnk .

Then xn =

∞ 

h kn ynk gnk and sup ynk ∞ ≤ 1. n

k=1

The norm estimate is clear. Similar to Proposition 1, we have the following result. Proposition 2 Let 0 < p ≤ ∞. Then H p (A; 1 ) = {(xn ) ∈ L p (M; 1 ) : (xn ) ⊂ H p (A)}.

(2.3)

(xn ) L p (M;1 ) = (xn ) H p (A;1 ) , ∀ (xn ) ∈ H p (A; 1 ).

(2.4)

Moreover,

Using Lemma 2 and Proposition 2 we obtain the following result. Theorem 2 Let 0 < p ≤ ∞. Then H p (A; 1 ) is a quasi-Banach (or Banach for p ≥ 1) space. (r,s)

3 Contractivity of Φ on H p

(A; ∞ ) and H p (A; 1 )

It is well-known that the conditional expectation Φ extends to a contractive projection from L p (M) onto L p (D) for every 1 ≤ p ≤ ∞. In general, Φ cannot be continuously extended to L p (M) for p < 1. In [3] proved that Φ is a contractive projection from H p (A) onto L p (D) for p < 1. We will prove that there is a contractive projection from (r,s) (r,s) H p (A; ∞ ) onto L p (D; ∞ ) for 0 < p, r, s ≤ ∞ [respectively, from H p (A; 1 ) onto L p (D; 1 ) for 0 < p ≤ ∞].

T. N. Bekjan et al.

Theorem 3 Let 0 < p, r, s ≤ ∞. Then

(Φ(h n )) L (r,s) (D; p

∞)

≤ (h n ) H (r,s) (A; ) , ∀ (h n ) ∈ H p(r,s) (A; ∞ ). ∞

p

(r,s)

Proof Let (h n ) ∈ H p (A; l∞ ). Then for ε > 0 there exist a ∈ Hr (A), b ∈ Hs (A) and a bounded sequence (xn ) ⊂ A such that for all n, h n = axn b, and

(h n ) H (r,s) (A; p

∞)

+ ε ≥ a r sup xn ∞ b s . n

Hence, by Corollary 2.2 and Theorem 2.1 in [3], Φ(h n ) = Φ(axn b) = Φ(a)Φ(xn )Φ(b), where Φ(a) ∈ L r (D), Φ(xn ) ∈ D, Φ(b) ∈ L s (D) and

Φ(a) r ≤ a r , Φ(xn ) ∞ ≤ xn ∞ , Φ(b) s ≤ b s . Therefore,

(Φ(h n )) L (r,s) (D; p

∞)

≤ Φ(a) L r (D) sup Φ(xn ) L ∞ (D) Φ(b) L s (D) n

≤ a Hr (A) sup xn H∞ (A) b Hs (A) n

≤ (h n ) H (r,s) (A; p

∞)

+ ε.

Letting ε → 0 we obtain the desired inequality. The following result proved in [2] (see Lemma 5.1), we will use it. Lemma 3 Let 0 < p ≤ ∞. Then  ∞ 1/2     2   |Φ(xn )|   n=1

L p (D )

 ∞ 1/2     2   ≤ |xn |  n=1

H p (A)

, ∀(xn ) ⊂ H p (A).

Theorem 4 Let 0 < p ≤ ∞. Then

(Φ(yn )) L p (D;1 ) ≤ (yn ) H p (A;1 ) , ∀(yn ) ∈ H p (A; 1 ).

(r,s)

The noncommutative H p

(A; ∞ ) and H p (A; 1 ) spaces

Proof Let (yn ) ∈ H p (A; 1 ), then for ε > 0, there are (u kn )k,n≥1 and (vnk )n,k≥1 in H2 p (A) such that yn =

∞ 

u kn vnk , ∀n ≥ 1,

k=1

and (

∞

∗ 21 k,n=1 u kn u kn ) ,

(

∞

1 ∗ 2 n,k=1 vnk vnk )

∈ L 2 p (M), and

1  ∞ 1  ∞ 2   ∗ 2   ∗   

(yn )n≥1 H p (A;1 ) + ε ≥  u kn u kn   vnk vnk   . p

k,n=1

k,n=1

p

Hence, by Theorem 2.1 and Corollary 2.2 in [3] Φ(yn ) = Φ

 ∞

 u kn vnk

k=1

=

∞ 

Φ(u kn vnk ) =

k=1

∞ 

Φ(u kn )Φ(vnk ), ∀n.

k=1

Since J (A) is a subdiagonal algebra of A, using Lemma 3 we obtain that

(Φ(yn ))n≥1 L p (D;1 )

    ∞  = Φ(u kn )Φ(vnk ) k=1   1/2    ∞ ∗ 2   ≤ |Φ(u kn )| 

n≥1 L p (D ;1 )

2p

k,n=1

 ∞     ∗ 2 1/2    ≤ |u kn | 

2p

k,n=1

   

  1/2   ∞  2   |Φ(vnk )|  

2p

k,n=1

 ∞ 1/2     2   |vnk |   k,n=1

2p

≤ (yn )n≥1 H p (A;1 ) + ε. Letting ε → 0 we obtain the desired inequality. 4 Duality Let 1 ≤ p < ∞. Set H p0 (A; ∞ ) = {x ∈ H p (A; ∞ ):Φ(xn ) = 0, ∀n} and H p0 (A; 1 ) = {y ∈ H p (A; 1 ):Φ(yn ) = 0, ∀n}. Proposition 3 Let 1 ≤ p < ∞ and

1 p

+

1 p

= 1. Then we have the following:

∞ τ (xn yn ) = 0, ∀(yn ) ∈ H p0 (A; 1 )} H p (A; ∞ ) = {(xn ) ∈ L p (M; ∞ ) : n=1 (4.1)

T. N. Bekjan et al.

and ∞ τ (xn yn ) = 0, ∀(yn ) ∈ H p (A; 1 )}. H p0 (A; ∞ ) = {(xn ) ∈ L p (M; l∞ ) : n=1 (4.2) ∞ τ (x y ) = 0, ∀(y ) ∈ Proof The inclusion H p (A; ∞ ) ⊂ {(xn ) ∈ L p (M; l∞ ): n=1 n n n 0 ∞ τ (x y ) = 0, ∀(y ) ∈ H p (A; 1 )} is clearly. Let (z n ) ∈ {(xn ) ∈ L p (M; l∞ ) : n=1 n n n H p0 (A; 1 )} and c ∈ A0 . For n ∈ N, set yk = 0(k = n) and yn = c, then (yk ) ∈ H p0 (A; 1 ). Hence for all n ∈ N,

τ (z n c) = 0, ∀c ∈ A0 . By (1.2) in [3] (see [19]), we get (z n ) ⊂ H p (A). Using Lemma 1 we obtain that (z n ) ∈ H p (A; ∞ ). The later equality follows from the continuity of Φ on H p (A; ∞ ). Proposition 4 Let 0 < p < q ≤ ∞. Then Hq (A; ∞ ) = H p (A; ∞ ) ∩ L q (M; ∞ ) and Hq0 (A; ∞ ) = H p0 (A; ∞ ) ∩ L q (M; ∞ ).

(4.3)

Proof We prove only the first equivalence. The proofs of the second equivalence is similar. It is obvious that Hq (A; ∞ ) ⊂ H p (A; ∞ ) ∩ L q (M; ∞ ). To prove the converse inclusion let (yn )n≥1 ∈ H p (A; ∞ ) ∩ L q (M; ∞ ), then (yn )n≥1 ∈ L q (M; ∞ ). By Proposition 3.3 in [3], (yn ) ⊂ H p (A) ∩ L q (M) = Hq (A). Applying Lemma 1 we find that (yn )n≥1 ∈ Hq (A; ∞ ). By Proposition 2, arguments similar to proofs of Proposition 3 and 4, we get the following results.

Proposition 5 Let 1 ≤ p < ∞, 1/ p + 1/ p = 1. Then  H p (A; 1 ) = x ∈ L p (M; 1 ) :

∞ 

τ (xn yn∗ )

= 0, f or all

(yn∗ )

∈

H p0 (A; ∞ )

 .

n=1

(4.4) Moreover,  H p0 (A; 1 )

= x ∈ L p (M; 1 ) :

∞ 

τ (xn yn∗ )

= 0, f or all

(yn∗ )

 ∈ H p (A; ∞ ) .

n=1

(4.5) Proposition 6 Let 0 < p < q ≤ ∞. Then Hq (A; 1 ) = H p (A; 1 ) ∩ L q (M; 1 ) and Hq0 (A; 1 ) = H p0 (A; 1 )∩ L q (M; 1 ). (4.6)

(r,s)

The noncommutative H p

(A; ∞ ) and H p (A; 1 ) spaces

Theorem 5 Let 1 ≤ p < ∞ such that 1/ p + 1/ p = 1. Then (i) (H p (A; 1 ))∗ = L p (M; ∞ )/J (H p0 (A; ∞ )) isometrically via the following duality bracket ((xn ), (yn )) =

∞ 

τ (yn∗ xn )

n=1

(ii)

for x ∈ H p (A; 1 ) and y ∈ H p (A; ∞ ), where J (H p0 (A; ∞ )) = {x ∗ : x ∈ H p0 (A; ∞ )}. (L p (M; 1 )/J (H p0 (A; 1 )))∗ = H p (A; ∞ ) isometrically via the following duality bracket ((xn ), (yn )) =

∞ 

τ (yn∗ xn )

n=1

for x ∈ H p (A; 1 ) and y ∈ H p (A; ∞ ), where J (H p0 (A; 1 )) = {x ∗ : x ∈ H p0 (A; 1 )}. Proof It is clear that (H p (A; 1 ))∗ = L p (M; ∞ )/(H p (A; 1 ))⊥ and (L p (M; 1 )/⊥ (H p (A; ∞ )))∗ = H p (A; ∞ ), where   ∞  τ (yn∗ xn ) = 0 ∀ (yn ) ∈ H p (A; 1 ) (H p (A; 1 ))⊥ = (xn ) ∈ L p (M; ∞ ) : n=1

and ⊥

  ∞  ∗ (H p (A; ∞ )) = (xn ) ∈ L p (M; 1 ) : τ (yn xn ) = 0 ∀ (yn ) ∈ H p (A; ∞ ) . n=1

On the other hand, by Proposition 3 and 5, we have that ⊥

(H p (A; ∞ )) = J (H p0 (A; 1 )), (H p (A; 1 ))⊥ = J (H p0 (A; ∞ )).

From this follow the desired results.

T. N. Bekjan et al.

Let M = L ∞ (T), A = H ∞ (T) and let  Φ(a) =

   adt 1, τ (a) = adt , ∀a ∈ M.

Then A is a finite subdiagonal algebra in M and A is maximal. Let 1 < p < ∞,

1/ p + 1/ p = 1. Then   L p (M; ∞ ) = (yn )n≥1 ⊂ L p (T) | sup |yn | ∈ L p (T) , n   H p (A; ∞ ) = (yn )n≥1 ⊂ H p (T) | sup |yn | ∈ H p (T) n

and      sup

(xn ) L p (M;∞ ) =  |x |  n n

L p (T)

     sup , (yn ) H p (A;∞ ) =  |y |  n n

H p (T)

.

If H p (A; 1 )∗ = H p (A; ∞ ), then L p (M; ∞ )/J (H p0 (A; ∞ ) is equivalent to H p (A; ∞ ). Hence the Hilbert transform H is bounded projection from L p (M; ∞ ) to H p (A; ∞ ), i.e.

sup |Hxn | H p (T) ≤ C p sup |xn | L p (T) , ∀(xn ) ∈ L p (T). n

n

This means that H ⊗ id is bounded on L p (∞ , T). By Lemma 2 in [8],we get ∞ ∈ U M D. This is a contraction. Thus, in general, H p (A; 1 )∗ = H p (A; ∞ ). 5 Interpolation In this section, we use the method of the proof of Proposition 2.5 in [13] to prove a result on complex interpolation of these vector valued noncommutative H p -spaces. For this need some preparations. Let 1 ≤ p ≤ ∞, n ∈ N and a = (ak )1≤k≤n be a n-tuple in H p (A), define

a H p (A,2

 n 1/2     2  , = |a | n  

a H p (A,2

  1/2    n ∗ 2  . = |a | n  

n,C )

n,R )

k=1

p

k=1

p

This gives two noms on the family of all n-tuples in H p (A). To see this, denoting by N = Mn (M) be the algebra of n × n matrices with entries from M and B = Mn (A) be the algebra of n × n matrices with entries from A. For x ∈ N entries xi, j , with n τ (xi,i ). Then define Φ(x) to be the matrix with entries Φ(xi, j ) and ν(x) = i=1

(r,s)

The noncommutative H p

(A; ∞ ) and H p (A; 1 ) spaces

(N , ν) is finite von Neaumann algebra, B is a finite subdiagonal algebra of (N , ν). Now, any n-tuple a = (an )1≤k≤n in H p (A) can be regarded as an element in H p (B) via the following map ⎛

a1 ⎜ a2 ⎜ a −→ T (a) = ⎜ . ⎝ ..

0 0 .. .

an

0

... ... ... ...

⎞ 0 0⎟ ⎟ .. ⎟ , .⎠ 0

that is, the matrix of T (a) has all vanishing entries except those in the first column which are the an ’s. Such a matrix is called a column matrix, and the closure in H p (B) of all n × n column matrices called the column subspace of H p (B) (when p = ∞, we take the w∗ -closure of all column matrices). Then

a H p (A,2

n,C )

Therefore . H p (A,2

n,C )

= |T (a)| H p (B) = T (a) H p (B) .

defines a norm on the family of all n-tuples of H p (A). The

corresponding completion is a Banach space, denoted by H p (A, 2n,C ). Similarly, we may show that . H p (A,2 ) is a norm on the family of all n-tuples in H p (A). As above, n,R

it defines a Banach space H p (A, 2n,R ), which now is isometric to the row subspace of H p (B) consisting of matrices whose nonzero entries lie only in the first row. Observe that the column and row subspaces of H p (B) are 1-complemented subspace (see [17]). We denote by H p (A; n∞ ) the subspace of H p (A; ∞ ) consisting of all finite sequences (x1 , x2 , . . . , xn , 0, . . .). Similarly, we introduce the subspace H p (A; n1 ) of H p (A; 1 ). Using Proposition 2.1 in [13] and Proposition 1 we obtain the following result. Proposition 7 Let 1 ≤ p ≤ ∞. (i) A sequence x = (xn ) in H p (A) belongs to H p (A, ∞ ) iff sup (xk )1≤k≤n H p (A,n∞ ) < ∞. n≥1

If this is the case, then

(xn ) H p (A,∞ ) = sup (xk )1≤k≤n H p (A,n∞ ) . n≥1

(ii) If p < ∞, then ∪n≥1 H p (A, n1 ) dense in H p (A, 1 ). Now, we can prove the following result.

T. N. Bekjan et al.

Theorem 6 Let 1 ≤ p0 < p1 ≤ ∞, 0 < θ < 1 and isometrically

1 p

=

1−θ p

+ θp . Then we have

H p (A; 1 ) = (H p0 (A; 1 ), H p1 (A; 1 ))θ and H p (A; ∞ ) = (H p0 (A; ∞ ), H p1 (A; ∞ ))θ , where ()θ is the complex interpolation functor. Proof We will use a method similar to the proof of Proposition 2.5 in [13]. By Theorem 4.3 in [15], it is known that {H p (B)}1≤ p≤∞ form an interpolation scale with respect to complex interpolation. Hence, {H p (A, 2n,C )}1≤ p≤∞ form an interpolation scale with respect to complex interpolation. The same is true for the row spaces. Note that by the definition of H p (A; 1 ), the bilinear map B : H p (A, 2n,R ) × H p (A, 2n,C ) → H p (A; n1 )

(u k j )k,n≥ j≥1 × (vk j )k,n≥ j≥1 →

 k≥1

u k1 vk1 ,

 k≥1

u k2 vk2 , . . . ,



 u kn vkn , 0, . . .

k≥1

is contractive [in fact, H p (A; n1 ) is just the quotient space of H p (A, 2n,R ) × H p (A, 2n,C ) by the kernel of B]. Thus by complex interpolation for bilinear maps (see Theorem 4.4.1 in [4]), we deduce that H p (A, 2n,R ) × H p (A, 2n,C ) → (H p0 (A; 1 ), H p1 (A; 1 ))θ is contractive. This yields H p (A; n1 ) ⊂ (H p0 (A; n1 ), H p1 (A; n1 ))θ , a contractive inclusion. Applying Proposition 7 we get H p (A; 1 ) ⊂ (H p0 (A; 1 ), H p1 (A; 1 ))θ , a contractive inclusion. Similarly, using the complex interpolation for trilinear maps, we obtain the following contractive inclusion H p (A; ∞ ) ⊂ (H p0 (A; ∞ ), H p1 (A; ∞ ))θ . Conversely, by Proposition 2.5 in [13], we have the following contractive inclusions (H p0 (A; 1 ), H p1 (A; 1 ))θ ⊂ L p (M; 1 ) and (H p0 (A; ∞ ), H p1 (A; ∞ ))θ ⊂ L p (M; ∞ ).

(r,s)

The noncommutative H p

(A; ∞ ) and H p (A; 1 ) spaces

Since (H p0 (A; 1 ), H p1 (A; 1 ))θ ⊂ H p0 (A; 1 ), (H p0 (A; ∞ ), H p1 (A; ∞ ))θ ⊂ H p0 (A; ∞ ), using Proposition 4 and 6 we obtain the desired contractive inclusions (H p0 (A; 1 ), H p1 (A; 1 ))θ ⊂ H p0 (A; 1 ), (H p (A; ∞ ), H p1 (A; ∞ ))θ ⊂ H p (A; ∞ ). Acknowledgments The author is grateful to the anonymous referee for making helpful comments and suggestions, which have been incorporated into this version of the paper. This work is supported by project 3606/GF4 of Science Committee of Ministry of Education and Science of Republic of Kazakhstan.

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