Analytic Hardy spaces on the quantum torus

Acta Mathematica Scientia 2011,31B(5):1985–1996 http://actams.wipm.ac.cn

ANALYTIC HARDY SPACES ON THE QUANTUM TORUS∗ Chen Zeqian (


)1

)1,2

Yin Zhi (

1. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China 2. Graduate University of Chinese Academy of Sciences, Beijing 100049, China

Abstract Analytic Hardy and BMO spaces on the quantum torus are introduced. Some basic properties of these spaces are presented. In particular, the associated H 1 -BMO duality theorem is proved. Finally, we discuss some possible extensions of the obtained results. Key words quantum torus; analytic Hardy spaces; BMO space; Hilbert transforms 2000 MR Subject Classification

1

46L52; 46J15

Introduction

In recent years, noncommutative analysis (in a wide sense) developed rapidly. The recent theory of martingales inequalities in noncommutative Lp -spaces is a good example for this development. Indeed, square functions associated to martingales and most of the classical martingale inequalities were successfully transferred to the noncommutative setting. We refer to a recent book by Xu [20] for an up-to-date exposition of theory of noncommutative martingales. Parallel to the theory of noncommutative inequalities, noncommutative harmonic analysis also made great advances. We refer the reader notably to the recent works by Junge-Le Merdy-Xu [9] on noncommutative diffusion semigroups, by Blecher and Labuschagne [3–5] and Bekjan-Xu [2] on noncommutative Hardy spaces, by Mei [12] and Chen [6] on operator-valued Hardy spaces, by Parcet [14] and Mei-Parcet [13] on noncommutative Caldr´ on-Zygmund and Littlewood-Paley theories, and so on. Our paper continues the line of this investigation. We will introduce analytic Hardy and BMO spaces on the quantum torus. Some basic properties of these spaces are presented. In particular, the associated H 1 -BMO duality theorem is proved. We would like to point out that the Hardy spaces introduced here are slightly distinct from noncommutative Hardy spaces in the sense of Arveson [1]. The remainder of this paper is organized as follows. In Section 1 we present some preliminaries on noncommutative Hardy and BMO spaces and the quantum torus. Analytic Hardy spaces on the quantum torus are then introduced and some basic properties of these spaces are ∗ Received

October 18, 2009. This work partially supported by NSFC (10775175).

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presented in Section 2. In Section 3, the “analytic” BMO spaces on the quantum torus will be introduced and the associated H 1 -BMO duality theorem will be proved. Finally, in Section 4 we will discuss some possible extensions of the results obtained in previous sections. In what follows, any notation and terminology not otherwise explained, are as those used in [8, 17] for classical harmonic analysis and in [15, 21] for noncommutative Lp -spaces.

2

Preliminaries

2.1

Noncommutative Hardy and BMO Spaces Let M be a finite von Neumann algebra equipped with a normal faithful tracial state τ and D be a von Neumann subalgebra of M. Let Φ : M → D be the (unique) normal faithful conditional expectation such that τ ◦ Φ = τ. Definition 2.1 A w∗ -closed subalgebra A of M is called a finite subdiagonal algebra of M with respect to D (or Φ) if (i) A + A∗ = {x + y ∗ : x, y ∈ A} is w∗ -dense in M; (ii) Φ is multiplicative on A, i.e., Φ(xy) = Φ(x)Φ(y) for any x, y ∈ A; (iii) A ∩ A∗ = D; where A∗ denotes the family of the adjoint elements of A, i.e., A∗ = {x∗ : x ∈ A}. The algebra D is called the diagonal of A. It was proved by Exel [7] that a finite subdiagonal algebra A is automatically maximal in the sense that, if B is another subdiagonal algebra with respect to D containing A, then B = A. From this maximality, it is concluded that A = {x ∈ M : τ (xa) = 0, ∀a ∈ A0 },

(2.1)

where A0 = {x ∈ A : Φ(x) = 0} (see [1]). For 0 < p ≤ ∞ we denote by Lp (M) the usual noncommutative Lp -space associated with (M, τ ). The norm of Lp (M) is denoted by · p . Recall that L∞ (M) = M, equipped with the operator norm. Definition 2.2 For 0 < p < ∞, the noncommutative Hardy space Hp (A) is defined as the closure of A in Lp (M). These are noncommutative extensions of the classical Hardy spaces on the unit circle T. It is well-known that Φ extends to a contractive projection from Lp (M) onto Lp (D) for every 1 ≤ p ≤ ∞. In general, Φ cannot be continuously extended to Lp (M) for 0 < p < 1. Surprisingly, Φ does extend to a contractive projection on Hp (A) as proved in [2]. For 1 ≤ p < ∞, it was proved by Saito [18] that Hp (A) = {x ∈ Lp (M) : τ (xa) = 0, ∀a ∈ A0 }.

(2.2)

Moreover, for 0 < p < q ≤ ∞, Hp (A) ∩ Lq (M) = Hq (A) and H0p (A) ∩ Lq (M) = H0q (A),

(2.3)

where H0p (A) is the closure of A0 in Lp (M) (see [2]). Clearly, each x ∈ {a + b∗ : a, b ∈ A} admits a unique decomposition x = a + b∗ + d, with a, b ∈ A0 , d ∈ D.

(2.4)

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Definition 2.3 The Hilbert transform H is defined by Hx = −i(a − b∗ ) for every x ∈ {a + b∗ : a, b ∈ A} with decomposition (2.4). Clearly, Φ(Hx) = 0 and x + iHx ∈ A; moreover, if x is self-adjoint, Hx is the unique self-adjoint element in {a + b∗ : a, b ∈ A} such that Φ(Hx) = 0 and x + iHx ∈ A. Note that L2 (M) = H02 (A) ⊕ L2 (D) ⊕ H2 (A)⊥ .

(2.5)

Since {a + b∗ : a, b ∈ A} is w∗ -dense in M, by (2.4) one easily checks that H2 (A)⊥ is the closure of {x∗ : x ∈ A0 } in L2 (M). Decomposition (2.5) shows that H extends to a contraction on L2 (M), again denoted by H. Moreover, it was proved in [16] that H extends to a bounded map on Lp (M) for any 1 < p < ∞; more precisely, one has

Hx p ≤ Cp x p , ∀x = a + b∗ , a, b ∈ A,

(2.6)

where Cp ≤ Cp2 /(p − 1) with a universal constant C. Definition 2.4 The noncommutative BMO space is defined as BMO(M) = {x + Hy : x, y ∈ L∞ (M)} equipped with the norm

z BMO = inf{ x ∞ + y ∞ : z = x + Hy, x, y ∈ L∞ (M)}. The noncommutative analytic BMO space BMO(M) is defined as BMO(M) = BMO(M) ∩ H2 (A) equipped with the norm · BMO . It was proved in [10] that the dual space of H1 (A) can be identified with BMO(M) under the pairing   z, x + Hy = τ z(x + Hy)∗ (2.7) for z ∈ H2 (M), and this identification satisfies 2

x + Hy BMO ≤ x + Hy (H 1 (M))∗ ≤ 2 x + Hy BMO 3

(2.8)

for x + Hy ∈ BMO(M). 2.2 The Quantum Torus Let T = {z ∈ C : |z| = 1} be the unit circle and T2 = {(t, s) : s, t ∈ T} the torus. dm = dθ/2π will denote the normalized Lebesgue measure on T and dm ⊗ dm the normalized Lebesgue product measure on T2 . In the sequel, unless specified otherwise, we always consider the Hilbert space H = L2 (T2 ). Denote by B(H) the space of all bounded operators on H, equipped with the operator norm. Definition 2.5 Let  ≥ 0. Define unitary operators U and V on H respectively by U f (s, t) = eis f (s, t) and V f (s, t) = eit f (s + , t), ∀(s, t) ∈ T2

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for f ∈ H. Let C (T2 ) be the closed span in B(H) of the operators U m V n for m, n = 0, ±1, ±2, · · · . C (T2 ) is a C∗ -algebra called noncommutative torus. Clearly, U V = e−i V U. (2.9) Equivalently, C (T2 ) is the C∗ -algebra generated by U and V. It is easy to check that, if  = 0, then C (T2 ) ∼ = C(T2 ). However, the C∗ -algebras C (T2 ) are generally not ∗-isomorphic for different values of . In particular, C (T2 )  C (T2 ) if  is a rational multiple of π and  is an irrational multiple of π (e.g., see [19]). For any (n, k) ∈ Z2 , let en,k : T2 → C be the exponential function given by en,k (s, t) = ei(ns+kt) , ∀(s, t) ∈ T2 . Then, {en,k } is the standard orthonormal basis of H with U en,k = en+1,k and V en,k = ein en,k+1 for any (n, k) ∈ Z2 . Any polynomial x in U and V can be written as a finite sum  x= am,l U m V l . In this case, xen,k =



eiln am,l en+m,k+l

(2.10)

(2.11)

(2.12)

for n, k = 0, ±1, ±2, · · · , and am,l = xe0,0 , em,l

(2.13)

for (m, l) ∈ Z2 . We regard the am,l ’s as the “Fourier coefficients” of x in (2.11). 2 ∗ 2 Definition 2.6 L∞  (T ) is defined as the w -closure of C (T ) in B(H). 2 For all positive values of , L∞  (T ) are von Neumann algebras and mutually ∗-isomorphic 2 [19], despite the fact that the corresponding C∗ -algebras C (T2 ) are not. The algebra L∞  (T ) ∞ 2 ∞ 2 is called the quantum torus. Clearly, if  = 0 then L (T ) = L (T ). 2 Also, for x ∈ L∞  (T ), define  SN (x) = am,l (x)U m V l , N ∈ N, (2.14) |m|,|l|≤N

where {am,l (x)} are the Fourier coefficients of x given by (2.13). This is a partial Fourier sum of x. Proposition 2.1 (see [19]) We have 2 (1) The partial Fourier sums sN (x) converge weak operator to x for all x ∈ L∞  (T ), i.e.,  x= am,l (x)U m V l , m,l

where the series on the right-hand side denotes the weak operator limit of {sN (x)}N ≥1 , which is called the noncommutative Fourier series of x.

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2 (2) An operator A ∈ B(H) belongs to L∞  (T ) if and only if

Aen,k , en+m,k+l = eiln Ae0,0 , em,l for all n, k, m, l ∈ Z. We define

 τ (x) = xe0,0 , e0,0 =

T2

(xe0,0 )(s, t)dm(s)dm(t)

(2.15)

2 for x ∈ L∞  (T ). Proposition 2.2 τ is a normal faithful finite normalized trace on the quantum torus ∞ L (T2 ). Proof See for instance [19]. It is easy to check that for any x ∈ L1 (M), am,l (x) is well defined and

am,l (x) = τ [x(U m V l )∗ ] = τ (xV −l U −m ), ∀(m, l) ∈ Z2 .

(2.16)

The Fourier series of x ∈ L1 (M) is the formal series 

am,l (x)U m V l .

(2.17)

m,l

It is not clear at present in which sense and for which x ∈ L1 (M) series (2.17) converges. The study of convergence of the Fourier series over the quantum torus will be carried elsewhere. However, the following analogue of the Riesz-Fisher theorem on the quantum torus holds true. Proposition 2.3 Series (2.17) is the Fourier series of some x ∈ L2 (M) if and only if 

|am,l |2 < ∞.

(2.18)

m,l

In this case

x 22 =



|am,l (x)|2 ;

(2.19)

m,l

moreover, series (2.17) converges to x unconditionally in L2 (M). Proof See [19].

3

Analytic H p Spaces on the Quantum Torus

2 Throughout, we always denote the quantum torus L∞  (T ) by M. In this section, we will give the definition of analytic H p spaces on the quantum torus and present some basic properties of these spaces. Definition 3.1 Set

A = {x ∈ M : am,l (x) = 0, m < 0 or l < 0}. For 0 < p < ∞, analytic Hardy spaces Hp (A) on the quantum torus is defined as the closure of A in Lp (M). For p = ∞ we simply set H∞ (A) = A for convenience.

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We would like to point out that these Hardy spaces are distinct from noncommutative Hardy spaces in Arveson’s sense [1], because here A + A∗ is not w∗ -dense in M (e.g., (i) of Definition 2.1 does not hold). But those spaces can be considered to be noncommutative extensions of the classical Hardy spaces on the torus T2 . Also, we denote (1) A1 = {x ∈ M : am,l (x) = 0, m < 0, l ∈ Z}, (2) A2 = {x ∈ M : am,l (x) = 0, l < 0, m ∈ Z}, (3) D1 = A1 ∩ A∗1 = {x ∈ M : am,l (x) = 0, m = 0, l ∈ Z}; (4) D2 = A2 ∩ A∗2 = {x ∈ M : am,l (x) = 0, l = 0, m ∈ Z};  (5) Φ1 (x) = a0,l (x)V l , l  (6) Φ2 (x) = am,0 (x)U m , m

(7) A1,0 = A1 ∩ KerΦ1 = {x ∈ M : am,l (x) = 0, m ≤ 0, l ∈ Z}; (8) A2,0 = A2 ∩ KerΦ2 = {x ∈ M : am,l (x) = 0, l ≤ 0, m ∈ Z}. In the sequel, we denote by Hp = Hp (A) and Lp = Lp (M) in short. We will use the following standard notation: For A ⊂ Lp , 1 ≤ p < ∞, let [A]p denote the closure of A in Lp . Thus, Hp = [A]p . Proposition 3.1 For 1 ≤ p ≤ ∞, the following three sets are all equal to Hp : (1) H1 ∩ Lp , (2) {x ∈ Lp : τ (xy) = 0, ∀y ∈ A1,0 or A2,0 }, (3) {x ∈ Lp : am,l (x) = 0, m < 0 or l < 0}. Proof For p = ∞, the result is clear. Given 1 ≤ p < ∞. By the noncommutative H¨older inequality, it is easy to check that Hp ⊂ (1) ⊂ (2) ⊂ (3). It remains to prove (3) ⊂ Hp . To this end, we use the transference method. For each z ∈ T2 , let πz be the isomorphism of M determined by πz (Um ) = z m Um = z1m1 z2m2 U1m1 U2m2 , ∀m = (m1 , m2 ) ∈ Z2 , where U = (U1 , U2 ) = (U, V ). Since πz is unitary (i.e., πz πz¯ = 1 for any z ∈ T2 ), it is concluded that

πz (x) p = x p , ∀x ∈ Lp (M). ˜(z) = πz (x) for z ∈ T2 . Then x˜ ∈ Lp (T2 , Lp (M)) and ˜ x p = x p , For x ∈ Lp (M), define x 2 that is, x → x˜ is an isometric embedding from Lp (M) into Lp (T , Lp (M)) and so an isometric imbedding from Hp into Hp (T2 , Lp (M)). Evidently, if x ∈ (3) then x˜ ∈ Hp (T2 , Lp (M)). Thus, x ∈ Hp . Proposition 3.2 For each k = 1, 2, the map Φk is the conditional expectation of M with respect to Dk and Ak is a finite subdiagonal algebra of M with respect to Dk . Consequently, for k = 1 or 2, Φk extends to a contractive projection on Lp (M) for every 1 ≤ p ≤ ∞. Moreover, Φk extends to a contractive projection on Hp (A) for 0 < p < 1. Proof By Definition 2.1, it is easy to check that Ak is a finite subdiagonal algebra of M with respect to Dk for k = 1, 2. Moreover, by Theorem 2.1 in [2] we obtain the second assertion.

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Proposition 3.3 For k = 1 or 2, τ (Φk (x)y) = τ (xΦk (y)), ∀x, y ∈ M. Proof τ (Φk (x)y) = τ (Φk (Φk (x)y)) = τ (Φk (x)Φk (y)) = τ (xΦk (y)). Proposition 3.4 For k = 1 or 2, the adjoint of the map Φk : Lp → Lp (1 ≤ p ≤ ∞) is a map Lq → Lq , where 1/p + 1/q = 1. This is an immediate consequence of Propositions 3.2 and 3.3.  For x = am,n (x)U m V n , we define mn



H1 (x) = −i

sgn(m)am,l (x)U m V l

m,l∈Z

and H2 (x) = −i



sgn(l)am,l (x)U m V l ,

m,l∈Z

respectively. We also define R1 (x) =



am,l (x)U m V l

m≥0,l∈Z

and R2 (x) = for x =

 m,l∈Z



am,l (x)U m V l

m∈Z,n≥0 m

l

am,l (x)U V , respectively. By Proposition 2.3 we immediately conclude that

both Hk and Rk are contractive operators on L2 (M). On the other hand, we have H1 H2 (x) = H2 H1 (x), ∀x ∈ L2 (M) and

(3.1)

 1 x + iHk (x) + Φk (x) (3.2) 2 for any x ∈ L2 (M). Thus, as far as boundedness is concerned, by Proposition 3.2, it suffices to consider Hk . Proposition 3.5 For k = 1 or 2, Rk (x) =





τ Hk (x)y = −τ xHk (y) , ∀x, y ∈ M. Proof For x, y ∈ M, we have  sgn(m)am,l (x)bj,k (y)e−ijl U m+j V l+k H1 (x)y = −i m,l,j,k

and xH1 (y) = −i



sgn(j)am,l (x)bj,k (y)e−ijl U m+j V l+k .

m,l,j,k

Hence, τ (H1 (x)y) = −i

 m,l

sgn(m)am,l (x)b−m,−l (y)eiml = −τ (xH1 (y))

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for any x, y ∈ M. Similarly, we can obtain the same result for H2 . Proposition 3.6 Hk (k = 1, 2) extends to a bounded map on Lp (M) for any 1 < p < ∞; more precisely, one has

Hk x p ≤ Cp x p , ∀x ∈ Lp (M),

(3.3)

where Cp ≤ Cp2 /(p − 1) with C being a universal constant. Proof We will use the trick due to Cotlar (e.g., Theorem 8.4 (i) in [15]). First, an immediate computation yields that

(Hk x)∗ Hk x = x∗ x + Hk x∗ Hk x + (Hk x)∗ x

(3.4)

for any x ∈ L2 (M). Using (3.4) and the noncommutative H¨ older inequality, we have

Hk x 22p = (Hk x)∗ Hk x p



≤ x∗ x p + Cp Hk x∗ Hk x + (Hk x)∗ x p ≤ x 22p + Cp x∗ Hk x + (Hk x)∗ x p ≤ x 22p + 2Cp x 2p Hk x 2p .

This concludes that 

Hk x 2p ≤ Cp + Cp2 + 1 x 2p . In particular, one has C2n+1 ≤ C2n + C22n + 1 for any n ∈ N. Now starting from C2 = 1 and by induction, we deduce that C2n ≤ 2n − 1. For all other values of p, we may apply interpolation to obtain Cp ≤ 2p − 1. Hence, inequality (3.3) is proved for p ≥ 2. For 1 < p < 2, we use duality. Indeed, since the adjoint of Hk is equal to −Hk by Proposition 3.5, then Hk p = Hk p for any 1 < p < 2. Corollary 3.1 Let R = R1 R2 . Then, for every 1 < p < ∞, R is a bounded map from Lp into Hp with the norm Cp = Cp4 /(p − 1)2 at most. This is an immediate consequence of (3.2) and Proposition 3.6. Corollary 3.2 Let 1 < p < ∞. Then,



τ Hk (x)y = −τ xHk (y) , k = 1, 2 for any x ∈ Lp (M) and y ∈ Lq (M), where 1/p + 1/q = 1. Proof Since H1 is bounded on Lp (M) for 1 < p < ∞, by the noncommutative H¨ older inequality and the L1 (M)-continuity of τ , we conclude the required result for H1 . Similarly, we can obtain the same result for H2 . Proposition 3.7 Suppose 1 ≤ r, p, q ≤ ∞ with 1/p + 1/q = 1/r. If x ∈ Hp and y ∈ Hq , then xy ∈ Hr and xy r ≤ x p y q . Proof Suppose x ∈ Hp and y ∈ Hq . It is concluded from the noncommutative H¨ older r inequality (e.g., [21]) that xy ∈ L and the norm estimate holds. It remains to show that xy ∈ Hr , for which it suffices, by Proposition 3.1 (3), to show that am,l (xy) = 0 whenever m < 0 or l < 0. Without loss of generality, suppose m < 0. There exists a sequence (yn ) ⊂ A

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such that yn − y q → 0. Thus, by the noncommutative H¨older inequality and Proposition 3.1 (2) we have am,l (xy) = lim am,l (xyn ) = 0. n

Theorem 3.1 Let 1 < p < ∞. The dual space of Hp can be identified with Hq via the canonical pairing: x, y = τ (xy ∗ ), ∀x ∈ Hp , ∀y ∈ Hq , where q is the conjugate index of p, i.e., 1/p + 1/q = 1. Proof For any y ∈ Hq , by Proposition 3.7 we conclude that x → x, y is a bounded functional on Hp with its norm less than y q . Conversely, suppose  is in the dual space of Hp . By the Hahn-Banach extension theorem,  can be extended to a bounded functional on Lp . By the duality of Lp , there exists an w ∈ Lq such that (x) = τ (xw∗ ), ∀x ∈ Lp . Set y = Rw. By Corollary 3.1 we have y ∈ Hq . Since x = Rx for any x ∈ Hp , by Corollary 3.2, we have, for any x ∈ Hp , (x) = Rx, w = x, R1 R2 w = x, y .

4

H 1 -BMO Duality on the Quantum Torus

In this section, we will introduce BMO space on the quantum torus and prove the corresponding H 1 -BMO duality. Definition 4.1 The noncommutative BMO space on the quantum torus is defined as 

BMO(M) = x + H1 y + H2 z + H1 H2 (w) : x, y, z, w ∈ L∞ (M) , equipped with the norm 

v BMO = inf x ∞ + y ∞ + z ∞ + w ∞ , where the infimum is taken over all x, y, z, w ∈ L∞ (M) such that v = x + H1 y + H2 z + H1 H2 (w). The noncommutative “analytic” BMO space BMO(M) on the quantum torus is defined as BMO(M) = BMO(M) ∩ H2 (A), equipped with the norm · BMO . Proposition 4.1 the norm 94 at most.

Let R = R1 R2 . Then, R is a bounded map from M to BMO(M) with

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Proof By Corollary 3.1 we conclude that R is a bounded map from L2 (M) into H2 (A). Thus, for x ∈ M, it is concluded from (3.2) that

 1 R2 (x) + iH1 R2 (x) + Φ1 R2 (x) 2   1  1 = x + iH2 (x) + Φ2 (x) + iH1 x + iH2 (x) + Φ2 (x) 4 4  1  + Φ1 x + iH2 (x) + Φ2 (x) 4  1 1 1 1 = x + Φ1 (x) + Φ2 (x) + Φ1 (Φ2 (x)) + iH1 (x) + iH2 (x) 4 4 4 4  1 1 − H1 H2 (x) + i H1 (Φ2 (x)) + H2 (Φ1 (x)) . 4 4

R(x) =

Hence, by Proposition 3.2 we have

Rx BMO ≤

 9 7 1

x ∞ + H1 (Φ2 (x)) BMO + H2 (Φ1 (x)) BMO ≤ x ∞ . 4 4 4

Theorem 4.1 The dual space of H1 (M) can be identified with BMO(M) under the pairing u, v = τ (uv ∗ ) for u ∈ H2 (M) and v ∈ BMO(M). Moreover, this identification satisfies 4

v BMO ≤ v (H1 )∗ ≤ 4 v BMO . 9 Proof Suppose v = x + H1 y + H2 z + H1 H2 (w) ∈ BMO(M). Note that for any u ∈ H (M), Hk (u) = −i(I − Φk )u, k = 1, 2. 2

Then, by Proposition 3.2, we have





|τ (uv ∗ )| ≤ |τ (ux∗ )| + |τ u(H1 y)∗ | + |τ u(H2 z)∗ | + |τ u(H1 H2 w)∗ |





= |τ (ux∗ )| + |τ (H1 u)y ∗ | + |τ (H2 u)z ∗ | + |τ (H1 H2 u)w∗ | ≤ u 1 x ∞ + (I − Φ1 )u 1 y ∞ + (I − Φ2 )u 1 z ∞ + (I − Φ1 )(I − Φ2 )u 1 w ∞

≤ 4 u 1 x ∞ + y ∞ + z ∞ + w ∞ . Thus, v = x + H1 (y) + H2 (z) + H1 H2 (w) is a bounded linear functional on H1 satisfying

v (H1 )∗ ≤ 4 v BMO . Conversely, if f ∈ (H1 )∗ , then by the Hahn-Banach theorem, we can extend f to some ∗ w ∈ L1 (M)∗ = L∞ (M) with f (H1 )∗ = w∗ ∞ . Now, for u ∈ H2 (M), we have f (u) = τ (uw∗ ) = u, w = Ru, w = u, Rw . So let Rw = v. Thus, f (u) = u, v for each u ∈ H2 (M). Finally, by Proposition 4.1, we have

v (H1 )∗ = f (H1)∗ = w ∞ ≥

4 4

Rw BMO = v BMO . 9 9

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Remark

In this section, we make some remarks on our results and possible extension to higher dimension quantum tori. (1) Our results can be extended to n-dimensional quantum torus (n ≥ 2). Let MΘ be an n-dimensional quantum torus associated with a n × n real skew-symmetric matrix Θ = (θkj ). Recall that MΘ is a von Neumann algebra generated by n unitaries U1 , · · · , Un satisfying the relation Uk Uj = e2πiθkj Uj Uk , j, k = 1, · · · , n. Consider a formal series 

cj1 ,···,jn U1j1 · · · Unjn ,

(j1 ,···,jn )∈Zn

where cj1 ,···,jn ∈ C is the Fourier coefficient of the series associated with the index (j1 , · · · , jn ). Let Pn be the polynomial algebra consisting of the above formal series with Fourier coefficients being zero for all but finite indexes. For such a series x we define τ (x) = c0,···,0 (x). Then τ can be extended to a normal faithful finite normalized tracial state on MΘ . We let

 A = x ∈ MΘ : cj1 ,···,jn (x) = 0 whenever some jk < 0 . Then, analytic Hardy spaces Hp on the n-dimensional quantum torus MΘ are defined as the closure of A in Lp (MΘ , τ ) for 0 < p < ∞. Similarly, we define the associated BMO space. By merely repeating the above proofs, we can show that the results obtained above all hold true on the n-dimensional quantum torus MΘ . We omit the details. (2) As pointed above, analytic Hardy spaces Hp on n-dimensional quantum torus are different from noncommutative Hardy spaces in Arveson’s sense, although there are some analogues between them. Motivated by our results, we may generalize the definition of Arveson’s noncommutative Hardy spaces to cover those Hardy spaces. The following is such a generalization. Let M be a von Neumann algebra with a normal faithful finite tracial state τ. Given n ≥ 2. Let A1 , · · · , An be n w∗ -closed unital subalgebras of M so that (i) For every k ∈ {1, · · · , n}, Ak + A∗k is w∗ -dense in M, where A∗k = {a∗ : a ∈ Ak }; (ii) For each k ∈ {1, · · · , n}, Φk is the expectation from M onto its von Neumann subalgebra Ak ∩ A∗k such that τ ◦ Φk = τ and Φk (xy) = Φk (x)Φk (y) for all x, y ∈ Ak . n  Set A = Aj . For 0 < p < ∞, we define Hp (A) to be the closure of A in Lp (M), and j=1

for p = ∞ we simply set H∞ (A) = A for convenience. These should be called n-dimensional noncommutative Hardy spaces, because analytic Hardy spaces Hp on n-dimensional quantum torus defined above are included in this class of spaces. Evidently, they can be regarded as noncommutative extensions of classical Hardy spaces Hp (Tn ) in the n-torus or Hp (Bn ) in the unit ball of Cn (see [17]).

1996

ACTA MATHEMATICA SCIENTIA

Vol.31 Ser.B

We define n-dimensional noncommutative BMO spaces in a similar way. By slightly modifying the proofs, we can prove most results obtained above, such as the duality theorem for Hp with 1 < p < ∞ and H1 -BMO duality theorem. The details are left to readers. References [1] Arveson W B. Analyticity in operator algebra. Amer J Math, 1967, 89: 578-642 [2] Bekjan T N, Xu Q. Riesz and Szeg¨ o type factorizations for noncommutative Hardy spaces. J Oper Theory, 2009, 62(1): 215–231 [3] Blecher D P, Labuschagne L E. Characterizations of noncommutative H ∞ . Integr Equ Oper Theory, 2006, 56: 301–321 [4] Blecher D P, Labuschagne L E. Applications of the Fuglede-Kadison determinant: Szeg¨ o’s theorem and outers for noncommutative H p . Trans Amer Math Soc, 2008, 360(11): 6131–6147 [5] Blecher D P, Labuschagne L E. A Beuring theorem for noncommutative Lp . J Oper Theory, 2008, 59(1): 29–51 [6] Chen Z. Hardy spaces of operator-valued analytic functions. Ilinois J Math, 2009, 53: 303–324 [7] Exel R. Maximal subdiagonal algebras. Amer J Math, 1988, 110: 775–782 [8] Garnett J B. Bounded Analytic Functions. New York: Springer, 2007 [9] Junge M, Le Merdy C, Xu Q. H ∞ -functional caculus and square functions on noncommutative Lp spaces. Ast´ erisque, 2006, 305 [10] Marsalli M, West G. Noncommutative H p spaces. J Oper Theory, 1988, 40: 339–355 [11] Marsalli M, West G. The dual of noncommutative H 1 . Indiana Univ Math J, 1998, 47: 489–500 [12] Mei T. Operator valued Hardy spaces. Memoir Amer Math Soc, 2007, 881 [13] Mei T, Parcet J. Pseudo-localization of singular integrals and noncommutative Littlewood-Paley inequalities. Int Math Res Not, 2009, 9: 1433–1487 [14] Parcet J. Pseudo-localization of singular integrals and noncommutative Calder´ on-Zygmund theory. J Funct Anal, 2009, 256: 509–593 [15] Pisier G, Xu Q. Noncommutative Lp -spaces//Handbook of the Geometry of Banach Spaces, Vol 2. Amsterdan: North-Holland, 2003: 1459–1517 [16] Randrianantoanina N. Hillbert transform associated with finite maxmal subdiagonal algebras. J Austral Math Soc Ser A, 1998, 65: 388–404 [17] Rudin W. Function Theory in the Unit Ball of Cn . New York: Springer-Verlag, 1980 [18] Saito K S. A note on invariant subspaces for finite maximal subdiagonal algebras. Proc Amer Math Soc, 1979, 77: 348–352 [19] Weaver N. Mathematical Quantization. Florida: CRC Press, 2001 [20] Xu Q. Noncommutative Lp -Spaces and Martingale Inequalities. Book Manuscript, 2007 [21] Xu Q, Bekjan T N, Chen Z. Introduction to Operator Algebras and Noncommutative Lp Spaces (Chinese). Beijing: Science Press