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An Equivalent Quasinorm for the Lipschitz Space of Noncommutative Martingales

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An Equivalent Quasinorm for the Lipschitz Space of Noncommutative Martingales Open Mathematics 2020; 18: 1281–1291 Research Article Congbian Ma* and Yanbo Ren An equivalent quasinorm for the Lipschitz space of noncommutative martingales https://doi.org/10.1515/math-2020-0072 received December 4, 2019; accepted August 10, 2020 Abstract: In this paper, an equivalent quasinorm for the Lipschitz space of noncommutative martingales is presented. As an application, we obtain the duality theorem between the noncommutative martingale Hardy space hc (resp. hr ) and the Lipschitz space λc (resp. λr ) for 0 p 1, β 1 1. p( ) p( ) β( ) β( ) << =−p c r We also prove some equivalent quasinorms for hp( ) and hp( ) for p = 1 or 2 <<∞p . Keywords: noncommutative space, martingale, Hardy space MSC 2020: 46L53, 46L52, 60G42 1 Introduction In the past two decades, due to the excellent work of Pisier and Xu on noncommutative martingale inequalities [1], the study of noncommutative martingale theory has attracted more and more attention. Especially in recent years, some meaningful research results on the noncommutative martingale theory have emerged continuously, and it has become a research hotspot in the field of noncommutative analysis. The Lipschitz space was first introduced in the classical martingale theory by Herz and plays an important role in it. For instance, the Lipschitz space is the generalization of the BMO space and the dual ( ) c r space of the Hardy space hp 0 <≤p 1 . The noncommutative Lipschitz spaces λ0( ) and λ0( ) were first introduced in [2], and with their help, the atomic decomposition for the Hardy space h1() was c r proved. In this paper, we study the noncommutative Lipschitz spaces λβ( ) and λβ( ) for β ≥ 0.We show that for q 0, β 1 , we have − ∞< < =−q c c r r λXβ()=()q and λXβ ()=(q ) with equivalent norms. As its application, we have the duality equalities for 0 p 1 and β 1 1 << =−p c ∗∗c r r (hλp())=()β and (())=() hλp β . This answers positively a question asked in [2]. The other main result of this paper concerns the equivalent c r quasinorms for hp( ) and hp( ) for p = 1 or 2 <<∞p . We prove the equalities c 2,c r 2,r hL1 ()=()1 and hL1 ()=()1 , c r which give a new characterization of h1 ( ) and h1 ( ). * Corresponding author: Congbian Ma, School of Mathematics and Information Science, Xinxiang University, Henan, People’s Republic of China, e-mail: [email protected] Yanbo Ren: School of Mathematics and Statistics, Henan University of Science and Technology, Henan, People’s Republic of China, e-mail: [email protected] Open Access. © 2020 Congbian Ma and Yanbo Ren, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 1282 Congbian Ma and Yanbo Ren This paper is organized as follows. Some definitions and notations are given in Section 2.An equivalent quasinorm for noncommutative martingale Lipschitz space is shown in Section 3. Equivalent c r quasinorms for hp( ) and hp( ) for p = 1 or 2 <<∞p are considered in Section 4. 2 Preliminaries Let be a von Neumann algebra acting on a Hilbert space H and τ be a normal faithful finite trace on . We call () , τ a noncommutative probability space. Let x be a positive operator on H. Then x admits a ∞ unique spectral decomposition: x =()∫ λexd λ . We will often use the spectral projection e(∞)λ, (x) 0 corresponding to the interval (λ, ∞). For 0 <≤∞p , let Lp() be the associated noncommutative Lp-space. Recall that the norm on Lp() is defined by p 1 ∥∥xτxxLp =((||))p ,, ∈p ( ) ∗ 1 where |xxx|=( )2 is the usual modulus of x. Note that if p =∞, L∞() is just with the usual operator norm. For more detailed discussions about noncommutative Banach function spaces, see [3–5]. Let us recall the general setup for noncommutative martingales. Let ( nn) ≥1 be an increasing filtration of von Neumann subalgebras of such that the union of n’s is weak*-dense in and n (with 0 = 1) the conditional expectation with respect to n. A sequence x =(xnn )≥1 is said to be adapted if xnn∈(L1 ) for all n ≥ 1, and predictable if xnn∈(L11 − ) for n ≥ 2. A noncommutative martingale with respect to the filtration ( nn) ≥1 is a sequence x =(xnn )≥1 in L1() such that nn()=xx+1 n for all n ≥ 1. If additionally, x =(xLnn )≥1 ⊂ p () for some 0 ≤<∞p , we call x an Lp()-martingale. In this case, we set ∥∥xxpnn =sup ∥ ∥p.If∥∥x p <∞, then x is called a bounded Lp()-martingale. Note that the space of all bounded Lp-martingales, equipped with ∥⋅∥p, is isometric to Lp() for p > 1. This permits us to not distinguish a martingale and its final value x∞ (if the latter exists). Let x =(xnn )≥1 be a noncommutative martingale with respect to ( nn) ≥1 with the usual convention that 0 = 1.Define dxxxnnn=−−1 for n ≥ 1 with the usual convention that x0 = 0. The sequence ddxx=(nn )≥1 is called the martingale difference sequence of x. In the sequel, for any operator x ∈(L1 ) we denote xnn=( x) for n ≥ 1. Let x =(xnn )≥1 be a finite martingale in L2(). We set 1 1 n 2 ∞ 2 2 2 scn, ()=xxsxx∑∑kk−1 (|d, |) c ()= nn−1 (| d |) k=1 n=1 and 1 1 n 2 ∞ 2 ∗ 2 ∗ 2 srn, ()=xxsxx∑∑k−1 (|d,k |) r ()= n−1 (| dn |) . k=1 n=1 These will be called the column and row conditioned square functions, respectively. Let 0 <<∞p .Define c r ( ) fi - ( ) c hp( ) resp. hp( ) as the completion of all nite L∞ martingales under the quasi norm ∥∥x hp() = c r ( r ) fi ( ) ∥()∥sxc p resp. ∥∥xsxhrp() =∥ ()∥p . For p =∞.Wede ne h∞( ) resp. h∞( ) as the Banach space of the - ∞ 2 ( ∞ ∗ 2) L∞ martingales x such that ∑n=1 nn−1(|dx |) respectively ∑n=1 n−1(|dxn |) converge for the weak operator topology. For more information of noncommutative martingales, see the seminal article of Pisier and Xu [3] and the sequels to it. c r The main object of this paper is the noncommutative Lipschitz spaces λβ and λβ( ). An equivalent quasinorm for the Lipschitz space of noncommutative martingales 1283 Definition 2.1. Let β ≥ 0. Set c c λxLxβ()={∈()∥∥2 : λβ() <∞} with β 1 β 2 1 c − −−2 2 ∥∥xτexτeτexxλβ() =max sup () ∥1 ()∥∞ , sup sup () (nn (|− |)) , e∈ 1 ne≥∈1 n where n denotes the lattice of projections of n. Similarly, we define r r ∗ c λxLxxβ( )={ ∈2 ( ): ∥ ∥λλβ() =∥ ∥β() <∞}. 2 fi [ ] The classical martingale space L1 which is de ned in 6 has the following noncommutative analogue. Definition 2.2. We define ∞ 2,c 1 LxLxyzyLzLz()=∈():, =∑ k ∈(22 mk ) ,,0 ∈() mk ()= 1 k k kk k=1 ∞ and∥∥xyzL 2,c() = inf∑ ∥k ∥∥22k ∥ <∞ , 1 k=1 fi where ( mkk) ≥1 and ( mkk) ≥1 are the subsequences of ( kk) ≥1 and ( kk) ≥1, the in mum runs over all decompositions of x as above. Similarly, define 2,r ∗ L xL: x2,rc x2, . 1 ( )={ ∈1 ( ) ∥ ∥LL1 () =∥ ∥1 () <∞} 2,c 2,r It is clear that L1 ( ) and L1 ( ) are Banach spaces. We will use the following definitions from [2,7]. Definition 2.3. [2] Let 02<≤p . Denote the index class W1 which consists of sequences (ωnn) ≥1 such that 21/−p + fi (ωn )n≥1 is nondecreasing with each ωnn∈(L1 ) invertible with bounded inverse and ∥∥≤ωn 1 1.De ne 1 2 ∞ c c 12−/p 2 Xp()=∈()∥∥xL2 :infd xX () = τ∑ || ωn | xn+1 | <∞. p W 1 n=1 12−/p For 2 <≤∞p or − ∞≤p <0. Denote W2 which consists of sequences (ωnn) ≥1 such that (ωn )n≥1 is + nondecreasing with each ωnn∈(L1 ) invertible with bounded inverse and ∥∥≤ωn 1 1.Define 1 ∞ 2 c 12p 2 X ()=∈()∥∥xL:supd xc = τ || ω−/ | x | <∞. p 2 Xp() ∑ n n+1 W2 n=1 Similarly, define r r ∗ c Xp( )={xL ∈2 ( ): ∥ x ∥XXp() =∥ x ∥p() <∞}. c Remark that for 2 <≤∞p , Xp( ) can be rewritten in the following form. Given (ωWnn)∈≥1 2, we set 1 2 2 2 gωωα α ,1, n n =−()n n−1 ∀≥ where α 1 1 . It is clear that =−2 p α ∞ 2 2 ( ) (gGn)∈αnnnαn = uu =( )≥0 :, uL ∈ ( ) τ∑ | un | ≤ 1. 3.1 n=1 1284 Congbian Ma and Yanbo Ren Then 1 ∞ 2 22 ∥∥xτuxxc =sup | | (|− |) . Xp() ∑ nn n ()∈uGnα n=1 We recall the definition of the space Lp(ℓ ; ∞), with 1 ≤≤∞p . A sequence (xnn) ≥1 in Lp() belongs to Lp(ℓ; ∞) if (xnn) ≥1 admits a factorization xn = ayn b, with a, bL∈(2p ) and (yLn)∈(n≥∞1 ). The norm of (xnn) ≥1 is then defined as ∥()xaybnn≥(ℓ)1; ∥ Lp ∞ =inf {∥∥ 2p sup ∥n ∥∞ ∥∥2p }. xaybn= n n≥1 + We usually write ∥(xxnn )≥(ℓ)1; ∥ Lp ∞ = ∥sup n ∥p. n Definition 2.4. [7] Let 2 <<∞p .Define the space c + 2 p Lp mo()=∈()∥ a L2 :sup nn (|−()|)∥<∞ a a { 2 } equipped with the norm 1 2 c +2 p ∥aaaa ∥Lmo() =max ∥ ∥ p , ∥ sup n (| − n ( )| )∥ . p ()2 c c Then (Lmop ()∥⋅∥, Lmop ()) is a Banach space. Similarly, we set r ∗ c Lp mo()={ a: a ∈ Lp mo ()} equipped with the norm r ∗ c ∥∥aaLmop () =∥ ∥ Lmop (). 3 An equivalent quasinorm for the Lipschitz space of noncommutative martingales c In this section, we prove the noncommutative equivalent quasinorms for Lipschitz spaces λβ( ) and r ( ) c ( r ) λβ( ) β ≥ 0 . As its application, we obtain the duality of the Hardy space hp( ) resp. hp( ) and the c ( r ) Lipschitz space λβ( ) resp.
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