An equivalent quasinorm for the Lipschitz space of noncommutative martingales

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 h p c ( ℳ ) {h}_{p}^{c}( {\mathcal M} ) (resp. h p r ( ℳ ) {h}_{p}^{r}( {\mathcal M} ) ) and the Lipschitz space λ β c ( ℳ ) {\lambda }_{\beta }^{c}( {\mathcal M} ) (resp. λ β r ( ℳ ) {\lambda }_{\beta }^{r}( {\mathcal M} ) ) for 0 < p < 1 0\lt p\lt 1 , β = 1 p − 1 \beta =\tfrac{1}{p}-1 . We also prove some equivalent quasinorms for h p c ( ℳ ) {h}_{p}^{c}( {\mathcal M} ) and h p r ( ℳ ) {h}_{p}^{r}( {\mathcal M} ) for p = 1 p=1 or 2 < p < ∞ 2\lt p\lt \infty .

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 Keywords: noncommutative space, martingale, Hardy space MSC 2020: 46L53, 46L52, 60G42

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  This answers positively a question asked in [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: λ 0 We will often use the spectral projection ( ) be the associated noncommutative L p -space. Recall that the norm on ( ) L p is defined by is just with the usual operator norm. For more detailed discussions about noncommutative Banach function spaces, see [3][4][5].
Let us recall the general setup for noncommutative martingales. Let ( ) ≥ n n 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 , then x is called a bounded ( ) L p -martingale. Note that the space of all bounded L p -martingales, equipped with ∥⋅∥ p , is isometric to ( ) L p for > p 1. This permits us to not distinguish a martingale and its final value ∞ x (if the latter exists). Let = ( ) ≥ x x n n 1 be a noncommutative martingale with respect to ( ) ≥ n n 1 with the usual convention that =  and These will be called the column and row conditioned square functions, respectively. Let 2 ) 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.
The main object of this paper is the noncommutative Lipschitz spaces λ β c and ( ) λ β r . where n denotes the lattice of projections of n . Similarly, we define The classical martingale space L 1 2 which is defined in [6] has the following noncommutative analogue.
are the subsequences of ( ) ≥ k k 1 and ( ) ≥ k k 1 , the infimum runs over all decompositions of x as above. Similarly, define .
are Banach spaces.
We will use the following definitions from [2,7]. . Define Similarly, define can be rewritten in the following form.  We recall the definition of the space ( ℓ ) . The norm of ( ) ≥ x n n 1 is then defined as Similarly, ( ) = ( ) λ X β r q r with the same equivalence constants.
The following Lemma is the key ingredient of our proof. Thus by (3.2), we get that  It is easy to see that = e e 0 0 , which implies that ≤ e e 0 does not hold. Using the preceding result, we have that The proof is complete. □ We will also need the following well-known lemma from [8]. Noting that ( ) ∈ ≥ u G n n α 0 , we have is the characteristic function at point λ 0 . It follows that  , we obtain . Taking supremum over all The proof is complete. □ Using the dual result in Theorem 3.2 in [2], we will describe the dual space of ( )( < < ) h p 0 1 p as the Lipschitz space. In this section, we first describe an equivalent quasinorm for ( ) h c 1 . As in the classical case, the spaces h 1 and L 1 2 are equivalent (see [6]). We will transfer this to the noncommutative martingales. Note that the set G α defined in (3.1) can be reduced to the following one: Thus, the set G α can be reduced to  G α . □ , 0 , 1. Proof.
Step 1: Let ∈ ( ) x X p c . Fix a positive integer N, we will show that ( ) ∈ ( ) s x L c N p , . Let ≤ ≤ n N 1 . Since the dual space of ( ) L n p 2 is ( )