Abstract
Let be a separable symmetric space on and the corresponding noncommutative space. In this paper, we introduce a kind of quasimartingale spaces which is like but bigger than and obtain the following interpolation result: let be the space of all bounded -quasimartingales and . Then, there exists a symmetric space on with nontrivial Boyd indices such that .
1. Introduction
Let be a symmetric space on with the Fatou property and . Kalton and Montgomery-Smith [1] proved that there exists a symmetric space with nontrivial Boyd indices such that . As is now well-known, the preceding interpolation result can automatically lift to the noncommutative setting (see [[2], Theorem 3.4]): let be a symmetric space on with the Fatou property and a semifinite von Neumann algebra. Then, the following are equivalent: (i)(ii)There exists a symmetric space on with nontrivial Boyd indices such that
In this paper, we replace the space in (2) with a bigger and more complex space (see Definition 3) and obtain a generalized interpolation result. Our main result can be stated as follows (see Section 2 for the unexplained notations).
Theorem 1. Let be a separable symmetric space on with . Then, there exists a symmetric Banach function space on SSS with nontrivial Boyd indices such that
2. Preliminaries
2.1. Noncommutative Spaces
Let be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace . We denote by the family of all -measurable operators. Note that is the spectral projection of associated with the interval . For , define its generalized singular number by
Note that the function from into is right continuous and nonincreasing. For the case that is the abelian von Neumann algebra with the trace given by integration with respect to the Lebesgue measure, is the space of all measurable functions, and is the decreasing rearrangement of the measurable function (see [3, 4]).
Recall that a Banach function space on is called symmetric if for any and any measurable function with , we have and . The Köthe dual of is the function space defined by setting
When equipped with the norm is a symmetric Banach function space. For any , we define the dilation operator on by
Define the lower and upper Boyd indices of by respectively. It is well-known that , and we shall say that has nontrivial Boyd indices, whenever . We refer to [1, 5] for unexplained terminology from function space theory.
For a given symmetric Banach function space on , we define the corresponding noncommutative space by setting equipped with the norm
It is well-known that is a Banach space and is referred to as the noncommutative symmetric space associated with corresponding to the function space . Note that if and , then is the usual noncommutative Lp-space associated with .
Recall that is a von Neumann algebra equipped with the trace: (see [6]). Now, let be the von Neumann algebra tensor product and the tensor trace. This gives rise to noncommutative spaces . Note that coincides with the space .
2.2. Noncommutative Martingales
A noncommutative probability space is a couple , where is a finite von Neumann algebra and is a normal faithful trace with . Let be an increasing sequence of von Neumann subalgebras of such that the union of the is in . Let be the conditional expectation with respect to .
Definition 2. A sequence in is called a sequence of martingale differences if for and if for all .
In this paper, we always consider noncommutative martingales associated with a noncommutative probability space unless explicit explanation.
2.3. Interpolation
Let be a compatible couple of quasi-Banach space. Its K-functional is defined by for and . Let be a symmetric Banach space on . Set
Then, the interpolation space is defined as equipped with the norm .
3. Main Result
The main result in this section is Theorem 1, which extends the result of Kalton and Montgomery-Smith [1] to a -quasimartingale spaces. We first introduce the quasimartingale spaces.
Definition 3. Let be a symmetric Banach function space on . A sequence is called a -quasimartingale with respect to if for and (with ) We set where and denote the standard basic sequence of .
If , is called a bounded -quasimartingale. The quasimartingale space is defined as the space of all bounded -quasimartingales, equipped with the norm .
Remark 4. (i)In the case for , we have that and (ii)Let be a bounded -quasimartingale. Set and . Then, is a predicable -quasimartingale with , and is a bounded E-martingale. Moreover, the decomposition, is unique (see Lemma 2.5, [7]).The following lemma is the key ingredient of our proof of Theorem 7.
Lemma 5. Let be a separable symmetric space on with . Then, with equivalent norms.
Proof. Let and . Let and be the decomposition of and as in (14). Then, is a bounded -martingale and is a bounded -martingale. Thus, there exist and such that
Now, we define a linear functional on by
Then, by Hölder’s inequality,
Thus, is continuous on and .
We pass to the converse inclusion. Let . Let be the restriction of on . Noting that , there exists an operator and such that
On the other hand, let be the space of all sequences such that is a predictable -quasimartingale with equipped with the norm . Define a functional on by
Then, by the inequality
we have is a continuous linear functional on and . Note that is isometric to the subspace of . By the Hahn-Banach theorem, extends to a functional on . Since , the representation theorem allows us to find a sequence such that
and
Set and . For any , noting that is predicable, it follows from (22) that
It is easy to see that is predictable with and
Set , where . Then, and
For any , let be its decomposition as in (14).
Noting that is a bounded E(M) martingale and , it follows from (19) and (24) that
The proof is completed.
The following lemma is about the duality theorem of interpolation spaces.
Lemma 6 (see [2]). Let be separable and be a couple of Banach spaces such that is dense in both and . Then,
Proof of Theorem 7. Let and be a decomposition of x, where Let be the decomposition of as in (14). Then, is a bounded -martingale, and is a bounded -martingale. Thus, there exist and such that Using Definition 3, we get that Set .Then, by the definition of -functionals and (30), Thus, taking infimum over all decomposition of , we obtain Therefore, using the equality , we have It follows from the equality that For any , we obtain which implies that Similarly, we have where and denote the conjugate index of and . By Lemma 5 and Lemma 6, we obtain that Thus, The proof is completed.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11801489, 11671308).