# An Interpolation Theorem for Quasimartingales in Noncommutative Symmetric Spaces

Authors: Congbian MA; Guoxi Zhao
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

LetE be a separable symmetric space on 0,∞ and EM the corresponding noncommutative space. In this paper, we introduce a kind of quasimartingale spaces which is like but bigger than EM and obtain the following interpolation result: let E^M be the space of all bounded EM-quasimartingales and 1<p<pE<qE<q<∞. Then, there exists a symmetric space F on 0,∞ with nontrivial Boyd indices such that E^M=L^pM,L^qMF,K.


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## Body

## 1. Introduction

LetE be a symmetric space on 0,∞ with the Fatou property and 1<p<pE<qE<q<∞. Kalton and Montgomery-Smith [1] proved that there exists a symmetric space F with nontrivial Boyd indices such that E=Lp0,∞,Lq0,∞F,K. As is now well-known, the preceding interpolation result can automatically lift to the noncommutative setting (see [[2], Theorem 3.4]): let E be a symmetric space on 0,∞ with the Fatou property and M,τ a semifinite von Neumann algebra. Then, the following are equivalent:
(i)
1<p<pE<qE<q<∞.(ii)
There exists a symmetric spaceF on 0,∞ with nontrivial Boyd indices such that
(1)EM=LpM,LqMF,K.In this paper, we replace the spaceEM in (2) with a bigger and more complex space E^M (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.
LetE be a separable symmetric space on 0,∞ with 1<p<pE<qE<q<∞. Then, there exists a symmetric Banach function space F on 0,∞ SSS with nontrivial Boyd indices such that
(2)E^M=L^pM,L^qMF,K.


## 2. Preliminaries

### 2.1. Noncommutative Spaces

LetM be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ. We denote by L0M the family of all τ-measurable operators. Note that χλ,∞x is the spectral projection of x∈L0M associated with the interval λ,∞. For x∈L0M, define its generalized singular number by
(3)μtx=infλ>0:τχλ,∞x≤t,t>0.Note that the functiont↦μtx from 0,∞ into 0,∞ is right continuous and nonincreasing. For the case that M is the abelian von Neumann algebra L0,∞ with the trace given by integration with respect to the Lebesgue measure, L0M is the space of all measurable functions, and μf is the decreasing rearrangement of the measurable function f (see [3, 4]).Recall that a Banach function spaceE,⋅E on 0,∞ is called symmetric if for any g∈E and any measurable function f with μf≤μg, we have f∈E and fE≤gE. The Köthe dual of E is the function space defined by setting
(4)E×=f∈L00,∞:∫0∞ftgtdt<∞:∀g∈E.When equipped with the normfE×≔sup∫0∞ftgtdt:gE≤1,E× is a symmetric Banach function space. For any s>0, we define the dilation operator Ds on E by
(5)Dsft=ft/s,t>0,f∈E.Define the lower and upper Boyd indices ofE by
(6)pE≔lims→∞logslogDsandqE≔lims→0+logslogDs,respectively. It is well-known that 1≤pE≤qE≤∞, and we shall say that E has nontrivial Boyd indices, whenever 1<pE≤qE<∞. We refer to [1, 5] for unexplained terminology from function space theory.For a given symmetric Banach function spaceE,⋅E on 0,∞, we define the corresponding noncommutative space by setting
(7)EM,τ=x∈L0M:µtx∈E,equipped with the norm
(8)xEM,τ≔µtxE.It is well-known thatEM,τ is a Banach space and is referred to as the noncommutative symmetric space associated with M,τ corresponding to the function space E,⋅E. Note that if 1≤p<∞ and E=Lp0,∞, then EM,τ=LpM,τ is the usual noncommutative Lp-space associated with M,τ.Recall thatl∞ is a von Neumann algebra equipped with the trace: γa=∑an,a=an∈l∞+ (see [6]). Now, let N=M⊗¯l∞ be the von Neumann algebra tensor product and ν=τ⊗γ the tensor trace. This gives rise to noncommutative spaces EM⊗¯l∞. Note that LpM⊗¯l∞ coincides with the space lpLpM.


### 2.2. Noncommutative Martingales

A noncommutative probability space is a coupleM,τ, where M is a finite von Neumann algebra and τ is a normal faithful trace with τ1=1. Let Mnn≥1 be an increasing sequence of von Neumann subalgebras of M such that the union of the Mn′s is weak∗dense in M. Let εn be the conditional expectation with respect to Mn.Definition 2.
A sequencex=xnn≥1 in L1M is called a sequence of martingale differences if xn∈EMn for n≥1 and if εnxn+1=0 for all n≥0.In this paper, we always consider noncommutative martingales associated with a noncommutative probability space unless explicit explanation.


### 2.3. Interpolation

LetX0,X1 be a compatible couple of quasi-Banach space. Its K-functional is defined by
(9)Ktx;X0,X1=infx0XO+tx1x1:x=x0+x1,x0∈X0,x1∈X1,for x∈X0+X1 and t>0. Let E be a symmetric Banach space on 0,∞. Set
(10)xX0,X1E,K=Ktx;X0,X1tE.Then, the interpolation spaceX0,X1E,K is defined as X0,X1E,K=x∈X0+X1:xX0,X1E,K<∞ equipped with the norm ⋅X0,X1E,K.


## 2.1. Noncommutative Spaces

LetM be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ. We denote by L0M the family of all τ-measurable operators. Note that χλ,∞x is the spectral projection of x∈L0M associated with the interval λ,∞. For x∈L0M, define its generalized singular number by
(3)μtx=infλ>0:τχλ,∞x≤t,t>0.Note that the functiont↦μtx from 0,∞ into 0,∞ is right continuous and nonincreasing. For the case that M is the abelian von Neumann algebra L0,∞ with the trace given by integration with respect to the Lebesgue measure, L0M is the space of all measurable functions, and μf is the decreasing rearrangement of the measurable function f (see [3, 4]).Recall that a Banach function spaceE,⋅E on 0,∞ is called symmetric if for any g∈E and any measurable function f with μf≤μg, we have f∈E and fE≤gE. The Köthe dual of E is the function space defined by setting
(4)E×=f∈L00,∞:∫0∞ftgtdt<∞:∀g∈E.When equipped with the normfE×≔sup∫0∞ftgtdt:gE≤1,E× is a symmetric Banach function space. For any s>0, we define the dilation operator Ds on E by
(5)Dsft=ft/s,t>0,f∈E.Define the lower and upper Boyd indices ofE by
(6)pE≔lims→∞logslogDsandqE≔lims→0+logslogDs,respectively. It is well-known that 1≤pE≤qE≤∞, and we shall say that E has nontrivial Boyd indices, whenever 1<pE≤qE<∞. We refer to [1, 5] for unexplained terminology from function space theory.For a given symmetric Banach function spaceE,⋅E on 0,∞, we define the corresponding noncommutative space by setting
(7)EM,τ=x∈L0M:µtx∈E,equipped with the norm
(8)xEM,τ≔µtxE.It is well-known thatEM,τ is a Banach space and is referred to as the noncommutative symmetric space associated with M,τ corresponding to the function space E,⋅E. Note that if 1≤p<∞ and E=Lp0,∞, then EM,τ=LpM,τ is the usual noncommutative Lp-space associated with M,τ.Recall thatl∞ is a von Neumann algebra equipped with the trace: γa=∑an,a=an∈l∞+ (see [6]). Now, let N=M⊗¯l∞ be the von Neumann algebra tensor product and ν=τ⊗γ the tensor trace. This gives rise to noncommutative spaces EM⊗¯l∞. Note that LpM⊗¯l∞ coincides with the space lpLpM.


## 2.2. Noncommutative Martingales

A noncommutative probability space is a coupleM,τ, where M is a finite von Neumann algebra and τ is a normal faithful trace with τ1=1. Let Mnn≥1 be an increasing sequence of von Neumann subalgebras of M such that the union of the Mn′s is weak∗dense in M. Let εn be the conditional expectation with respect to Mn.Definition 2.
A sequencex=xnn≥1 in L1M is called a sequence of martingale differences if xn∈EMn for n≥1 and if εnxn+1=0 for all n≥0.In this paper, we always consider noncommutative martingales associated with a noncommutative probability space unless explicit explanation.


## 2.3. Interpolation

LetX0,X1 be a compatible couple of quasi-Banach space. Its K-functional is defined by
(9)Ktx;X0,X1=infx0XO+tx1x1:x=x0+x1,x0∈X0,x1∈X1,for x∈X0+X1 and t>0. Let E be a symmetric Banach space on 0,∞. Set
(10)xX0,X1E,K=Ktx;X0,X1tE.Then, the interpolation spaceX0,X1E,K is defined as X0,X1E,K=x∈X0+X1:xX0,X1E,K<∞ equipped with the norm ⋅X0,X1E,K.


## 3. Main Result

The main result in this section is Theorem1, which extends the result of Kalton and Montgomery-Smith [1] to a EM-quasimartingale spaces. We first introduce the quasimartingale spaces.Definition 3.
LetE be a symmetric Banach function space on 0,∞. A sequence x=xnn≥1 is called a EM-quasimartingale with respect to Mnn≥1 if xn∈EMn for n≥1 and (with ε0=0,x0=0)
(11)∑n=1∞εn−1dxn⊗enEM⊗¯l∞<∞.
We set(12)xE^M≔supnynEM+∑n=1∞εn−1dxn⊗enEM⊗¯l∞,where γn=∑k=1ndxk−εk−1dxk and enn≥1 denote the standard basic sequence of l∞.IfxE^M<∞, x is called a bounded EM-quasimartingale. The quasimartingale space E^M is defined as the space of all bounded E^M-quasimartingales, equipped with the norm ⋅E^M.Remark 4.
(i)
In the caseE=Lp for 1≤p<∞, we have that E^M=Lp^M and
(13)xLp^M≔supnynLpM+∑n=1∞εn−1dxnLpMp1/p(ii)
Letx=xnn≥1 be a bounded EM-quasimartingale. Set zn=∑k=1nEk−1dxk and yn=xn−znn≥1. Then, z=znn≥1 is a predicable EM-quasimartingale with z1=0, and y=ynn≥1 is a bounded EM-martingale. Moreover, the decomposition,
(14)x=y+z,is unique (see Lemma 2.5, [7]).
The following lemma is the key ingredient of our proof of Theorem7.Lemma 5.
LetE be a separable symmetric space on 0,∞ with 1<pE≤qE<∞. Then,
(15)E∧M∗=E∧×M,with equivalent norms.Proof.
Letμ=μnn≥1∈E∧×M and x=xnn≥1∈E^M. Let μn=νn+ωn and xn=yn+znn≥1 be the decomposition of μ and x as in (14). Then, y=ynn≥1 is a bounded EM-martingale and ν=νnn≥1 is a bounded E^M-martingale. Thus, there exist y∞∈EM and ν∞∈E×M such that
(16)yn→EMy∞,νn→E×Mν∞.
Now, we define a linear functional onE^M by
(17)lμx=τν∞y∞+∑n=1∞τdωndzn.
Then, by Hölder’s inequality,(18)lμx≤ν∞E×My∞EM+τ⊗γ∑n=1∞dωn⊗en∑n=1∞dzn⊗en≤ν∞E×My∞EM+∑n=1∞dωn⊗enEM⊗¯l∞∑n=1∞dzn⊗enE×M⊗¯l∞≤μE∧×MxE^M.
Thus,lμx is continuous on E^M and lμ≤μE∧×M.
We pass to the converse inclusion. Letl∈E∧M∗. Let l1 be the restriction of l on EM. Noting that EM∗=E×M, there exists an operator ν∈E×M and νE×M≤l such that
(19)l1a=τaν,a∈EM.
On the other hand, letFEM be the space of all sequences db=dbnn≥1 such that b=bnn≥1 is a predictable EM-quasimartingale with b1=0 equipped with the norm dbFEM=∑n=1∞dbn⊗enEM⊗¯l∞. Define a functional on FEM by
(20)l2db=lb,db=dbnn≥1∈FEM.
Then, by the inequality(21)l2db≤lbE^M=ldbFEM,we have l2 is a continuous linear functional on FEM and l2≤l. Note that FEM is isometric to the subspace of EM⊗¯l∞. By the Hahn-Banach theorem, l2 extends to a functional on EM⊗¯l∞. Since EM⊗¯l∞∗=E×M⊗¯l∞, the representation theorem allows us to find a sequence Wn′n≥1∈E×M⊗¯l∞ such that
(22)l2s=∑n=1∞vwn′⊗ensn⊗en=∑n=1∞τwn′sn,snn≥1∈EM⊗¯l∞,and
(23)∑n=1∞wn′⊗enE×M⊗¯l∞≤l2.
Setω1=0 and wn=∑k=1nεk−1wk′n≥2. For any db=dbnn≥1∈FEM, noting that db=dbnn≥1 is predicable, it follows from (22) that
(24)l2db=∑n=1∞τεn−1wn′dbn=∑n=1∞τdbnεn−1wn′=∑n=1∞τdwndbn.
It is easy to see thatω=ωnn≥1 is predictable with ω1=0 and
(25)∑n=1∞dwn⊗enE×M⊗¯l∞=∑n=1∞εn−1wn′⊗enE×M⊗¯l∞≤∑n=1∞wn′⊗enE×M⊗¯l∞≤l2.
Setμn=νn+ωnn≥1, where νn=ℇnνn≥1. Then, μ=μnn≥1∈E∧×M and
(26)μE∧×M=vE×M+∑n=1∞dwn⊗enE×M⊗¯l∞≤2l.
For anyx=xnn≥1∈E^M, let xn=yn+znn≥1 be its decomposition as in (14).Noting thaty=ynn≥1 is a bounded E(M) martingale and dz=dznn≥1∈FEM, it follows from (19) and (24) that
(27)lx=ly+lz=τy∞v∞+∑n=1∞τdwndzn.The proof is completed.The following lemma is about the duality theorem of interpolation spaces.Lemma 6 (see [2]).
LetE be separable and X1,X2 be a couple of Banach spaces such that X1∩X2 is dense in both X1 and X2. Then,
(28)X1,X2E,K∗=X2∗,X1∗E×,K.Proof of Theorem 7.
Letx∈L^pM,L^qMF,K and x=x0+x1 be a decomposition of x, where x0∈LpM,x1∈LqM. Let xnk=ynk+znkn≥1 be the decomposition of xkk=0,1 as in (14). Then, y0 is a bounded LpM-martingale, and y1 is a bounded LqM-martingale. Thus, there exist y∞0∈LpM and y∞1∈LqM such that
(29)yn0LpM→y∞0,yn1LpM→y∞1,y∞0LpM=supnyn0LpM,y∞1LpM=supnyn1LpM.
Using Definition3, we get that
(30)x0L^pM=y∞0LpM+dz0LpM⊗¯l∞,x1L^pM=y∞1LqM+dz1LqM⊗¯l∞.
Sety=y∞0+y∞1,z=z0+z1.Then, by the definition of K-functionals and (30),
(31)Kty;LpM,LqM+Ktdz;LpM⊗¯l∞,LqM⊗¯l∞≤y∞0LpM+ty∞1LqM+dz0LpM⊗¯l∞+tdz1LqM⊗¯l∞=x0L^pM+tx1L^qM.
Thus, taking infimum over all decomposition ofx, we obtain
(32)Kty;LpM,LqM+Ktdz;LpM⊗¯l∞,LqM⊗¯l∞≤Ktx;L^pM,L^qM.
Therefore, using the equalityxX0,X1F,K=Ktx;X0,X1/tF, we have
(33)yLpM,LqMF,K+dzLpM⊗¯l∞,LqM⊗¯l∞F,K≤2xL^pM,L^qMF,K.
It follows from the equalityEM=LpM,LqMF,K that
(34)xE^M=yEM+dzEM⊗¯l∞≤2xL^pM,L^qMF,K.
For anyx∈L^pM,L^qMF,K, we obtain x∈E^M which implies that
(35)E^M⊃L^pM,L^qMF,K.
Similarly, we have(36)E∧×M⊃L^q′M,L^p′MF×,K,where p′ and q′ denote the conjugate index of p and q. By Lemma 5 and Lemma 6, we obtain that
(37)E^M=E∧×M∗⊂L^q′M,L^p′MF×,K∗=L^pM,L^qMF,K.
Thus,(38)E^M=L^pM,L^qMF,K.
The proof is completed.



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