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# Complex interpolation of noncommutative Hardy spaces associated semifinite von Neumann algebra

Turdebek N. Bekjan College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China.  and  Kordan N. Ospanov Faculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, Astana 010008, Kazakhstan. ospanov
###### Abstract.

We studied complex interpolation noncommutative Hardy space associated with semi-finite von Neumann algebra and extend Pisier’s interpolation theorem for this case.

###### Key words and phrases:
subdiagonal algebra, complex interpolation, noncommutative Hardy space, semifinite von Neumann algebra
###### 2010 Mathematics Subject Classification:
Primary 46L52; Secondary 47L05.

## 1. Introduction

In , Pisier give a new proof of the interpolation theorem of Peter Jones (see  or , p.414). He obtained the complex case of Peter Jones’ theorem as a consequence of the real case. The Pisier’s method does to extend to the noncommutative case and the case of Banach space valued -spaces (see and in ). In Pisier and Xu[PX1] obtained noncommutative version of P. Jones’ theorem for noncommutative Hardy spaces associated with a finite subdiagonal algebra in Arveson’s sense  (It is stated in  without proof, see the remark following Lemma 8.5 there). The first named author , using Pisier’s method, proved that the real case of Peter Jones’ theorem for noncommutative Hardy spaces associated semifinite von Neumann algebra holds (also see ).

This paper is devoted to the study of complex interpolation of noncommutative Hardy spaces associated semifinite von Neumann algebra. Using the Pisier method in , we proved the complex case of Peter Jones’ theorem for noncommutative Hardy spaces associated semifinite von Neumann algebra.

## 2. Preliminaries

We use standard notation and notions from noncommutative -spaces theory (see e.g. [13, 16]).Throughout this paper, we denote by a semifinite von Neumann algebra on the Hilbert space with a normal faithful semifinite trace . A closed densely defined linear operator in with domain is said to be affiliated with if and only if for all unitary operators which belong to the commutant of . If is affiliated with , then is said to be -measurable if for every there exists a projection such that and (where for any projection we let ). The set of all -measurable operators will be denoted by . The set is a -algebra with sum and product being the respective closure of the algebraic sum and product. For a positive self-adjoint operator (the spectral decomposition) affiliated with , we set

 τ(x)=supnτ(∫n0λdeλ)=∫∞0λτ(eλ).

For is defined as the set of all -measurable operators affiliated with such that

 ∥x∥p=τ(|x|p)1p<∞.

In addition, we put and denote by , the usual operator norm. It is well-known that is a Banach space under satisfying all the expected properties such as duality.

In this paper, denotes the closed linear span of in (relative to the w*-topology for ) and is the family of the adjoints of the elements of .

Henceforth we will assume that is a von Neumann subalgebra of such that the restriction of to is still semifinite. Let be the (unique) normal positive faithful conditional expectation of with respect to such that .

###### Definition 2.1.

A w*-closed subalgebra of is called a subdiagonal algebra of with respect to (or ) if

1. is w*-dense in ,

is then called the diagonal of .

It is proved by Ji  that a semifinite subdiagonal algebra is automatically maximal, i.e., is not properly contained in any other subalgebra of which is subdiagonal algebra respect to .

Since is semifinite, we can choose an increasing family of of -finite projections in such that strongly, where 1 is identity of (see Theorem 2.5.6 in ). Throughout, the will be used to indicate this net.

Let . We define the distribution function of by

 λt(x)=τ(e(t,∞)(|x|)),t≥0,

where is the spectral projection of corresponding to the interval . Define

 μt(x)=inf{s>0:λs(x)≤t},t>0.

The function is called the generalized singular numbers of . For more details on generalized singular value function of measurable operators, see .

For , , the noncommutative Lorentz space is defined as the space of all measurable operators such that

Equipped with , is a quasi-Banach space. Moreover, if and , then can be renormed into a Banach space. More precisely,

 x↦(∫∞0[t−1+1/p∫t0μs(x)ds]qdtt)1/q

gives an equivalent norm on . It is clear that (for details see [7, 18] ).

The space is usually called a weak -space, . Its quasi-norm admits the following useful description in terms of the distribution function:

 ∥∥x∥∥p,∞=sups>0sλs(x)1/p

(see Proposition 3.1 in ).

For , we define the noncommutative Hardy-Lorentz space

 Hp,q(A)=closure ofA∩Lp,q(M)in Lp,q(M).
 Hp,q0(A)=closure ofA0∩Lp,q(M)in Lp,q(M).

## 3. Complex interpolation

Let (respectively, ) denote the open strip (respectively, the closed strip ) in the complex plane. Let be a compatible couple of complex Banach spaces. Let be the family of all functions satisfying the following conditions:

1. is continuous on and analytic in ;

2. for and the function is continuous from to , ;

3. , .

We equip with the norm:

 ∥∥f∥∥F(X0,X1)=max{supt∈R∥∥f(it)∥∥X0,supt∈R∥∥f(1+it)∥∥X1}.

Then becomes a Banach space. Let . The complex interpolation space is defined as the space of all those for which there exists such that . Equipped with

 ∥x∥θ=inf{∥∥f∥∥F(X0,X1):f(θ)=x,f∈F(X0,X1)},

becomes a Banach space.

Let be the algebra of infinite diagonal matrices with entries from and be the algebra of infinite diagonal matrices with entries from , i.e.

 N=⎧⎪ ⎪⎨⎪ ⎪⎩x:x=⎛⎜ ⎜⎝x10…0x2…⋮⋮⋱⎞⎟ ⎟⎠,xi∈M,i∈N⎫⎪ ⎪⎬⎪ ⎪⎭

and

 B=⎧⎪ ⎪⎨⎪ ⎪⎩a:a=⎛⎜ ⎜⎝a10…0a2…⋮⋮⋱⎞⎟ ⎟⎠,ai∈A,i∈N⎫⎪ ⎪⎬⎪ ⎪⎭.

For with entries , define to be the diagonal matrix with entries and

 ν(x)=∑i≥1τ(xi).

Then is a simifinite von Neaumann algebra and is a simifinite subdiagonal algebra of respect to .

Using same method as in the proof of Lemma 4.1 of , we obtain that

###### Lemma 3.1.

Let . If , then there is a bounded nature inclusion

 (L1,∞(N)/J(H1,∞0(B)),N/J(B0))1−θ,∞=Hp,∞(B) (3.1)

with equivalent norms.

###### Proof.

Let . Set . It is clear that . Using Theorem 4.1 in  and Theorem 6.3 in , we know that

 (L1(N),N)η,r=Lr(N),(L1(N),N)γ,q=Lq(N)

and

 (H1(B),B)η,r=Hr(B),(H1(B),B)γ,q=Hq(B).

On the other hand, (i) and (iii) of Lemma 6.5 in , we have that is -closed with respect to and is -closed with respect to . Hence by Theorem 1.2 in , we obtain is -closed with respect to . Since on can extend to a contractive projection from onto for every , we deduce that is -closed with respect to . By Holmstedt’s formula (see , p. 52-53), we get is -closed with respect to . Therefore,

 (J(H1,∞0(B)),J(B0))1−θ,∞=J(Hq,∞0(B)).

Since

 (L1,∞(N)/J(H1,∞0(B)),N/J(B0))

is a compatible couple (see , p. 351-352), the space

 (L1,∞(N)/J(H1,∞0(B)),N/J(B0))1−θ,∞

can be identified with the space

 Lp,∞(N)/J(Hp,∞0(B)).

By Theorem 4.1 in  and Theorem 6.3 in , for and with , we have that

 (Lp0(N),Lp1(N))ϑ,∞=Lp,∞(N),(Hp0(B),Hp1(B))ϑ,∞=Hp,∞(B).

Applying Theorem 4.2 in  and interpolation, we deduce that

 Lp,∞(N)/J(Hp,∞0(B))

can be identified with . Hence, (3.1) holds. ∎

For , we define by

 Kp(x)=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝x00…02−1px0…003−1px…⋮⋮⋮⋱⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,∀x∈Lp(M).

Then

 ∥x∥p=∥Kp(x)∥p,∞. (3.2)

Indeed, it is clear that for any ,

 ∥Kp(x)∥∞=∥x∥∞.

If and , then

 ∥x∥pp=τ(|x|p)=∫+∞0spdτ(es(|x|))=supt>0∑j≥1jtpτ(e(j1pt,(j+1)1pt](|x|))=supt>0∑j≥1tpτ(e(j1pt,∞)(|x|))=supt>0∑j≥1tpτ(e(t,∞)(|j−1px|))=supt>0tpν(e(t,∞)(Kp(x)))=supt>0tpλt(Kp(x))=∥Kp(x)∥pp,∞.

Hence, has norm 1. Note that maps into . Let

 ˜Kp:Lp(M)/Hp0(A)→Lp,∞(N)/Hp,∞0(B)

be the mapping canonically associated to . We have following result (see Lemma 4.2 in ).

###### Lemma 3.2.

Let and . Then defines a contraction from

 (L1(M)/J(H10(A)),M/J(A0))1−θ

into .

###### Proof.

Let

 X=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝x00…02−1px0…003−1px…⋮⋮⋮⋱⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠:∀x∈L1(M)⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭.

We equip with the norm

 ∥∥ ∥ ∥ ∥ ∥ ∥∥⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝x00…02−1px0…003−1px…⋮⋮⋮⋱⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠∥∥ ∥ ∥ ∥ ∥ ∥∥X=∥x∥1.

Then becomes a Banach space. It is clear that and this inclusion has norm one. Let

 X0=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝x00…02−1px0…003−1px…⋮⋮⋮⋱⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠:∀x∈J(H10(M))⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭.

Then we have that

 X/X0⊂L1,∞(N)/J(H1,∞0(B)) (3.3)

with norm one.

For any , let

 Kz(x)=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝x00…02z−1x0…003z−1x…⋮⋮⋮⋱⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,∀x∈L1(M)∩M.

Then is an analytic families of linear operators on

 L1(M)/J(H10(A))∩M/J(A0)

into

 X/X0+N/J(B0).

From (3.2), it follows that if , is a contraction from

 L1(M)/J(H10(A))

into and if , it is a contraction from into . Hence, by Stein’s interpolation theorem for analytic families of operators (see Theorem 1 in ), we obtain that is a contraction from

 (L1(M)/J(H10(A)),M/J(A0))1−1p

into

 (X/X0,N/J(B0))1−1p.

By Theorem 4.7.1 in , is a contraction from

 (L1(M)/J(H10(A)),M/J(A0))1−1p

into

 (X/X0,N/J(B0))1−1p,∞.

By (3.3), is a contraction from

 (L1(M)/J(H10(A)),M/J(A0))1−1p

into

 (L1,∞(N)/J(H1,∞0(B)),N/J(B0))1−1p,∞.

Using Lemma 3.1, we obtain the desired result.

###### Theorem 3.3.

Let and . Then

 Hp(A)=(H1(A),A)θ. (3.4)
###### Proof.

Let be the conjugate of , so that . The inclusions

 (H1(A),A)θ⊂(L1(M),M)θ=Lp(M)

and

 (H1(A),A)θ⊂{x∈Lp(M):x⊥J(Hq0(A))}

imply . To prove the converse we dualize. Hence we have to prove that

 (L1(M)/J(H10(A)),M/J(A0))1−1q⊂Hq(A). (3.5)

Let . By (3.2) and Lemma 3.2, we have that

 \par∥x∥q=∥∥ ∥ ∥ ∥ ∥ ∥∥⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝x00…02−1qx0…003−1qx…⋮⋮⋮⋱⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠∥∥ ∥ ∥ ∥ ∥ ∥∥q,∞≤∥x∥(L1(M)/J(H10(A)),M/J(A0))1−1q.

Hence, we get (3.5). ∎

Let be the space defined in Definition 5.1 in . By Theorem 5.1 in , we have that . We use the reiteration theorem (Theorem 4.6.1 in ) and Wolff’s theorem (Theorem 2 in ) to obtain that

###### Corollary 3.4.

Let and . Then

 Hp(A)=(H1(A),BMO(M))θ. (3.6)

### Acknowledgement

T.N. Bekjan is partially supported by NSFC grant
No.11771372.

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