Some Results on Noncommutative Hardy-Lorentz Spaces

Han and Shao Journal of Inequalities and Applications (2015)2015:120 DOI 10.1186/s13660-015-0642-3

R E S E A R C H Open Access Some results on noncommutative Hardy-Lorentz spaces

Yazhou Han* and Jingjing Shao

*Correspondence: [email protected] Abstract College of Mathematics and A M Systems Science, Xinjiang Let be a maximal subdiagonal algebra of a ﬁnite von Neumann algebra .For University, Urumqi, 830046, China 0

1 Introduction The concept of maximal subdiagonal algebras A, which appeared earlier in Arveson’s paper [], unifies analytic function spaces and nonselfadjoint operator algebras. In fact, subdiagonal algebras are the noncommutative analogue of weak* Dirichlet algebras. In the case that M has a finite trace, Hp(A) may be defined to be the closure of A in the noncom-

mutative Lp space Lp(M). Subsequently, Arveson’s pioneering work has been extended to diﬀerent cases by several authors. For example, Marsalli and West []obtainedaseries of results including a Riesz factorization theorem, the dual relations between Hp(A)and Hq(A), and Labuschagne [] proved the universal validity of Szegö’s theorem for ﬁnite

subdiagonal algebras. Recently, the noncommutative Hp spaces have been developed by Blecher, Bekjan, Labuschagne, Xu and their coauthors in a series of papers. The noncommutative Hardy spaces have received a lot of attention since Arveson’s pioneering work. Most results on the classical Hardy spaces on the torus have been estab- lished in this noncommutative setting. Here we mention only two of them directly related with the objective of this paper. After the fundamental work of Arveson, Labuschagne [] proved the noncommutative Szegö type theorem for ﬁnite subdiagonal algebras, and Blecher and Labuschagne [] gave several useful variants of this theorem for Lp(M). In [], Bekjan and Xu presented the more general form of Szegö type factorization theorem: Let 

p M p A ⊕ p ⊕ p A L ( )=H ( ) L (D) JH ( ). (.)

This result is proved in []forp =,in[] for the general case as above.

© 2015 Han and Shao; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 2 of 19

In this article we introduce the noncommutative Hardy-Lorentz spaces Hp,ω(A). If ω ≡ , the noncommutative Hardy-Lorentz spaces Hp,ω(A) correspond to the noncommutative Hardy spaces Hp(A). By adapting the ideas and techniques in [, , ], we es- tablish the Riesz and Szegö factorizations on these spaces. We also present some results of inner-outer type factorization and Jordan morphism according to noncommutative Hardy-Lorentz spaces. The remainder of this article is organized as follows. After a short introduction to this article, Section  consists of some preliminaries and notations, including the noncommutative weighted Lorentz spaces and their elementary properties. Section  presents the Riesz and Szegö factorization of noncommutative Hardy-Lorentz spaces. Section  contains some results of outer operators according to noncommutative Hardy-Lorentz spaces. The last section is devoted to Jordan morphism on these spaces.

2Preliminaries Throughout this paper M will be a finite von Neumann algebra with a faithful normal tracial state τ . We refer to [, ] for the theory of noncommutative integration. Let M be a von Neumann algebra acting on a Hilbert space H.WedenotebyP(M)thecomplete lattice of all (self-adjoint) projections in M. The closed densely defined linear operator x in H with domain D(x)issaidtobeaffiliatedwithM if and only if yx ⊆ xy for all y ∈ M. When x is affiliated with M, x is said to be τ -measurable if for every ε >thereexists e ∈ P(M)suchthate(H) ⊆ D(x)andτ(e⊥)<ε (where for any projection e,welete⊥ =

–e). The set of all τ -measurable operators will be denoted by L(M). The set L(M) is an ∗-algebra with sum and product being the respective closure of the algebraic sum

and product. The topology of L(M) is determined by the convergence in measure. For 

 p p p L (M)= x ∈ L(M : τ |x| < ∞ .

We deﬁne

 p p p x p = τ |x| , x ∈ L (M).

p Then (L (M); · p) is a Banach (or quasi-Banach for p <)space.Asusual,weput ∞ L (M; τ)=M and denote by · ∞ (= · ) the usual operator norm.

For x ∈ L(M), we deﬁne λt(x)=τ e(t,∞) |x| and μt(x)=inf s >:λs(x) ≤ t ,

where e(t,∞)(|x|) is the spectral projection of |x| associated with the interval (t, ∞). The

function t → λt(x) is called the distribution function of x and t → μt(x)isthegeneralized

singular number of x.Wewilldenotesimplybyλ(x)andμ(x) the functions t → λt(x)and

t → μt(x),respectively.Itiseasytocheckthatbotharedecreasingandcontinuousfrom the right on (, ∞)(cf. []). It will be sometimes convenient to write L = L(I)forbrevity,whereI = [, ]. When ω is a nonnegative, integrable function on [, ] and not identically zero, we say that ω is a Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 3 of 19

t ∞ ≤ ≤ weight. For a given weight ω,wewriteW(t)=  ω(s) ds < , t . We also agree that ‘decreasing’ or ‘increasing’ will mean ‘nonincreasing’ or ‘nondecreasing’,respectively.

Let L be the set of all Lebesgue measurable functions on I.Forf ∈ L,wedeﬁneits nonincreasing rearrangement as ∗ f (t)=inf s >:df (s)=m r : f (r) > s ≤ t , t >,

p where m denotes the Lebesgue measure on I.TheLorentzspaceω,

space of L such that

  p ∗ p f p = f (t) ω(t) dt < ∞.(.) ω 

It is clear that

∞  p p– f p = pt W d (t) dt . ω f 

Let  < p < ∞,wedeﬁne

  p p ∗∗ p = f ∈ L : f p = f (t) ω(t) dt < ∞ , ω  ω  ∗∗  t ∗ t ∈ where f (t)= t  f (s) ds.Letω be a weight in I,wewriteW(t)=  ω(s) ds  if there ≤  ∞ exists some constant C such that W(t) CW(t),  < t <  . For any  < p < ,itiswell p known that ω is a quasi-Banach space if and only if W(t) ∈  (cf. []). Since we deal p further with the space ω which is at least a quasi-Banach space, we will assume in what

follows that W(t) ∈ . p Let ω be a quasi-Banach space. For  < s < ∞, we deﬁne the dilation operator Ds on p ω by

(Dsf )(t)=f (st)χ[,](ts), t ∈ [, ].

p Deﬁne the lower Boyd index α p of by ω ω

log s log s α p = lim = sup ω →∞ s log D  s> log D  s s

p and the upper Boyd index β p of by ω ω

log s log s β p = lim = inf . ω s→ log D  

Note that –  α p = sup p >:∃c >,∀

and –  β p = inf p >:∃c >,∀

p It is clear that  ≤ α p ≤ β p ≤∞and α rp = rα p , β rp = rβ p , r >.If is a Banach ω ω ω ω ω ω ω function space, then  ≤ α p ≤ β p ≤∞. For further results about the Boyd index of ω ω quasi-Banach spaces, the reader is referred to [, ].

We consider a couple (E, E) of topological vector spaces E and E, which are both

continuously embedded in a topological vector space E. The morphisms T :(E, E) → → (E, E)inE are all bounded linear mappings from E +E to E +E such that TE : E E → and TE : E E. A (quasi-)Banach space A is said to be an intermediate space between E and E if E is continuously embedded between E ∩ E and E + E.ThespaceE is called

an interpolation space between E and E if, in addition, T :(E, E) → (E, E)implies T : E → E. Further details may be found in [, ]. p p Let  < p < ∞,wesaythatω has order continuous norm if for every net (fi)inω such

that fi ↓ wehave fi p,ω ↓ . It follows from Proposition .. and Theorem .. of [] p that the norm on ω is order continuous. Then it follows from Theorem . of []that

r p q r ≤ r ≤∞ ω is an interpolation space for the couple (L , L ), where  < p < αω βω < q .Let ∞ p p 

suchthat t s ω(t) dt Bt W(t), t I. Wereferto[, ]forthesespaces. ∗ A (quasi-)Banach function space E is called symmetric if for f ∈ L and g ∈ E with f ≤ ∗ ∈ ≤ ∈ g ,wehavef E and f E g E. It is called fully symmetric if, in addition, for f L and ∈ t ∗ ≤ t ∗ ∈ ≤ ≤ ∞ g E with  f (s) ds  g (s) ds,wehavef E and f E g E.If p < and ω is a p non-increasing weighted function, it is clear that ω is a fully symmetric Banach function p space. Let  < p < ∞,thenω is a fully symmetric quasi-Banach function space.

Let x ∈ L(M)and

  p p x = x p = μ (x) ω(t) dt . (.) p,ω ω(M) t 

By a simple computation we derive

∞  p p– x p,ω = pt W λt(x) dt . 

p The noncommutative Lorentz space ω(M)isdeﬁnedasthespaceofallτ -measurable

operators affiliated with a finite von Neumann algebra M such that x p,ω < ∞.Ifω ≡ , p then the noncommutative Lorentz space ω(M) is the usual noncommutative Lp space p q – p q,p Lp(M). If ω = t , p, q >,thenω(M)=L (M). For further results about the noncommutative Lorentz spaces Lq,p(M), the reader is referred to []. If  < p < ∞,wedefine

  p p ∗∗ p (M)= x ∈ L (M): x p = x (t) ω(t) dt < ∞ , ω  ω(M)  ∗∗  t where x (t)= t  μs(x) ds. Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 5 of 19

p p Since ω and ω are quasi-Banach spaces, then the following result follows from The- orem  of [].

p p Proposition . Let 

We should introduce the Köthe dual spaces (associate spaces) generalizing the deﬁnition p that can be found in [] in the context of classical Lorentz space ω,

p If x ∈ ω(M) ,itisclearthat x p = sup xy = sup τ(xy) : y ≤  . ω(M)  p,ω y p,ω≤

Lemma . Let M have no minimal projection for every measurable function f with

lim df (t)=, t→∞

∗ then there exists x ∈ L(M) such that μt(x)=f (t).

–n Proof By Lemma . of [], there exist Qk–n ∈ Mproj such that τ(Qk–n )=k and

–n ≤ –n –n ≤ –n ∈ N –n Qk  Qk  , k k ,wheren, k, ni, ki , i =,.LetQλ = sup k ≤λ Qk , n λ ∈ [, ]. Then {Qλ}λ∈[,] is an increasing family of projections in M,anditisclearthat M ∈ τ(Qλ)=λ, Q =andQ = . Therefore, it is a spectral family of .Letf L with ∞ ∗ →∞ limt df (t)=.Forx =  f (λ) dQλ,wehave τ e(t,∞)(x) = τ dQλ = dλ = df ∗ (t)=df (t) → , t →∞, {λ≥:f ∗(λ)>t} {λ≥:f ∗(λ)>t}

∗ which implies x ∈ L(M)andμt(y)=f (t), t >.

p Lemma . Let 

p Proof For x ∈ ω(M) with x p M ≤ . Since μ (x) → , t →∞,then|x| admits ω( ) t | | ∞ the Schmidt decomposition x =  μt(x) det,whereet = eμt(x)–, t >,ande– = ∞ δ μ ≥ μ χ δ | | μ ≥ μ (cf. []). Given >,itisclearthat t(x) δ(x) [ ,δ).Thus x =  t(x) det δ(x)q, ∞  p where q = χ δ det.Sinceτ(q)<∞,weobtainq ∈ ω(M) . Therefore, x p M =  [  ,δ) ω( ) – |x| p M ≥ μδ(x) q p M and q p ≥ μδ(x), which complete the proof. ω( ) ω( ) ω(M)

p Proposition . Let 

Proof It is clear that · p is subadditive, homogenous and positive. If x p =, ω(M) ω(M) p p then xy  =foreveryy ∈ ω(M). Since eA(|x|) ∈ ω(M), we have xeA(|x|)=,where A is a subset of (, ∞), which implies that x =.TheproofofTheorem.of[]shows Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 6 of 19

that it is suﬃcient to prove the noncommutative form of the Riesz-Fischer theorem, i.e., we have an estimate ∞ ∞ x ≤ x p M , x ≥ , n =,,,..., n n ω( ) n p n= ω(M) n=

∞ ∞ whenever the right-hand side is ﬁnite. Let x p M < ∞,then x converges n= n ω( ) n= n M n { }∞ to some x in L( ). Indeed, set zn = k= xk,itisclearthat zn n= is a Cauchy sequence p M { }∞ M in ω( ) . It follows from Lemma . that zn n= converges to some x in L( ). Since ∞ ∞ x y p ≤ x p < ∞,then n= n ω(M) n= n ω(M) ∞ ∞ ∞ ∞ x y = x y ≤ x y ≤ x p M < ∞ n n n  n ω( ) n=  n=  n= n= p ∞ p holds for each y ∈ (M)with y p ≤ . Thus x ∈ (M) and ω ω(M) n= n ω ∞ ∞ x ≤ x p M < ∞. n n ω( ) p n= ω(M) n=

p Proposition . Let M have no minimal projection, then the associate space ω(M) is p a noncommutative Banach function space. For x ∈ ω(M) ,

 x p = sup μ (x)μ (y) dt : y p ≤  .(.) ω(M) t t ω(M) 

∈ p M ∈ p M ∈  M Proof Letx ω( ) and y ω( ), then xy L ( ). By Theorem . of [], we have | | ≤ ∞ τ( xy )  μt(x)μt(y) dt.Thus

 x p ≤ sup μ (x)μ (y) dt : y p ≤  . ω(M) t t ω(M) 

p To prove the other inequality, let y ∈ (M)with y p ≤ . By Proposition . of ω ω(M) [], we obtain

 μt(x)μt(y) dt = sup τ |x|y :y ∈ L(M),y ∼ y  ∗ ≤ sup τ |x|y = sup τ u xy y p ≤ y p ≤ ω(M) ω(M) ≤ sup τ |xy| = x p . ω(M) y p ≤ ω(M)

∞ Hence, x p ≥ sup{ μ (x)μ (y) dt : y p ≤ }. ω(M)  t t ω(M)

p Proposition . Let x ∈ ω(M) , then we have x p M = μt(x) p . ω( ) ω(I)

p p Moreover,(ω) (M)=ω(M) . Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 7 of 19

Proof This result follows immediately from Lemma . and Proposition ..First,letM p p have no minimal projection. For every f ∈ ω, by Lemma .,thereexistsy ∈ ω(M)such that μ(y)=f ∗(t), t > . Thus, by Proposition .,weobtain

 x p = sup μ (x)μ (y) dt : y p ≤  ω(M) t t ω(M) 

 ∗ = sup μ (x)f (t) dt : f p ≤  t ω  = μt(x) p . ω(I)

If M has minimal projections, we consider the von Neumann algebra tensor product M⊗¯ L∞([, ], m)denotedbyM, equipped with the tensor product trace τ ⊗ m,then M has no minimal projection. By the trivial fact μ(x)=μ(x ⊗ ) and argument of above, we have x p = μ (x) p , which implies that ω(M) t ω(I) p M ∈ M ∈ p p M ω( ) = x L( ):μt(x) ω = ω ( ).

p ∗ Deﬁnition . Let  < p ≤∞.Ifl ∈ ω(M) ,thenl is called normal if xα ↓ holdsin p ω(M)impliesl(xα) → .

If l ∈ M∗ then, by Theorem . of [], l is normal if and only if l is ultra-weak topology on M. Then a similar discussion of Lemma . of [] leads to the following lemma.

p ∗ Lemma . Let <α p ≤ β p < ∞,

 Proof Let e ∈ M and x > xα ↓ holdinM,thenexαe ↓ holdsinL (M), and so  μt(exαe) ↓ holdsinL .Thus,

    | |      ↓ μt(exα)=μt(xαe)=μt xαe = μt exαe = x μt(exαe) .   By Theorem .. of [], we have ex p =( μ (ex )pω(t)) p ↓ . Therefore, α ω(M)  t α le(xα) → . This tells us that le is a normal linear functional on M.

p ∗ Proposition . Let <α p ≤ β p < ∞,

p Proof It follows from Theorem . of []thatω is an interpolation space for the couple  ∞ ∞ p  p  (L , L ). Thus L → ω → L ,andsoM → ω(M) → L (M), where ‘ →’denotesa continuous embedding. It follows that there exists some constant C such that

– C x  ≤ x p,ω ≤ C x , x ∈ M

and W(t) ≤ C, t ∈ [, ]. We apply Lemma . and some techniques from the proof of Theorem . in [].Theresultfollowsinthesamewayasin[].

Proposition . Let <α p ≤ β p < ∞,

p M ∗ p M p M q M q –q (i) ω( ) = ω( ) =(ω) ( )=ω( ), where ω(t)=t W(t) ω(t), t >. (ii) If ei is an increasing sequence of projections in M converging strongly to , we have p limi→∞ xei – x p,ω =, limi→∞ eix – x p,ω =, ∀x ∈ ω(M). p ∗ (iii) ω(M) separates points.

p Proof (i): Combining Proposition . with Remark .. of [], we obtain ω(M) = p M q M ∈ p M ∗ ∈ p M (ω) ( )=ω( ). For l ω( ) ,byProposition.,thereexistsy ω( ) such that

l(x)=ly(x)=τ(xy), ∀x ∈ M.

s s p ∗ s p Let l = l – ly,wehavel ∈ ω(M) and l (x)=,x ∈ M.Forx ∈ ω(M), it is well known

that τ(e  ∞ (|x|)) < ∞, n =,,,....Letxn = xe  (|x|), n =,,...,thenx–xn ∈ M.This ( n , ) [, n ] s s s implies that |l (x)| = |l (xn)|≤ l xn p,ω, n =,,....Notethat|xn|≤|x| and μ(xn) → ,

n →∞. Then, by Theorem .. of [], we have xn p,ω = μ(xn) p,ω → , n →∞,and | s | | s | s so l (x) = l (xn) = . Therefore, l =,thatis,l = ly. (ii): Without loss of generality, we ≤ ∈ p M ∞ suppose  x ω( ). Let x =  λdeλ be the spectral decomposition of x.Wewrite  qi =–ei,thenqi converges strongly to  and qi > qi+, n =,,.... Set xi =(xqix)  , n = ∞  → ,,....Sinceτ() < ,thenxi  in the measure topology. Thus, by Theorem .. of []andμt(xn) ≤ μt(x), we have xn p,ω = μt(xn) p ω(I)    = μt x p ↓n . n ω(I)

 It follows that xqn p,ω = (xqnx)  p,ω = xn p,ω → , n →∞. (iii): Suppose that there p p ∗ exists  = x ∈ ω(M)suchthatl(x)=holdsforalll ∈ ω(M) .Then,forev- ∈ p M p M q M ery y ω( ) ,wehavely(x)=τ(xy) = . By (i), we have ω( ) = ω( ), then p ∞ | | ∈ M ∀ ∞ | | | | ∀ e(a, )( x ) ω( ) , a >.Thusτ(xe(a, )( x )) = le(a,∞)( x )(x)=, a >,whichim- p ∗ plies that x = , contradiction. Therefore, ω(M) separates points.

The identity in M is denoted by , and we denote by D a von Neumann subalgebra of M; moreover, we let E : M → D be the unique normal faithful conditional expectation such that τ ◦ E = τ . A ﬁnite subdiagonal algebra of M with respect to E (or D)isaw∗-closed subalgebra A of M satisfying the following conditions: (i) A + JA is w∗-dense in M; (ii) E is multiplicative on A, i.e., E(ab)=E(a)E(b) for all a, b ∈ A; (iii) A ∩ JA = D. D is then called the diagonal of A,whereJA = {x∗ : x ∈ A}.WesaythatA is a maximal subdiagonal algebra in M with respect to E in case that A is not properly contained in any other subalgebra of M which is subdiagonal with respect to E.ItisprovedbyExel(Theo- rem , []) that a ﬁnite subdiagonal algebra A is automatically maximal. This maximality yields the following useful characterization of A: A = x ∈ M : τ(xa)=,∀a ∈ A ,

where A = A ∩ ker E (see []). Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 9 of 19

p p If K is a subset of ω(M), [K]p,ω will denote the closure of K in ω(M)(withrespect to the w∗-topology in the case of p = ∞). Since M is a ﬁnite von Neumann algebra, then M ∞ M ⊆ p M ≤∞ = ω ( ) ω( ),  < p .

Definition . Let M be a finite von Neumann algebra, we define noncommutative p,ω A A p,ω A A weighted Hardy spaces by H ( )=[ ]p,ω and H ( )=[ ]p,ω.

In what follows, we will keep all previous notations throughout the paper, and C will al- ways denote a constant, which may be diﬀerent in diﬀerent places. Unless otherwise stated, p it will be assumed throughout that Lorentz spaces ω satisfy the following property: for ∈ ∈ p t ∗ ≤ t ∗ f L and g ω with  f (s) ds  g (s) ds,wehave

∈ p ≤ f ω and f p,ω C g p,ω.(.)

For two nonnegative (possibly inﬁnite) quantities A and B,byA B we mean that there exists a constant C >suchthatA ≤ CB.

Remark . ≡ p,ω A p A p,ω A p A (i) If ω ,thenH ( )=H ( ) and H ( )=H ( ).In[], Section  it is shown that for  ≤ p ≤∞, p p H (A)=[A]p = x ∈ L (M):τ(xy)=for all y ∈ A

and p A A ∈ p M ∈ A H ( )=[ ]p = x L ( ):τ(xy)=for all y .

Subsequently, Bekjan and Xu (Proposition ., []) showed that q A p A ∩ q M q A p A ∩ q M ≤∞ H ( )=H ( ) L ( ) and H ( )=H ( ) L ( ),where

3 Riesz and Szegö factorization of noncommutative weighted Hardy spaces Proposition . Let 

Moreover, if <α p ≤ β p < ∞,

r A ∩ p M p,ω A H ( ) ω( )=H ( ), r A ∩ p M p,ω A H( ) ω( )=H ( ).

Proof Since  < α p ≤ β p < ∞ and A is an ideal of A,then ω ω  A ⊆ ∈ p M ∀ ∈ A x ω( ):τ(xy)=, y  .

p,ω For x ∈ H (A), there exist xn ∈ A such that x – xn p,ω → , n →∞and τ(xny)=,∀y ∈

A, n =,,....Hence,xn → x in the measure topology. This means that xny → xy in the p  measure topology holds for every y ∈ A.ByRemark.(iii), we obtain ω(M) ⊆ L (M). Using Theorem . of [], we obtain lim τ(xny – xy) ≤ lim τ |xny – xy| lim xny – xy p,ω =, n→∞ n→∞ n→∞

which implies that τ(xy)=,andso p,ω A ⊆ ∈ p M ∀ ∈ A H ( ) x ω( ):τ(xy)=, y  .

p p,ω Conversely, let x ∈{x ∈ ω(M):τ(xy)=,∀y ∈ A} and x ∈/ H (A). By (i) and (iii) of p p,ω Proposition .,thereexistsy ∈ ω(M) such that τ(xy) =and τ(ya)=,∀a ∈ H (A). ∗∗ ≤ ∗∗ ∗∗ q   Since (f + g) f + g , ω is Banach function spaces over [, ], where q + p =.By q ⊆  q M ⊆  M Corollary II. . of [], we have ω L , which implies that ω( ) L ( ). Combining ∈ p M q M ⊆  M this with Proposition .,weobtainy ω( ) = ω( ) L ( ). Note that τ(ya)=, ∀ ∈ A ⊆ p,ω A ∈  A a H ( ). Then by Proposition . of []andRemark.(i), we have y H( ) and E(y)=.Set≤ s < α p ,thenx ∈{z ∈ L (M):τ(za)=,∀a ∈ A } = Hs(A). By Corol- ω s  lary.of[], we deduce τ(xy)=τ(E(xy)) = τ(E(x)E(y))=.Thisisacontradiction,and so the ﬁrst equality holds. A similar argument to the proof of above shows that p,ω A ⊆ ∈ p M ∀ ∈ A H ( ) x ω( ):τ(xy)=, y .

p On the other hand, let x ∈{x ∈ ω(M):τ(xy)=,∀y ∈ A}.SinceD ⊆ A,thenτ(xy)= p  τ(E(x)y)=,∀y ∈ D. It follows that E(x)=.Sincex ∈ ω(M) ⊆ L (A), it follows from

Corollary . of []thatτ(xy)=τ(E(xy)) = τ(E(x)E(y)) =  holds for all y ∈ A.Thisimplies p,ω that x ∈ H (A), and so there exist xn ∈ A, n =,,...,suchthat x – xn p,ω → , n →∞. p On the other hand, Remark .(v) shows that E is bounded on ω(M). It follows from the fact E(x)=that E E E → →∞ xn – (xn)–x p,ω xn – x p,ω + (xn)– (x) p,ω , n .

E ∈ A ∈ p,ω A Since xn – (xn) ,thenx H ( ), which implies the second equality holds. r p p,ω For the third equality, it is clear that H (A) ∩ ω(M) ⊇ H (A). Conversely, if r ≥  r p p r and x ∈ H (A) ∩ ω(M), then x ∈{z ∈ ω(M):τ(za)=,∀a ∈ A} since H (A)={z ∈ p p,ω Lr(A):τ(za)=,∀a ∈ A} and Lr(A) ⊇ ω(M). By the ﬁrst equality, we have x ∈ H (A). Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 11 of 19

p On the other hand, if r <andx ∈ Hr(A) ∩ (M), there exists  ≤ s < α p such that ω ω s M ⊆ M r A ∩ p M r A ∩ ω( ) Ls( ), and so, by Proposition . of [], we have H ( ) ω( )=H ( ) p s p p,ω Ls(M) ∩ ω(M)=H (A) ∩ ω(M)=H (A). Similarly, the fourth equality holds.

p Theorem . Let 

Proof By the fact α rp = rα p , β rp = rβ p , r >and<α p ≤ β p < ∞,wehave< ω ω ω ω ω ω p α q ≤ β q < ∞.First,let,α q >,andletx ∈ (M)withx– ∈ ω ω ω ω ω q (M). Take r , r with  ≤ r < α p ,≤ r < α q .ByTheorem.of[], there exist a ω    ω  ω ∗ ∗ unitary u ∈ M and h ∈ Hr (A)suchh = u x and h– ∈ Hr (A). It is clear that h = u x ∈ p p,ω – q,ω ω(M). It follows from Proposition . that h ∈ H (A). Similarly, h ∈ H (A). On the other hand, if min(p, q) ≤ ormin(α p , α p ) ≤ , there exists an integer n such ω ω that min(np, nq)>andmin(nα p , nα q )=min(α np , α nq )>.Letx = v|x| be the polar ω ω ω ω   decomposition of x.Wewritex = v|x| n , xk = |x| n ,≤ k ≤ n,thenx = xx ···xn.Thus, ∈ np M – ∈ nq M ∈ M ∈ np,ω A xk ω ( )andxk ω ( ). Therefore, there exist un and hn H ( )such – ∈ nq,ω A that xn = unhn and xn H ( ). Repeating this argument, we again get the same factorization for xn–un: xn–un = un–hn–,andthenforxn–un–,andsoon.Inthisway,we np,ω obtain the factorization x = uh ···hn, u ∈ M,whereu ∈ M is a unitary and hk ∈ H (A) – ∈ nq,ω A ··· such that hk H ( ). We write h = h hn,thenx = uh is the desired factorization.

Remark . ∈ ∞ M M – ∈ ∞ M (i) Let x ω ( )= be an invertible operator such that x ω ( ). Theorem . of [] implies that there exist a unitary u ∈ M and h ∈ A such that x = uh, h– ∈ A. p (ii) Let 

with  ≤ r < α p .ByTheorem.of[], ω ω there exist a unitary u ∈ M and h ∈ Hr(A) such that h = u∗x and h– ∈ A.Since ∗ p p,ω h = u x ∈ ω(M), it follows from Proposition . that h ∈ H (A). Therefore, by adapting the proof of Theorem ., we complete the proof.

Proposition . Let 

p,ω A ∩ q M q,ω A p,ω A ∩ q M q,ω A H ( ) ω( )=H ( ), H ( ) ω( )=H ( ).

q p p,ω Proof Since  < p < q < ∞,thenω(M) ⊆ ω(M), which implies that H (A) ∩ q q,ω p,ω q ∗  ω(M) ⊇ H (A). Conversely, let x ∈ H (A) ∩ ω(M). We write a =(x x +) ,then q – q,ω a ∈ ω(M)anda ∈ M. Applying Remark .(ii) to a,thereexistu ∈ M and h ∈ H (A) such that a = uh and h– ∈ A.Sinceh∗h = x∗x+, then there exists v ∈ M with v ≤such that x = vh.Sinceα rp = rα p , r >and<α q , then there exists  < α p .Let

Theorem . Let 

,there exist y H ( ) and z H ( ) such that x = yz, where p = q + r .

Proof The case where max{q, r} = ∞ is trivial. Thus we assume max{q, r} < ∞.Letx ∈ p,ω ∗  p – H (A)andε >.Wewritea =(x x + ε)  ,thenx ∈ ω(M)anda ∈ M.Letv ∈ M be a p p contraction operator such that x = va. Now applying Remark .(ii) to a r ,wehavea r = uz, p – r,ω q z ∈ M,whereu is a unitary in M and z ∈ H (A). Let y = va q u ∈ ω(M), then x = yz and so y = xz–.Sincex ∈ Hp,ω(A)andz– ∈ A,wehavey ∈ Hp,ω(M). By Proposition . and the fact p < q,weobtainy ∈ Hq,ω(A).

   p ω Remark . Let  < p, q ≤∞,<α p ≤ β p < ∞ and = + .Foreveryx ∈ H , (A) ω ω p q r and ε >,lety, z be as in Theorem .,then y q,ω z r,ω ≤ C( + ε) x p,ω for some constant C (independent of x). Indeed, the case p = ∞ follows from Theorem . of []since ∞ M M ω ( )= . The other case follows from Theorem ..

4 Outer operators according to Hp,ω(A)

Let x be a τ -measurable operator with limt→∞ μt(x) = . The Fuglede-Kadison determinant (x)isdeﬁnedby ∞ (x)=exp τ log |x| = exp log tdν|x|(t) , 

+ where dν|x| denotes the probability measure on R which is obtained by composing the spectral measure of |x| with the trace τ . We refer the reader to [, ] for more information on determinant.

q Proposition . Let 

Proof We shall prove only the third equivalence. The other cases are similar. Since ω ω ω p < q,thenHq, (A)isdenseinHp, (A). It follows from [AhA] q = Hq, (A)that ω(M) ω p [AhA] p = Hp, (A). Conversely, since α p >,thereisr >suchthat (M) ⊆ ω(M) ω ω r ∞ Lr(M), and so [AhA]r = H (A). Using Proposition . of [], we have [AhA]∞ = H (A), q,ω which means that [AhA]q,ω = H (A).

ω Deﬁnition . Let  < p < ∞ and  < α p ≤ β p < ∞.Anoperatorh ∈ Hp, (A) is called ω ω p,ω p,ω left outer, right outer or bilaterally outer according to [hA]p,ω = H (A), [Ah]p,ω = H (A) p,ω or [AhA]p,ω = H (A).

Remark . (i) Proposition . justifies the relative independence of the index p in Definition .. ∞ M M (ii) Since ω ( )= ,Proposition.of[] and Proposition . imply that Definition . coincides with the definition in the sense of [].

p ω Proposition . Let <α p ≤ β p < ∞ and h ∈ H , (A). ω ω Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 13 of 19

(i) If h is left or right outer, then (h)=(E(h)). Conversely, if (h)=(E(h)) and (h)>, then h is left and right outer (so bilaterally outer too). (ii) If A is antisymmetric and h is bilaterally outer, then (h)=(E(h)).

p ω r Proof Since α p > , then there is r >suchthatH , (A) ⊆ H (A). Applying Theorem . ω of []toh ∈ Hp,ω(A) ⊆ Hr(A), we obtain that (i) and (ii) hold.

In the classical function algebra setting, one assumes that D = A ∩ JA is one-dimen- sional, which forces E = τ(·). If in our setting this is the case, then we say that A is antisymmetric. It is worth remarking that the antisymmetric maximal subdiagonal subalgebras of commutative von Neumann algebras are precisely the weak* Dirichlet algebras. It is clear that A is antisymmetric if and only if dim D =(equivalently,D = C). The following corollary is a consequence of Proposition ..

ω Corollary . Let h ∈ Hp, (A),  < α p ≤ β p < ∞. ω ω (i) If (h)>, then h is left outer if and only if h is right outer. (ii) Assume that A is antisymmetric, then the following properties are equivalent: (a) h is left outer; (b) h is right outer; (c) h is bilaterally outer; (d) (E(h)) = (h)>.

p We will say that h is outer if it is at the same time left and right outer. If h ∈ Hω(A) with (h)>,thenh is outer if and only if (h)=(E(h)). Also in the case that A is antisymmetric, h with (h) >  is outer if and only if it is left, right or bilaterally outer. The following corollary is an immediate consequence of Corollary . of []andthe proof of Proposition ..

ω Corollary . Let 

p Theorem . Let 

,then ω ω ω there exist a unitary u ∈ M and an outer h ∈ Hp,ω(A) such that x = uh.

p Proof First, let α p >andp >,thereisα p > r >suchthatM ⊆ (M) ⊆ L (M)and ω ω ω r p,ω r p A ⊆ H (A) ⊆ H (A). Applying Theorem . of []tox ∈ ω(M) ⊆ Lr(M), there exist aunitaryu ∈ M and an outer h ∈ Hr(A)suchthatx = uh. It follows from Proposition . ∗ p r p,ω that h = u x ∈ ω(M) ∩ H (A)=H (M). By adapting the proof of Theorem .,we complete the proof.

p Corollary . Let <α p ≤ β p < ∞ and x ∈ (M) with 

, ω ω ω then there exist a unitary u ∈ A (inner) and an outer h ∈ Hp,ω(A) such that x = uh.

p Proof Let α p >andp >,thereexistsα p > r >suchthatM ⊆ (M) ⊆ L (M)and ω ω ω r p,ω r p A ⊆ H (A) ⊆ H (A). Applying Corollary . of []tox ∈ ω(M) ⊆ Lr(M), there exist a unitary u ∈ A and an outer h ∈ Hr(A)suchthatx = uh. Thus, Proposition . implies that ∗ p r p,ω h = u x ∈ ω(M) ∩ H (A)=H (M). By adapting the end of the proof of Theorem .,  we get the desired factorization of x. For simplicity we consider only the case α p > and ω  Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 14 of 19

 | | p >  .Letx = v x be the polar decomposition of x. Using a similar discussion of above to  p,ω  |x|  ,thereexistaunitaryu ∈ A and an outer h ∈ H (A)suchthat|x|  = uh since   p,ω (|x|  )>.Similarly,wehavev|x|  u = uh,whereu ∈ A and an outer h ∈ H (A).

Therefore, u = u and h = hh yield the desired factorization of x.

p ω Corollary . Let 

,then h ω ω is outer if and only if for any x ∈ Hp,ω(A) with |x| = |h|, we have (E(x)) ≤ (E(h)).

ω Proof Let h be outer and x ∈ Hp, (A)with|x| = |h|.Taking

5 Jordan morphism ∗ ∗ x+x x–x ∈ A Let x be an operator, we write Re x =  and Im x = i .Ifu Re ,thenu = Re x for ∈ A  E ∗ ∈ A E some x .Leta = x –  (x – x ), it is clear that a , u = Re a and (Im a) = . There- fore, there exists u = Im a ∈ Re A such that a = u + iu ∈ A and E(u)=E(Im a)=.By a similar discussion of [], we have u ∈ Re A is unique. Thus, we can deﬁne u = Im a, where a ∈ M is the unique element of M with u = Re a and E(Im a) = . It is obvious that ∼: u →u is real linear. We called u the conjugate of u. We deﬁne the Herglotz transform p p H : ω(M) → ω(M)byH(u)=u + iu.ItisclearthatH is real linear.

p Theorem . Let be a fully symmetric quasi-Banach function space and <α p ≤ ω ω β p < ∞. The real linear maps ω

∼: Re A → Re A, H : Re A → A

extend to real linear maps

∼ p M sa → p M sa p M sa → p,ω A : ω( ) ω( ) , H : ω( ) H ( ).

p sa p,ω If x ∈ ω(M) , then H(x)=x + ix ∈ H (A) and E(x)=.Both ∼ and H are bounded.

Proof Let r , r >with

∼: Lri (M)sa → Lri (M)sa, H : Lri (M)sa → Hri (A)

and E(x)=,H(x)=x + ix ∈ Hri (A), i = , . Now consider the standard complexiﬁcation ∼ : Lri (M) → Lri (M), i =,,thatis,x = Rex + iImx.Fromthemethodin[], we know that x, y ≺ x + iy, x, y ∈ Lri (M)sa, i =,.Thus, ∼(x + iy) = x + iy r x + iy r , ri i i

r and so ∼ is bounded on L i (M), i =,.Since<α p ≤ β p < ∞, it follows from The- ω ω p r r orem . of []thatω(M) is an interpolation space for the couple (L  (M), L  (M)), p which implies that ∼ is bounded on ω(M). Thus, both ∼ and H are bounded. By the r r sa p r discussion of above we know that H(x) ∈ H  (M), ∀x ∈ L  (M) .Sinceω(M) ⊆ L  (M) Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 15 of 19

r p and H is bounded, we have H(x) ∈ H  (M), E(x)=andH(x) ∈ ω(M) hold for all p sa p,ω x ∈ ω(M) . Combining this with Proposition .,weobtainH(x) ∈ H (A).

Using the same method as that in Theorem . of [], we obtain the following result.

p Theorem . Let be a fully symmetric quasi-Banach function space and <α p ≤ ω ω β p < ∞, then ω

p M p,ω A ⊕ p D ⊕ p,ω A ω( )=H ( ) ω( ) JH ( ).

→  E → E →  E The relevant projections are x  [x + ix – (x)]; x (x); x  [x – ix – (x)], where p p ∼ : ω(M) → ω(M) is the standard complexiﬁcation of ∼, that is,x = Re x + iIm x.

Let B, B be two Banach algebras, a linear map ϕ : B → B is a homomorphism if

ϕ(ab)=ϕ(a)ϕ(b), a, b ∈ B. We say that a linear map ϕ : B → B is an anti-morphism

if ϕ(ab)=ϕ(b)ϕ(a), a, b ∈ B. Given a unital Banach algebra B with identity , under the term irreducible representation of B, we understand a continuous homomorphism π : B → B(X), where B(X) is the set of all bounded linear operators on some Banach space X such that π(B) is an irreducible subalgebra of B(X) in the sense of admitting only trivial

invariant subspaces. A Jordan morphism between two von Neumann algebras M and

M is a linear mapping ϕ : M → M which preserves the Jordan product, i.e., ϕ(ab +

ba)=ϕ(a)ϕ(b)+ϕ(b)ϕ(a) for all a, b ∈ M. For further information, we refer the reader to [].

Lemma . Let  ≤ p < ∞ and let ϕ : A → A be a Jordan morphism such that π ◦ ϕ is either a morphism or an anti-morphism for each irreducible representation π of M. p,ω p,ω If ϕ extends to a bounded map ϕ : H (A) → H (A) with norm ϕ p,ω, then for any p ,ω p ,ω integer 

Proof Let ε >andx ∈ A be given. Using Remark .,weobtainh,...,hk ∈ A so that

x = hh ···hk with

k k– hn p,ω ≤ C( + ε) x p . k ,ω n=

By Lemma . of [], we have k k k ≤ ϕ hn ϕ(hn) + ϕ(hk+–n) . n= n= n=

p Thus, by the fact that ω is a fully symmetric quasi-Banach function space, we deduce k k k ϕ(x) p = ϕ hn ≤ ϕ(hn) + ϕ(hk+–n) k ,ω n= p n= n= p k ,ω k ,ω k k ≤ C ϕ(hn) + ϕ(hk+–n) n= p n= p k ,ω k ,ω Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 16 of 19

k k ≤ k ≤ k k C ϕ(hn) p,ω C ϕ p,ω hn p,ω n= n= k k k– ≤ C ϕ p,ω C( + ε) x p . k ,ω

This implies that the required estimate holds.

Lemma . Let ϕ : A → A be a continuous identity-preserving Jordan morphism extend- p,ω p,ω ing continuously to a map ϕ : H (A) → H (A),  ≤ p < ∞ with norm ϕ p,ω. For any  k ∈ N and any x ∈ A such that ϕ(x) is normal, we have ϕ(x) pk,ω ≤ ( ϕ p,ω) k x pk,ω. Let p 

  ≤  k ∈ N ∈ A ϕ(x) pk ,ω C ϕ p,ω x pk ,ω, k , x ,(.)

where C is a constant. If, moreover, ϕ satisﬁes the condition

∗ k k ∗ k k ϕ x + |x| ≥ ϕ x + ϕ(x) , k ∈ N, x ∈ M,(.)

we obtain the estimate

 k ϕ ≤ ϕ  k ∈ A ∈ N (x) pk ,ω C  p,ω x p ,ω, x , k ,(.)

where C is a constant.

Let B be a linear complement of D in A.ItiswellknownthatJA = D ⊕ JB and hence that A + JA = B ⊕ D ⊕ JB.Letϕ : A → M be a contractive identity-preserving Jordan morphism, we may extend ϕ to a map ϕ on A + JA by setting ϕ = ϕ on A and deﬁning the action on JB by ϕ(x)=ϕ(x∗)∗, x ∈ JB.Sinceϕ is order-preserving on D, it follows that ϕ preserves adjoints. Indeed, φis order-preserving, positive and contractive on A + JA.For further information, we refer the reader to [].

Proof Our proof follows Labuschagne’s argument of (Lemma ., []). Given some x ∈ A such that ϕ(x)isnormal,then k   ϕ ϕ k ϕ k k (x) pk,ω = (x) p,ω = (x) p,ω

  ϕ k k ≤ ϕ k k = x p,ω p,ω x p,ω

 k ≤ ϕ p,ω x pk,ω, k ∈ N.

Let  < p < ∞,<α p ≤ β p < ∞,andletϕ be an identity-preserving contractive Jor- ω ω dan morphism on A. From the preceding discussion (above the proof of this lemma), ϕ uniquely extends to a positive map on A + JA.Moreover,ϕ then acts positively on the von D p,ω A p,ω A ⊕ p D D Neumann algebra .ByTheorem.,wehaveH ( )=H ( ) ω( ). Since is p p,ω p dense in ω(D), then the continuous action of ϕ on H (A) (and hence also on L (M)) Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 17 of 19

p,ω p ensures that the unique extension of ϕ to H (A) acts positively on ω(D). The extension p of ϕ to a hermitian map ϕ on ω(M) can now be deﬁned by ∗ ∗ ∈ p,ω A ∈ p,ω A ϕ(x)=ϕ x , x JH ( )andϕ(x)=ϕ(x), x H ( ). (.)

Since this construction canonically contains that construction in the discussion above this lemma, ϕ constructed as (.) will restrict to A + JA, which yields precisely the map in

the preceding discussion (above the proof of this lemma). From Theorem .,the · p,ω

boundedness of ϕ follows from the · p,ω boundedness of ϕ and of the conjugate linear ∗ ∗  → ≤ ∞ | |  | |q | |q q | |q ≤ map x x .Let q < ,wewrite x q =( ( x + x )) .Itiswellknownthat x  q q p |x|q,andso|x|≤ |x|q.Ifϕ acts positively on all of ω(M), it will surely map · ∞ boundedly M into M. By Lemma . of [], we get ∗ ∗ ∗ ∗ ϕ(x) ϕ(x)+ϕ(x)ϕ(x) ≤ ϕ x x + xx , x ∈ M.

Since ϕ(y) ≤ ϕ(y)foreachy ∈ M+,wehave

   k ϕ(y) = ϕ(y)  ≤ ϕ y  ≤ ϕ y k , k ∈ N, y ∈ M+. (.) pk ,ω pk–,ω pk–,ω p,ω

Now note that    ∗  ∗  ϕ(x) = ϕ(x) + ϕ x ≤ ϕ |x| + x = ϕ |x| , x ∈ M,   

which implies that    ϕ(x) = ϕ(x)  ≤ ϕ |x|  .  pk ,ω  pk–,ω  pk–,ω

Combining this with (.), we get k–   k  ≤ | | k–  | | k ∈ M ∈ N ϕ(x)  pk ,ω ϕ x  p,ω = ϕ x  p,ω , x , k .

 Consequently, given x ∈ A, the positivity of ϕ and the fact that |ϕ(x)|≤  |ϕ(x)| imply that  ϕ ϕ ≤  ϕ (x) pk ,ω = (x) pk ,ω  (x)  pk ,ω    k k  k k ≤  ϕ  ≤  ϕ | |   (x)  p,ω  x  p,ω     k k ≤  ϕ | | ≤  ϕ k  p,ω x  pk ,ω C p,ω x p ,ω.

 | |≤ k | | The final inequality is a consequence of Lemma . of []. Observe that x   x k , | | | | a similar proof using x k instead of x  will suffice in the case that ϕ satisfies condition (.).

≤∞ Lemma . Let 

case x p,ω = limn→∞ x qn,ω. Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 18 of 19

p qn Proof Let x ∈ ω(M) ⊆ ω (M), then

 qn qn qn x = μt(x) ω(t) dt ≤  p μt(x) p qn,ω p–q ,ω q ,ω  n n q (  –  ) qn n qn p = x p,ωW() .

≤ Thus, lim supn→∞ x qn,ω x p,ω. By the Levi lemma and the dominated convergence theorem, we have

lim x qn = lim μ (x)qn dt + lim μ (x)qn dt →∞ qn,ω →∞ t →∞ t n n {t∈[,]:μ (x)>} n {t∈[,]:μ (x)≤} t t p p p = μt(x) dt + μt(x) dt = x p,ω. {t∈[,]:μt(x)>} {t∈[,]:μt(x)≤} ≤ ∞ ∞ Lemma . Let p < be given and let (qn) be a sequence of reals in (p, ) decreasing ∈ ∞ qn ⊆ p M to p. Given x n= ω ω( ), we have x p,ω = limn→∞ x qn,ω.

qn qn+ p Proof Let x ∈ ω ⊆ ω ⊆ ω(M). Let E = {t ∈ [, ] : μt(x) ≥ } and E = {t ∈ [, ] : p qn p qn μt(x)<},thenμt(x) ≤ μt(x) , t ∈ E and μt(x) ≥ μt(x) , t ∈ E.Thisimpliesthat

qn p lim μt(x) ω(t) dt = μt(x) ω(t) dt n→∞ E E

and

qn p lim μt(x) – μt(x) ω(t) dt = ω(t) dt =, n→∞ E E

which yield the desired result.

Using the same method as that in Theorem . of [], we obtain the following result.

p Theorem . Let ω be a fully symmetric quasi-Banach function space, and let ϕ : A → A be a contractive identity-preserving Jordan morphism such that: (a) π ◦ ϕ is either a homomorphism or an anti-morphism for each irreducible representation of M ⊇ A; and (b) for some 

q H ( ); (ii) for any r > q, ϕ restricts to a bounded map ϕ : Hr,ω(A) → Hq,ω(A). (iii) for any q > s ≥ , ϕ extends uniquely to a bounded map ϕ : Hq,ω(A) → Hs,ω(A). p p If ϕ : ω(M) → ω(M) satisﬁes the slightly stronger condition (.) in Lemma ., then ϕ : Hq,ω(A) → Hq,ω(A) itself is a bounded everywhere deﬁned operator.

Competing interests The authors declare that they have no competing interests. Han and Shao Journal of Inequalities and Applications (2015)2015:120 Page 19 of 19

Authors’ contributions The main idea of this paper was proposed by the corresponding author YZH. YZH and JJS prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the ﬁnal manuscript.

Acknowledgements The authors are grateful to the referee for useful suggestions. This work was partially supported by NSFC Grant No. 11371304 and NSFC Grant No. 11401507.

Received: 10 October 2014 Accepted: 25 March 2015

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