Noncommutative H^p spaces associated with type 1 subdiagonal algebras
aa r X i v : . [ m a t h . OA ] J a n NONCOMMUTATIVE H p SPACES ASSOCIATED WITH TYPE 1SUBDIAGONAL ALGEBRAS
RUIHAN ZHANG AND GUOXING JI ∗ A bstract . Let A be a type 1 subdiagonal algebra in a σ -finite von Neumann algebra M with respect to a faithful normal conditional expectation Φ . We consider a Riesz typefactorization theorem in noncommutative H p spaces associated with A . It is shown thatif 1 ≤ r , p , q < ∞ such that r = p + q , then for any h ∈ H r , there exist h p ∈ H p and h q ∈ H q such that h = h p h q . Beurling type invariant subspace theorem for noncommutative L p (1 < p < ∞ ) space is obtained. Furthermore, we show that a σ -weakly closed subalgebracontaining A of M is also a type 1 subdiagonal algebra. As an application, We prove thatthe relative invariant subspace lattice Lat M A of A in M is commutative.
1. I ntroduction
Riesz factorization theorem and Beurling invariant subspace theorem in the classicalHardy spaces are well-known. There were many very interesting extensions to abstractfunction algebras. Arveson in [2] introduced the notion of subdiagonal algebras as thenoncommutative analogue of the classical Hardy space H ∞ ( T ). It is worth noting that therewere several successful examples showing that some famous mature theorems about theclassical Hardy spaces have been extended to the noncommutative Hardy spaces basedon subdiagonal algebras(cf. [3, 4, 5, 6, 13, 14, 15, 17, 18]). Marsalli and West [24] gave aRiesz factorization theorem for finite noncommutative H p spaces. One important extensionis due to Blecher and Labuschagne on Beurling type invariant subspace theorems for afinite subdiagonal algebra in [7]. Recently, Labuschagne in [23] extended their results tononcommutative H for maximal subdiagonal algebras in a σ -finite von Neumann algebrabased on noncommutative H p spaces in [13]. We also discussed Riesz type factorizationtheorem in noncommutative H and Beurling type invariant subspace theorem for L ( M )associated with a type 1 subdiagonal algebra in the sense that every right invariant subspaceof a maximal subdiagonal algebra in noncommutative H is of Beurling type in [15, 16]. DoRiesz factorization theorem and Beurling type invariant subspace theorem when 1 < p < ∞ Mathematics Subject Classification.
Key words and phrases. von Neumann algebra; type 1 subdiagonal algebra; noncommutative H p space;invariant subspace. ∗ Corresponding authorThis research was supported by the National Natural Science Foundation of China(No. 11771261). hold for type 1 subdiagonal algebras? We consider these problems in this paper. We firstlyrecall some notions.Let M be a σ -finite von Neumann algebra acting on a complex Hilbert H . Z ( M ) = M ∩ M ′ is the center of M , where M ′ is the commutant of M . If Z ( M ) = C I , the multiplesof identity I , then M is said to be a factor. We denote by M ∗ and M p the space of all σ -weakly continuous linear functionals of M and the set of all projections in M respectively.Let Φ be a faithful normal conditional expectation from M onto a von Neumann subalgebra D . Arveson [2] gave the following definition. A subalgebra A of M , containing D , is calleda subdiagonal algebra of M with respect to Φ if(i) A ∩ A ∗ = D ,(ii) Φ is multiplicative on A , and(iii) A + A ∗ is σ -weakly dense in M .The algebra D is called the diagonal of A . we may assume that subdiagonal algebras are σ -weakly closed without loss generality(cf.[2]).We say that A is a maximal subdiagonal algebra in M with respect to Φ in case that A is not properly contained in any other subalgebra of M which is subdiagonal with respectto Φ . Put A = { X ∈ A : Φ ( X ) = } and A m = { X ∈ M : Φ ( AXB ) = Φ ( BXA ) = , ∀ A ∈ A , B ∈ A } . By [2, Theorem 2.2.1], we recall that A m is a maximal subdiagonalalgebra of M with respect to Φ containing A . A is said to be finite if there is a faithfulnormal finite trace τ on M such that τ ◦ Φ = τ . Finite subdiagonal algebras are maximalsubdiagonal(cf.[8]).We next recall Haagerup’s noncommutative L p spaces associated with a general vonNeumann algebra M (cf.[9, 26]). Let ϕ be a faithful normal state on M and let { σ ϕ t : t ∈ R } be the modular automorphism group of M associated with ϕ by Tomita-Takesaki theory.We consider the crossed product N = M ⋊ σ ϕ R of M by R with respect to σ ϕ . We denoteby θ the dual action of R on N . Then { θ s : s ∈ R } is an automorphisms group of N .Note that M = { X ∈ N : θ s ( X ) = X , ∀ s ∈ R } . N is a semifinite von Neumann algebraand there is the normal faithful semifinite trace τ on N satisfying τ ◦ θ s = e − s τ, ∀ s ∈ R . According to Haagerup [9, 26], the noncommutative L p space L p ( M ) for each 0 < p ≤ ∞ is defined as the set of all τ -measurable operators x a ffi liated with N satisfying θ s ( x ) = e − sp x , ∀ s ∈ R . ONCOMMUTATIVE H p SPACES 3
There is a linear bijection between the predual M ∗ of M and L ( M ): f → h f . If we definetr( h f ) = f ( I ) , f ∈M ∗ , then tr( | h f | ) = tr( h | f | ) = | f | ( I ) = k f k for all f ∈ M ∗ and | tr( x ) | ≤ tr( | x | )for all x ∈ L ( M ). As in [9], we define the operator L A and R A on L p ( M )(1 ≤ p < ∞ )by L A x = Ax and R A x = xA for all A ∈ M and x ∈ L p ( M ). Note that L ( M ) is a Hilbertspace with the inner product ( a , b ) = tr( b ∗ a ), ∀ a , b ∈ L ( M ) and A → L A ( resp. A → R A )is a faithful representation (resp. anti-representation) of M on L ( M ). We may identify M with L ( M ) = { L A : A ∈ M} .We recall that { σ ϕ t : t ∈ R } is the modular automorphism group of M associated with ϕ . Let h be the noncommutative Radon-Nikodym derivative of the dual weight of ϕ on N with respect to τ . Then h is the image( h ϕ ) of ϕ in L ( M ) and we have that the followingrepresentation of σ ϕ t (cf.[21, 22]):(1.1) σ ϕ t ( X ) = h it Xh − it , ∀ t ∈ R , ∀ X ∈ M . It is known that the noncommutative H p space H p ( M ) and H p ( M ) in L p ( M ) for any 1 ≤ p < ∞ is H p = H p ( M ) = [ h θ p A h − θ p ] p and H p = H p ( M ) = [ h θ p A h − θ p ] p for any θ ∈ [0 , p = ∞ , then we identify H ∞ as A and H ∞ as A .We introduce the notion of column L p -sum in the noncommutative L p -space L p ( M )(1 ≤ p ≤ ∞ ) for a σ -finite von Neumann algebra M given in [19] by Junge and Sherman. Forany subset S ⊆ L p ( M ), [ S ] p denotes the closed linear span of S in L p ( M ). If X is aclosed subspace of L p ( M ), and { X i : i ∈ Λ } is a family of closed subspaces of X such that X = [ W { X i : i ∈ Λ } ] p with the property that X ∗ j X i = { } if i , j , then we say that X is theinternal column L p -sum ⊕ coli ∈ Λ X i . If p = ∞ , we assume that X and { X i : i ∈ Λ } are σ -weaklyclosed, and the closed linear span is taken with the σ -weak topology. For symmetry, if X j X ∗ i = { } if i , j and X = [ ∨{ X i : i ∈ Λ } ] p , then we say that X is the internal row L p -sum ⊕ rowi ∈ Λ X i .Following [7, 23], we define the right wandering subspace of a right invariant subspace M ⊆ L ( M ), that is, MA ⊆ M , to be the space W = M ⊖ [ MA ] . We say that M is of type1 if W generates M as an A -module (that is, M = [ W A ] ). We say that M is of type 2 if W = { } . Note that every right invariant subspace M is an L -column sum M = N ⊕ col N ,where N i is of type i for i = , RUIHAN ZHANG AND GUOXING JI if M is of type 1, then M is of the Beurling type, that is, there exists a family of partialisometries { U n : n ≥ } in M satisfying U ∗ n U m = n , m and U ∗ n U n ∈ D such that M = ⊕ coln U n H . We refer to [7, 23] for more details. Symmetrically, we may consider leftinvariant subspaces for A and a type 1 left invariant subspace M may be represented as M = ⊕ rown ≥ H V n by a family of partial isometries { V n : n ≥ } satisfying V n V ∗ m = n , m and V n V ∗ n ∈ D if n = m and this fact will be used frequently. Motivated by the column(resp.row)-sum as above, we call that a family of partial isometries U = { U n : n ≥ } in M is column(resp. row) orthogonal if If U ∗ n U m = U n U ∗ m =
0) for n , m and U ∗ n U n ∈ D (resp. U n U ∗ n ∈ D ) for all n (cf.[16]).In this paper, we continuously consider type 1 subdiagonal algebras introduced in [15].Let A be a type 1 subdiagonal algebra with respect to Φ with diagonal D , that is, everyinvariant right invariant subspace M ⊆ H of A is of type 1. Note that there is a family ofcolumn orthogonal partial isometries U = { U n : n ≥ } such that H = ⊕ coln U n H and thosepartial isometries together with the diagonal D consist of generators of A (cf.[15]). We willfixed those partial isometries throughout this paper. We give a Riesz type factorizationtheorem for non-commutative H P (1 ≤ p < ∞ ) which says that every element in H r is aproduct of two elements in H p and H q respectively with r = p + q for any 1 ≤ r , p , q < ∞ in Section 2. This gives an answer of a problem in [15]. Moreover, we consider a Beurlingtype invariant subspace theorem in L p ( M ) when 1 < p < ∞ in Section 3. In Section4, We determine the structure of those σ -weakly closed subalgebras containing the type1 subdiagonal algebra A in M . It is shown that those subalgebras are of type 1. As anapplication, we show that the relative invariant subspace lattice Lat M A = { E ∈ M p :( I − E ) AE = , ∀ A ∈ A } of A in the von Neumann algebra M is commutative in Section 5.2. R iesz factorization in noncommutative H p spaces for type subdiagonal algebras In [15], we consider the Riesz factorization in noncommutative H for type 1 subdiagonalalgebra A . It is shown that for any h ∈ H , there are two elements h i ∈ H ( i = ,
2) suchthat h = h h . Let 1 ≤ r , p , q < ∞ such that r = p + q and h ∈ H r . Does h = h p h q for some h p ∈ H p and h q ∈ H q ? We consider this problem in this section. For any subset S ⊆ L p ( M )and K ⊆ L q ( M ), denote by SK = [ { sk : s ∈ S , k ∈ K } ] r , the closed subspace generatedby the subset { sk : s ∈ S , k ∈ K } in L r ( M ). We firstly recall that an element h ∈ H p isright(resp. left) outer in [ h A ] p = H p (resp. [ A h ] p = H p ). Lemma 2.1.
Let A be a type 1 subdiagonal algebra. Then for any nonzero f ∈ L ( M ) ,there exist a left outer element g ∈ H and a contraction B ∈ M such that f = gB. ONCOMMUTATIVE H p SPACES 5
Proof.
We consider the type 1 subdiagonal algebra A ∗ and ( H ) ∗ ,. Then for any f ∗ ∈ L ( M ), there exist a right outer element g ∗ ∈ ( H ) ∗ and a contraction B ∈ M such that f ∗ = B ∗ g ∗ by [15, Theorem 3.5]. Then f = gB . Note that [ g ∗ A ∗ ] = ( H ) ∗ . Thus [ A g ] = H ,that is, g ∈ H is left outer. (cid:3) Let M ⊆ L p ( M ) be a closed subspace. If MA ⊆ M (resp. AM ⊆ M ), then we say that M is a right(resp. left) invariant subspace. Lemma 2.2.
Let A be a type 1 subdiagonal algebra and < p < ∞ . If M ⊆ H p is a rightinvariant subspace such that M H s = H , then M = H p , where p + s = .Proof. Let p + q = s + t =
1. Then + s = q and p + = t . Put M ⊥ = { g ∈ L q ( M ) : tr( gm ) = , ∀ m ∈ M } . It is su ffi cient to show that M ⊥ = ( H p ) ⊥ = H q . It istrivial that M ⊥ ⊇ H q . Take any g ∈ M ⊥ . Let g = V | g | be the polar decomposition. Then g = V | g | q | g | qs = g g , where g = V | g | q ∈ L ( M ) and g = | g | qs ∈ L s ( M ). Then g = h B for a left outer element h ∈ H and a contraction B ∈ M by Lemma 2.1. Put h = Bg ∈ L s ( M ). Now tr( mg ) = tr( mg g ) = tr( mh h ) = m ∈ M . On the other hand,[ M h ] t = [ MA h ] t = [ M H ] t = [ M h A ] t = [ M h s h p A ] t = [[ M h s A ] H p ] t = [ H H p ] t = H t .Thus tr( xh ) = x ∈ H t . It follows that h ∈ ( H t ) ⊥ = H s . Therefore g = h h ∈ H q .Consequently, M = H p . (cid:3) Theorem 2.3.
Let A be a type 1 subdiagonal algebra and η ∈ L p ( M )(1 ≤ p < ∞ ) anonzero vector. Then for any δ > , there exist a contraction B ∈ M and a right outerelement h ∈ H p with k h k p < k η k p + δ such that η = Bh. If η ∈ H p , then we may have B ∈ A .Proof. If p =
2, then the result is proved in [15, Theorem 3.5]. Suppose that 2 < p < ∞ and we take some t > = p + t . Then η h t ∈ L ( M ).Put x = ( η ∗ η + ε h p ) h t . Then we have that x ∈ L ( M ) such that ε h ≤ x ∗ x and h t η ∗ η h t ≤ x ∗ x . By the proof of [15, Theorem 3.5], there are injective contractions A , C ∈M with dense ranges, a unitary operator U ∈ M and a right outer element ξ ∈ H , suchthat ε − Ax = h , η h t = C x and x = U ξ . Note that ξ = U ∗ x = U ∗ ( η ∗ η + ε h p ) h t is a rightouter element in H . Then H = [ ξ A ] = [ U ∗ ( η ∗ η + ε h p ) h t A ] = [ U ∗ ( η ∗ η + ε h p ) H r ] .Put h = U ∗ ( η ∗ η + ε h p ) . We note that h ∈ H p . In fact, H = [ hh t A ] = [ h A ] p H t and H q = H t H . Thus tr( hxy ) = x ∈ H t and y ∈ H , which implies that h ∈ ( H q ) ⊥ = H p .Therefore M = [ h A ] p ⊆ H p and M H t = [ M h t ] = [ h A h t ] = [ ξ A ] = H . By Lemma 2.2, M = H p and h is right outer. Note that η h t = CU ∗ ( η ∗ η + ε h p ) h t = Chh t . We then have RUIHAN ZHANG AND GUOXING JI η = Ch . Note that k h k p = k U ∗ ( η ∗ η + ε h p ) k p = (cid:18) tr(( η ∗ η + ε h p ) p ) (cid:19) p → k η k p ( ε → ε > k h k p < k η k p + δ .If 1 ≤ p <
2, then p = + s for some s ≥
2. Let η = V | η | = V | η | p | η | ps = η η s , where η = V | η | p ∈ L ( M ), η s = | η | ps ∈ L s ( M ) with k η k = k η k p p and k η s k s = k η k ps p . We choosetwo positive numbers δ , δ s such that ( k η k + δ )( k η s k s + δ s ) < k η k k η s k s + δ = k η k p + δ . By[15, Theorem 3.5] and the proof as above, η s = Ch s for a right outer element h s ∈ H s with k h s k s < k η s k s + δ s and a contraction C ∈ M . Now η C ∈ L ( M ). We again have η C = Bh for a right outer element h ∈ H with k h k < k η k + δ and a contraction B ∈ M . It iselementary that h = h h s ∈ H p is right outer such that k h k p ≤ k h k k h s k s < k η k k η s k s + δ = k η k p + δ .If η ∈ H p , then BhA = η A ∈ H p for all A ∈ A . Thus BH p ⊆ H p and B ∈ A from [13,Theorem 2.7]. (cid:3) We now have the following Riesz type factorization theorem in noncommutative H p spaces for 1 ≤ p < ∞ whose proof is similar to [15, Theorem 3.6]. To completeness, wegive the detail. Theorem 2.4.
Let ≤ r , p , q < ∞ such that r = p + q and let A be a type 1 subdiagonalalgebra. Then for any h ∈ H r and ε > , there exist h p ∈ H p , h q ∈ H q with k h t k t < k h k rt r + ε for t = p , q such that h = h p h q . If h ∈ H r , we may choose one of h t in H t for t = p , q.Proof. Take 1 < s ≤ ∞ such that r + s =
1. Let h = V | h | = V | h | rp | h | rq be the polardecomposition. Then | h | rt ∈ L t ( M ) with k| h | rt k t = k h k rt r for t = p , q . By Theorem 2.3, thereexist a contraction B ∈ M and a right outer element h q ∈ H q with k h q k q < k| h | rq k q + ε = k h k rq r + ε such that | h | rq = Bh q . Put h p = V | h | rp B . We show that h p ∈ H p . In fact, tr ( hg ) = g ∈ H s since h ∈ H r . Then tr ( hg ) = tr ( h p h q g ) = g ∈ H s . Note that h q isright outer. Then [ h q H s ] p ′ = [ h q A H s ] p ′ = H q H s = H p ′ , where p ′ > p + p ′ = h p ∈ H p with k h p k p ≤ k| h | rp k p < k h k rp r + ε . Moreover, if h ∈ H p , then tr ( hg ) = g ∈ H s . It easily follows that h p ∈ H p in this case.On the other hand, by taking the pair ( h ∗ , ( H r ) ∗ ) if h ∈ ( H r ), we may write h ∗ in the form h ∗ = g q g p with g q ∈ ( H q ) ∗ and g p ∈ ( H p ) ∗ . Then we have h = h p h q with h p ∈ H p and h q ∈ H q by setting h q = g ∗ q and h p = g ∗ p . (cid:3)
3. B eurling type invariant subspace theorem for type subdiagonal algebras Let A be a type 1 subdiagonal algebra. For any positive integer n ≥
1, let A n be the σ -weakly closed ideal of A generated by { a a · · · a n : a j ∈ A } . It is elementary that a ONCOMMUTATIVE H p SPACES 7 right invariant subspace M ⊆ L ( M ) is of type 1(resp. type 2) if and only if ∩ n ≥ [ MA n ] = { } (resp. M = [ MA ] ). Based on this fact, we introduce similar notions in L p spaces. Definition 3.1.
Let A be a type 1 subdiagonal algebra and M a right(resp. left) invari-ant subspace in L p ( M ). Then we say that M is of type 1 if ∩ n ≥ [ MA n ] p = { } (resp. ∩ n ≥ [ A n M ] p = { } ) and of type 2 if M = [ MA ] p (resp. M = [ A M ] p ).We recall that there exists a column orthogonal family of partial isometries { U n : n ≥ } in M such that(3.1) H = ⊕ coln ≥ U n H when A is of type 1.We note that a Beurling type invariant subspace theorem is given in [16] when p = p = ∞ . We now consider this theorem in noncommutative L p ( M ) for 1 < p < ∞ . Lemma 3.2.
Let A be a type 1 subdiagonal algebra. If M ⊆ L p ( M ) ( < p < ∞ ) is a rightinvariant subspace of type 2, then there exists a projection E ∈ M such that M = EL p ( M ) .Proof. By [15, Theorem 2.7], A = ∨{ D U i U i · · · U i n : i j ≥ } , where { U i : i ≥ } is a family of column orthogonal partial isometries as in (3 . M is D rightinvariant and [ MA ] p = M . Since ∨{ M U n : n ≥ } ⊇ ∨{ M U i U i · · · U i n : i k ≥ , n ≥ } , M = ∨{ M U n : n ≥ } . For any m , n ≥ x ∈ M , we have R U ∗ m R U n x = xU n U ∗ m ∈ M since U n U ∗ m ∈ D from [15, Proposition 2.3]. Then M is right M invariant. That is, M isan M right module. Then by [19, Theorem 3.6], there is a projection E ∈ M such that M = EL p ( M ). In fact, we also have that the right submodule M is column summand in L p ( M ) from the proof of this theorem(cf.[19, p ]). It follows that the projection onto M commutes with the right multiplications by M . Thus there is a projection E ∈ M such that M = EL p ( M ) by [19, Corollary 2.6]. (cid:3) For a family of column orthogonal partial isometries W = { W n : n ≥ } , we define aright invariant subspace M p W = ⊕ coln ≥ W n H p in L p ( M ) for any 1 ≤ p ≤ ∞ as in [16]. If1 ≤ r < ∞ and r = p + q for some p , q ≥
1, then M r W = [( ⊕ n ≥ W n H p ) H q ] r = M p W H q . Thefollowing theorem is motivated from [16, Theorem 2.2]. Theorem 3.3.
Let A be a type 1 subdiagonal algebra and let M ⊆ L p ( M )(1 < p < ∞ ) bea nonzero closed right invariant subspace of A . ( i ) There are right invariant subspaces M i of type i ( i = , such that M = M ⊕ col M . ( ii ) If M ⊆ L p ( M ) is a right invariant subspace of type 1, then there exists a family ofcolumn orthogonal partial isometries W such that M = M p W . RUIHAN ZHANG AND GUOXING JI
Proof. ( i ) Put M = ∩ n ≥ [ MA n ] p . Then M ⊆ M is a right invariant subspace of type 2.By Lemma 3.1, there is a projection E ∈ M , such that M = EL p ( M ) ⊆ M . Put M = ( I − E ) M ⊆ M . This mean that M is closed and right invariant such that M = M ⊕ col M .Note that ∩ n ≥ [ M A n ] p ⊆ ∩ n ≥ [ MA n ] p ⊆ M ∩ M = { } . Then M is of type 1.( ii ) Firstly, we assume that 1 < p < r > p = + r . Put P = {W = { W n } n ≥ ⊆ M : column orthogonalpartial isometries such that M W H r ⊆ M } . Note that P is non-empty. In fact, we show that for any nonzero h ∈ M , [ h A ] p = ( V H ) H r for a partial isometry V ∈ M with V ∗ V ∈ D . Write h = ( u | h | p ) | h | pr . Then | h | pr ∈ L r ( M ) byTheorem 2.3, there exist a contraction B ∈ M and a right outer element h r ∈ H r such that | h | pr = Bh r . Put h = u | h | p B . We have h = h h r . This implies that M ⊇ [ h A ] p = [ h h r A ] p = [ h H r ] p = [ h A h r A ] p = [ h A ] H r since [ h r A ] r = H r . Note that [ h A ] ⊆ L ( M ) is a right invariant subspace. By [15, Lemma3.3], [ h A ] = V H ⊕ col N for a partial isometry V ∈ M such that V ∗ V ∈ D and a rightinvariant subspace N of type 2. Note that[ N H r ] p = [[ N A ] H r ] p = [[ N H r ] p A ] p . It follows that [ N H r ] p = ∩ n ≥ [( N H r ) A n ] p ⊆ M = { } . Thus N = { } and [ h A ] = V H with V ∗ V ∈ D .Note that h r is right outer. This means that [ h A ] p = [ h h r A ] p = [ h H r ] p = [ h A H r ] p = [ h A ] H r = ( V H ) H r and { V } ∈ P .We define a partial order in P by W ≤ V if M W ⊆ M V for any W , V ∈ P . Let {W λ : λ ∈ Λ } ⊆ P be a totally ordered family in P . Put N = ∨{ M W λ : λ ∈ Λ } . Notethat N ⊆ L ( M ) is a right invariant subspace in L ( M ). Then N = M W ⊕ c N for afamily of column orthogonal partial isometries W and a right invariant subspace N oftype 2 in L ( M ). Then [ N H r ] p is also a type 2 right invariant subspace in L p ( M ). Thisimplies that [ N H r ] p = ∩ n ≥ [( N H r ) A n ] p ⊆ ∩ n ≥ [ MA n ] p = { } . Hence N = { } . Since M W λ H r ⊆ M , N H r ⊆ M . Thus W ∈ P with W λ ≤ W for any λ ∈ Λ . That is, W is an upper bound of {W λ : λ ∈ Λ } . By Zorn’s lemma, there exists a maximal element W ∈ P . We show that M = M p W . Otherwise, assume that there is an h ∈ M such that h < M p W . Then [ h A ] p = ( V H ) H r for a partial isometry V ∈ M with V ∗ V ∈ D as above.Since h < M p W , V H * M W . Now ˜ N = ∨{ M W , V H } is also a right invariant subspace in L ( M ) with ˜ N H r ⊆ M . In this case, we have ˜ N = M V for a family of column orthogonal ONCOMMUTATIVE H p SPACES 9 partial isometries
V ∈ P . It is trivial that M W $ M V by a similar treatment. This is acontradiction. Thus M = M W H r = M p W .We have that for a general right invariant subspace M ⊆ L p ( M ), there is a right invariantsubspace N ⊆ L ( M ) such that M = N H r . In fact, by ( i ), M = M ⊕ col M = M W H r ⊕ col EL ( M ) H r = ( M W ⊕ col EL ( M )) H r = N H r , where N = M W ⊕ col EL ( M ).Let 2 < p < ∞ and let p and q be conjugate exponents, that is, p + q =
1. Then1 < q < = p + r as well as q = + r for some 1 < r < ∞ . Note that M H r ⊆ L ( M )is right invariant of type 1. Thus there is a family of column orthogonal partial isometries W = { W n : n ≥ } such that M H r = M W = M p W H r .We next show that M = M p W . For any x ∈ M , it is elementary that xh r = ⊕ coln ≥ W n W ∗ n xh r with W ∗ n xh r ∈ H since W ∗ n xh r ∈ H p . Then W ∗ n x ∈ H p and x = ⊕ coln ≥ W n W ∗ n x ∈ M p W .Therefore M ⊆ M p W .For any closed subspace K ⊆ L p ( M ), we put K ⊥ = { y ∈ L q ( M ) : tr ( xy ) = , ∀ x ∈ K } . Itis known that K ⊥ is left invariant when K is right invariant. By symmetry, we easily have M ⊥ = H r N = ⊕ rown ≥ H q V n ⊕ row L q ( M ) F for a left invariant subspace N = ⊕ rown ≥ H V n ⊕ row L ( M ) F ⊆ L ( M ) since 1 < q < M H r ) ⊥ = N . It is clear that ( M H r ) ⊥ ⊇ N . Take any y ∈ ( M H r ) ⊥ . Thentr( xhy ) = x ∈ M and h ∈ H r . It follows that H r y ⊆ M ⊥ = ⊕ rown ≥ H q V n ⊕ row L q ( M ) F .By a similar treatment as above, we have h r y = ⊕ rown ≥ h r yV ∗ n V n ⊕ row h r yF and thus y = ⊕ rown ≥ yV ∗ n V n ⊕ row yF ∈ N .Thus M ⊥ = H r N = H r ( M H r ) ⊥ = H r ( M p W H r ) ⊥ = ( M p W ) ⊥ and M = M p W . (cid:3) Let M ⊆ L p ( M ) be a right invariant subspace. Then M = M p W ⊕ col EL p ( M ) for afamily of column orthogonal partial isometries W and a projection E in M . It follows that M ∞ = M ∞W ⊕ E M is a σ -weakly closed right invariant subspace in M . Corollary 3.4.
Let A be a type 1 subdiagonal algebra. Then there is a lattice isomorphismfrom the lattice of all right invariant subspaces of A in M onto that in L p ( M ) for ≤ p < ∞ .Proof. Let M ⊆ M be a σ -weakly closed right invariant subspace. Then M p = [ M h p ] p = M H p ⊆ L p ( M ) is a right invariant subspace. We show this correspondence is a bijection.Assume that M H p = N H p for a σ -weakly closed right invariant subspace N ⊆ M . Then wehave two column orthogonal families of partial isometries W and V and two projections E , F ∈ M such that M = M ∞W ⊕ col E M and N = M ∞V ⊕ col F M by [16, Theorem 2.2]. Forany A ∈ M , Ah p ∈ M H p = N H p . Then Ah p = ⊕ n ≥ V n V ∗ n Ah p ⊕ FAh p with V ∗ n A ∈ A and FAh p ∈ FL p ( M ). It follows that A = ⊕ V n V ∗ n A + FA ∈ N . That is, M ⊆ N . By symmetry,we have M = N . On the other hand, for any right invariant subspace M p ⊆ L p ( M ), thereis a family of column orthogonal partial isometries W and a projection E ∈ M such that M p = M p W ⊕ EL p ( M ) from Theorem 2.3. Put M = M ∞W ⊕ col E M . Then M p = M H p . Thusthe map: M → M H p is a bijection from the lattice of all σ − weakly closed right invariantsubspaces in M onto that of all right closed invariant subspaces in L p ( M ). (cid:3)
4. S ubalgebras containing a type subdiagonal algebra We gave a necessary and su ffi cient condition for a type 1 subdiagonal algebra A to be amaximal subalgebra of M in [16]. In fact, we may determine all σ − weakly closed subal-gebras containing A in M . Let B be a σ − weakly closed subalgebra of M containing A .Then [ B h ] is a right invariant subspace in L ( M ). As in [16], let p and q be the projec-tions from L ( M ) onto L ( D ) and the wandering subspace [ B h ] ⊖ [ B h A ] of [ B h ] respectively. It is known that both p and q are in the commutant ( R ( D )) ′ of R ( D ). For anytwo projections E and F in a von Neumann algebra M , E F means that there is a partialisometry V ∈ M such that V ∗ V = E and VV ∗ ≤ F . If E F and F E , then E ∼ F . Foran operator A , R ( A ) denotes the range of A . Lemma 4.1.
Let A be a type 1 subdiagonal algebra and B as above. Then q ≤ p.Proof. Note that B is two-side A invariant subspace. Then there exists a family of columnorthogonal partial isometries W and a projection E in M such that B = ⊕ coln ≥ W n A ⊕ col E M as well as [ B h ] = ⊕ coln ≥ W n H ⊕ col EL ( M ) by Corollary 3.1. This means that EL ( M ) = ∩ n ≥ [ B h A n ] . Since [ B h A n ] is left B invariant for any n , EL ( M ) is also left B in-variant. This implies that EBE = EB for any B ∈ B . In particular, ED = DE for all D ∈ B ∩ B ∗ . We now have W n ∈ B ∩ B ∗ for any n ≥
1. In fact, it is trivial that W n ∈ B forany n ≥
1. On the other hand, W ∗ n B = W ∗ n W n A ⊆ A ⊆ B since W ∗ n W n ∈ D . Then W ∗ n ∈ B for any n ≥
1. Since EW n = n , EW n = W n E = ⊕ n ≥ W n W ∗ n = I − E . Put N = ⊕ coln ≥ W n H ⊆ [ B h ] . Then N is the type 1 part of the right invariant subspace [ B h ] .Take any subprojections p ≤ p and q ≤ q such that p ∼ q in ( R ( D )) ′ . We beginby proving that p ≤ q . Put M = [ R ( p ) A ] and N = [ R ( q ) A ]. Let w ∈ ( R ( D )) ′ bea partial isometry such that w ∗ w = q and w w ∗ = p ≤ p as in [16, Proposition 2.5].Then N = W ∗ H for a partial isometry W ∈ M such that H = W N ⊕ col ( I − WW ∗ ) H by [16, Proposition 2.5] again. It follows that N = W ∗ H = W ∗ W N . Note that N = [ (cid:18) [ B h ] ⊖ [ B h A ] (cid:19) A ] = [ R ( q ) A ] . We have N = [ R ( q ) A ] ⊕ col [( R ( q − q )) A ] = ONCOMMUTATIVE H p SPACES 11 N ⊕ col [( R ( q − q )) A ] . We similarly obtain that H = [ R ( p ) A ] ⊕ col [( R ( p − p )) A ] = M ⊕ col [( R ( p − p )) A ] . It follows that W ∗ H = W ∗ M = N . Then we have M = W N since W ∗ M = W ∗ H = W ∗ W N . We also get WW ∗ H = WW ∗ M = M ⊆ H and so that WW ∗ ∈ D . Note that [ B h ] = [ R ( q ) A ] ⊕ col EL ( M ). We have [ R ( q ) M ] = ( I − E ) L ( M )since ⊕ coln ≥ W n W ∗ n = I − E . It is elementary that W is an isometry on [ R ( q ) M ] while W x = x ∈ [ R ( q ) M ] ⊥ . It follows that W ( EL ( M )) = W E =
0. Thus W [ B h ] = W [ R ( q ) A ] = W [ R ( q ) A ] = W N = M ⊆ H ⊆ [ B h ] . We now have W ∈ B by [16, Proposition 2.3].On the other hand, W ∗ H = W ∗ M = N ⊆ N = ⊕ coln ≥ W n H . Then W ∗ h = ⊕ coln ≥ W n ξ n for some ξ n ∈ H . It follows that W ∗ n W ∗ h = W ∗ n W n ξ n ∈ H , which implies that W ∗ n W ∗ ∈ A ⊆ B . Moreover, W n W ∗ n W ∗ h = W n ξ n . So we get W ∗ h = ⊕ coln ≥ W n ξ n = ⊕ coln ≥ W n W ∗ n W ∗ h .Then W ∗ = ⊕ coln ≥ W n W ∗ n W ∗ since h is separating. We now have W ∗ ∈ B . It follows that M = WW ∗ M = WW ∗ H ⊆ WW ∗ [ B h ] ⊆ W [ B h ] = W N = M . Thus WW ∗ H = WW ∗ [ B h ] = M with the wandering subspace R ( p ). Take any ξ ∈ R ( p ). We have h ξ, Bh A i = h WW ∗ ξ, Bh A i = h ξ, WW ∗ Bh A i = B ∈ B and A ∈ A . Moreover, we have ξ ∈ R ( p ) ⊆ R ( p ) = L ( D ) ⊆ H ⊆ [ B h ] .It follows that R ( p ) ⊆ [ B h ] ⊖ [ B h A ] = R ( q ). Therefore p ≤ q .We take a maximal family of mutually orthogonal subprojections { p n : n ≥ } of p suchthat p n ∼ q n for some subprojections q n ≤ q for all n ≥ R ( D )) ′ by Zorn’s lemma. Thenwe must have p n ≤ q for all n ≥ p = p − ( ⊕ n ≥ p n ) and q = q − ( ⊕ n ≥ p n ),then p and q are in ( R ( D )) ′ and either p or q is 0 by [20, Theorem 6.2.7].If q =
0, then we have q ≤ p . If p =
0, then we obtain that p ≤ q . Next we show that p = q . Note that N = [ R ( q ) A ] = [ R ( p ) A ] ⊕ col [( R ( q − p )) A ] = H ⊕ col [( R ( q − p )) A ] .It trivial that L ( M ) = [ R ( p ) M ] is orthogonal with [( R ( q − p )) M ] . Then we obtain that[( R ( q − p )) M ] = p = q . Consequently, we have q ≤ p . (cid:3) Theorem 4.2.
Let A be a type 1 subdiagonal algebra and B a σ -weakly closed subalgebracontaining A of M . Then the following assertions hold: (1) There exists a projection E ∈ Lat M A ∩ D such that B = E M + ( I − E ) A . (2) B is a type 1 subdiagonal algebra with respect to the unique faithful normal expec-tation Ψ from M onto the diagonal B ∩ B ∗ of B such that ϕ ◦ Ψ = ϕ .Proof. (1) As in the proof of Lemma 4.1, there exists a family of column orthogonal partialisometries W and a projection E in M such that B = ⊕ coln ≥ W n A ⊕ col E M and [ B h ] = ⊕ coln ≥ W n H ⊕ col EL ( M ). Put N = ⊕ coln ≥ W n H ⊆ [ B h ] be the type 1 part of [ B h ] . Let p and q as in Lemma 4.1. We have q ≤ p . This means that N = [ R ( q ) A ] ⊆ [ R ( p ) A ] = H .Thus we have W n ∈ A since N = ⊕ coln ≥ W n H ⊆ H . Since W ∗ n B = W ∗ n W n A ⊆ A , we get W ∗ n ∈ A . Therefore W n , W ∗ n ∈ D and E ∈ D ⊆ A from the fact that ⊕ coln ≥ W n W ∗ n = I − E ⊆ D .Note that E ∈ Lat M B ⊆ Lat M A , E ∈ D and B = ⊕ coln ≥ W n A ⊕ col E M . We have ( I − E ) B = ( I − E )( ⊕ coln ≥ W n A ) = ⊕ coln ≥ W n A ⊆ ( I − E ) A ⊆ ( I − E ) B and so ⊕ coln ≥ W n A = ( I − E ) A . Thus, B = E M + ( I − E ) A .(2) By (1), ˜ D = B ∩ B ∗ = E M E + ( I − E ) D . We next claim that B is σ ϕ t -invariant.In fact, E is in the center of D . Thus we have σ ϕ t ( E ) = E for all t ∈ R . It follows that σ ϕ t ( B ) = B and σ ϕ t ( ˜ D ) = ˜ D for all t ∈ R . By [25, Chapter IX, Theorem 4.2], there existsunique faithful normal conditional expectation Ψ from M onto ˜ D such that ϕ ◦ Ψ = ϕ . Weshow that B is a maximal subdiagonal algebra of M with respect to Ψ . In fact, B = { B ∈ B : Ψ ( B ) = } = E M ( I − E ) + ( I − E ) A . An elementary calculation shows that B is anideal of B . We thus have that Ψ is multiplicative on B . It is maximal subdiagonal by [27,Theorem 1.1].Finally we show that B is of type 1. Note that E is in the center of D , we have Φ ( ET ( I − E )) = T ∈ M . Moreover, E ( A + A ∗ )( I − E ) = E A ( I − E ) ⊆ A . We have E M ( I − E ) ⊆ A . It follows that B ⊆ A . Consequently, ∩ n ≥ [ h B n ] ⊆ ∩ n ≥ [ h A n ] = { } .Therefore, B is a type 1 subdiagonal algebra from [15, Proposition 2.2]. (cid:3)
5. T he relative invariant subspace lattice of a type subdiagonal algebra We recall that the invariant subspace lattice of a subalgebra
A ⊆ B ( H ) is the set Lat A = { E ∈ B ( H ) p : ( I − E ) AE = , ∀ A ∈ A} . If A ⊆ M , then
Lat M A = { E ∈ M p : ( I − E ) AE = , ∀ A ∈ A} = Lat
A ∩ M p is the relative invariant subspace lattice of A in M . A subspacelattice is a nest if it is linearly ordered by inclusion. Gilfeather and Larson in [10] studiedthe relative invariant subspace lattices of certain subalgebras in a von Neumann algebraand asked: If A is a subalgebra in a factor von Neumann algebra M such that A + A ∗ isdense in M in some topology for which multiplication is separately continuous, is Lat M A a nest? Although this question was negatively answered for the case in which M = B ( H )in [1], it is still interesting to consider those subalgebras with additional properties. Forexample, how is the relative invariant subspace lattice of a maximal subdiagonal algebra?In [12], we shown that the relative invariant subspace lattice of a finite subdiagonal algebrais commutative. As an application of Theorem 4.2, we prove that the relative invariantsubspace lattice Lat M A of a type 1 subdiagonal algebra A in M is commutative.We note that the centralizer M ϕ of von Neumann algebra M associated with ϕ is definedto be the set M ϕ = { A ∈ M : ϕ ( AB ) = ϕ ( BA ) , ∀ B ∈ M} (cf.[20, 25]). In fact, M ϕ is just the ONCOMMUTATIVE H p SPACES 13 fixed point algebra of M with respect to { σ ϕ t } t ∈ R : M ϕ = { A ∈ M : σ ϕ t ( A ) = A , ∀ t ∈ R } . Weremark that M ϕ is a finite von Neumann algebra. Theorem 5.1.
Let A be a type 1 subdiagonal algebra of M . Then the relative invariantsubspace lattice Lat M A of A in M is commutative.Proof. Take any Q ∈ Lat M A and let B be the σ -weakly closed subalgebra generated by Q and A . Then B is a σ ϕ t -invariant type 1 subdiagonal algebra by Theorem 4.2. Now B canbe expressed as B = ∨{ Q A Q } + ∨{ Q A ( I − Q ) } + ∨{ ( I − Q ) A ( I − Q ) } . Since Q ∈ Lat M A , Q A ∗ ( I − Q ) =
0. It follows that Q M ( I − Q ) = ∨{ Q ( A + A ∗ )( I − Q ) } = ∨{ Q A ( I − Q ) } .Thus we have Q M ( I − Q ) ⊆ B and Ψ ( QA ( I − Q )) = A ∈ M . This impliesthat ϕ ( QA ( I − Q )) = ϕ ◦ Ψ ( QA ( I − Q )) = A ∈ M . That is Q M ( I − Q ) ⊆ ker ϕ . Since ϕ is a state, we also have that ( I − Q ) M Q ⊆ ker ϕ . For every T ∈ M , ϕ ( T Q ) = ϕ ( QT Q ) + ϕ (( I − Q ) T Q ) and ϕ ( QT ) = ϕ ( QT Q ) + ϕ ( QT ( I − Q )). Then we have ϕ ( T Q ) = ϕ ( QT ). It implies that Q ∈ M ϕ . Therefore, Lat M A ⊆ M ϕ and we in turn obtainthat Lat M A ⊆ Lat M ϕ ( A ∩ M ϕ ). Note that M ϕ is a finite von Neumann algebra and A ∩ M ϕ is a finite subdiagonal algebra by [17, Corollary 2.5]. Thus Lat M ϕ ( A ∩ M ϕ ) is commutativeby [12, Theorem 2.1]. It is trivial that Lat M A is also commutative. (cid:3) Corollary 5.2.
Let A be a type 1 subdiagonal algebra of M . Then for any Q ∈ Lat M A ,Q M ( I − Q ) ⊆ A .Proof. Note that Q ∈ M ϕ and Q M ( I − Q )) ⊆ ker ϕ . Thus tr( T h Q ) = tr( QT h ) = ϕ ( QT ) = ϕ ( T Q ) = tr( T Qh ) for all T ∈ M . That is, tr( T ( h Q − Qh )) = T ∈ M . Thus Qh = h Q . Furthermore, tr( Ah QT ( I − Q )) = tr( AQh T ( I − Q )) = tr(( I − Q ) AQh T ) = A ∈ A . It follows that QT ( I − Q ) ∈ ( H ) ⊥ = A for all T ∈ M . (cid:3) Similar to [12, Theorem 2.4], we next show that
Lat M A is a complete lattice generatedby a nest in Lat M A together with the lattice of projections in the center of M . Theorem 5.3.
Let A be a type 1 subdiagonal algebra of M . Then the relative invariantsubspace lattice Lat M A of A in M is the complete lattice generated by a nest in Lat M A together with the lattice of projections in the center of M .Proof. It is known that
Lat M A is commutative by Theorem 5.1. Take any E , F ∈ Lat M A .Then E M ( I − E ) ⊆ A by Corollary 5.2. Thus we get ( I − F ) E M ( I − E ) F ⊆ ( I − F ) A F = { } and we in turn have C E ( I − F ) C ( I − E ) F =
0, where C P denotes the central carrier of a projection P . Note that
Lat M A contains the lattice of projections in the center of M . We next showthat Lat M A satisfies the condition of [11, Theorem 4.2]. In fact, for E , F ∈ Lat M A , we have C E ( I − F ) C ( I − E ) F =
0, then E ( I − F ) ≤ C E ( I − F ) while ( I − E ) F ≤ I − C E ( I − F ) . It follows that thereflexive operator algebra AlgLat M A = { T ∈ B ( H ) : ( I − P ) T P = , ∀ P ∈ Lat M A } with commutative subspaces lattice Lat M A is a nest subalgebra of a von Neumann algebra R with a nest N in Lat M A by [11, Theorem 4.2]. Thus AlgLat M A = Alg
N ∩ R . Notethat R ⊇ AlgLat M A ⊇ AlgLat M A ∩ M ⊇ A . Then R ⊇ [ A + A ∗ ] ∞ = M . Therefore, AlgLat M A ∩ M = Alg
N ∩ R ∩ M = Alg
N ∩ M . Similar to [12, Theorem 2.4], we havethe desired result. (cid:3)
Corollary 5.4.
Let A be a type 1 subdiagonal algebra of M . If M is a factor, then therelative invariant subspace lattice Lat M A of A is a nest. R eferences [1] M. Anoussis, A. Katavolos and M. Lambrou, On the reflexive algebra with two invariant subspaces,
J.Operator Theory, (1993), 267-299.[2] W. B. Arveson, Analyticity in operator algebras , Amer. J. Math. (1967), 578-642.[3] D. P. Blecher and L. E. Labuschagne, Characterizations of Noncommutative H ∞ , Integr. Equ. Oper.Theory (2006), 301-321.[4] D. P. Blecher and L. E. Labuschagne, Noncommutative function theory and unique extension , StudiaMath. (2007),177-195.[5] D. P. Blecher and L. E. Labuschagne,
Applications of the Fuglede-Kadison determinant: Szeg ¨ o’s theo-rem and outers for noncommotative H p , Trans. Amer. Math. Soc. (2008),6131-6147.[6] D. P. Blecher and L. E. Labuschagne, von Neumann algebraic H p theory , Function spaces,89-114,Contemp. Math. , Amer. Math. Soc., Providence, RI, 2007.[7] D. P. Blecher and L. E. Labuschagne, A Beurling theorem for noncommutative L p , J. Operator Theory (2008), 29-51.[8] R. Exel, Maximal subdiagonal algebras , Amer. J. Math. (1988), 775-782.[9] U. Haagreup, L p -spaces associated with an arbitray von Neumann algebra , alg` e bres d’op` e rateurs etleurs applications en physigue math` e matique (Collloques internationaux du CNRS, No.274, Marseille20–24, Juin, 1977), 175–184, Editions du CNRS, Paris, 1979.[10] F. Gilfeather and D. R. Larson, Structure in reflexive subspace lattices , J. London Math. Soc., (1982),117-131.[11] F. Gilfeather and D. R. Larson, Nest-subalgebras of von Neumann algebras , Adv. Math., (1982),171-199.[12] G. X. Ji, Relative lattices of certain analytic operator algebras , Houston J. Math. (2002), 183-191.[13] G. X. Ji, A noncommutative version of H p and characterizations of subdiagonal algebras , Integr. Equ.Oper. Theory (2012), 191-202.[14] G. X. Ji, Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra ,Sci. China Math. (2014), 579-588.[15] G. X. Ji, Subdiagonal algebras with Beurling type invariant subspaces , J. Math. Anal. Appl.(2)480(2019), 123409[16] G. X. Ji,
Maximality and finiteness of type 1 subdiagonal algebras , Proc. Amer. Math. Soc. DOI:https: // doi.org / / proc / ONCOMMUTATIVE H p SPACES 15 [17] G. X. Ji, T. Ohwada and K.-S. Saito,
Certain structure of subdiagonal algebras , J. Operator Theory (1998), 309-317.[18] G. X. Ji and K.-S. Saito, Factorization in subdiagonal algebras , J. Funct. Anal. (1998), 191-202.[19] M. Junge and D. Sherman,
Noncommutative L p modules , J. Operator Theory (2005), 3-34.[20] T. R. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras II , Academic Press,San Diego, 1986.[21] H. Kosaki,
Positive cones associated with a von neumann algebra , Math. Scand. (1980), 295-307.[22] H. Kosaki, Positive cones and L p -spaces associated with a von neumann algebra , J. Operator Theory (1981), 13-23.[23] L. E. Labuschagne, Invariant subspaces for H spaces of σ -finite algebras , Bull. Lond. Math. Soc. (2017), 33-44.[24] M. Marsalli and G. West, Noncommutative H p spaces , J. Operator Theory (1998), 339-355.[25] M. Takesaki, Theory of Operator Algebras II , Springler-Verlag Berlin Heidelberg New York, 1980.[26] M. Terp, L p -spaces associated with von Neumann algebras , Report No.3, University of Odense, 1981.[27] Q. Xu, On the maximality of subdiagonal algebras , J. Operator Theory (2005), 137-146.S chool of M athematics and S tatistics , S haanxi N ormal U niversity , X ian , 710119, P eople ’ s R epublic of C hina Email address : [email protected] S chool of M athematics and S tatistics , S haanxi N ormal U niversity , X ian , 710119, P eople ’ s R epublic of C hina Email address ::