A Beurling-Blecher-Labuschagne theorem for Haagerup noncommutative L p spaces
aa r X i v : . [ m a t h . OA ] J un A BEURLING-BLECHER-LABUSCHAGNE THEOREM FORHAAGERUP NONCOMMUTATIVE L p SPACES
TURDEBEK N. BEKJAN AND MADI RAIKHAN
Abstract.
Let M be a σ -finite von Neumann algebra, equipped with a nor-mal faithful state ϕ , and let A be maximal subdiagonal subalgebra of M . Weprove a Beurling-Blecher-Labuschagne theorem for A -invariant subspaces of L p ( A ) when 1 ≤ p < ∞ . As application, we give a characterization of outeroperators in Haagerup noncommutative H p -spaces associated with A . Introduction
Let M be a finite von Neumann and A be its Arveson’s maximal subdiagonalsubalgebras (see [1]). In [6], Blecher and Labuschagne extend the classical Beurl-ing’s theorem to describe closed A -invariant subspaces in noncommutative space L p ( M ) with 1 ≤ p ≤ ∞ . Sager [20] extends the work of Blecher and Labuschagnefrom a finite von Neumann algebra to semifinite von Neumann algebras, proveda Beurling-Blecher-Labuschagne theorem for A -invariant spaces of L p ( M ) when0 < p ≤ ∞ . The Beurling theorem has been generalized to the setting of unitarilyinvariant norms on finite and semifinite von Neumann algebras (see [3], [9], [21]).When A is subdiagonal subalgebra of σ -finite von Neumann M , Labuscha- gne[17] showed that a Beurling type theory of invariant subspaces of noncommutative H ( A ) spaces holds true. A motivation for this paper is to extend the result in[17] to the setting of the Haagerup noncommutative L p -spaces for 1 ≤ p < ∞ .Blecher and Labuschagne [7] studied outer operators of the noncommutative H p spaces associated with Arveson’s subdiagonal subalgebras. They proved inner-outer factorization theorem and characterization of outer elements for the case1 ≤ p < ∞ (for the case p <
1, see [2]). In [8], they extend their general-ized inner-outer factorization theorem in [7] and establish characterizations ofouter operators that are valid even in the case of elements with zero determi-nant. In this paper, we apply our Beurling-Blecher-Labuschagne theorem for A -invariant subspaces of Haagerup noncommutative spaces L p ( A ) to prove aBlecher-Labuschagne theorem for outer operators in Haagerup noncommutative H p ( A ).The organization of the paper is as follows. In Section 2, we give some defini-tions and related results of Haagerup noncommutative L p -spaces and H p -spaces.A version of Blecher and Labuschagne’s Beurling’s theorem for Haagerup non-commutative L p -spaces is presented in Section 3. In Section 4, we give charac-terizations of outer elements in Haagerup noncommutative H p -spaces. Mathematics Subject Classification.
Primary 46L52; Secondary 47L05.
Key words and phrases. subdiagonal algebras, Beurling’s theorem, invariant sub-spaces,Haagerup noncommutative H p -space. Preliminaries
Our references for modular theory are [18, 22], for the Haagerup noncommu-tative L p -spaces are [11, 23] and for the Haagerup noncommutative H p -spacesare [13, 14]. Let us recall some basic facts about the Haagerup noncommutative L p -spaces and the Haagerup noncommutative H p -spaces, and fix the relevant no-tation used throughout this paper. Throughout this paper M will always denotea σ -finite von Neumann algebra on a complex Hilbert space H , equipped with adistinguished normal faithful state ϕ . Let { σ ϕt } t ∈ R be the one parameter modularautomorphism group of M associated with ϕ . We denote by N = M ⋊ σ ϕ R thecrossed product of M by { σ ϕt } t ∈ R . It is well known that N is the semi-finite vonNewmann algebra acting on the Hilbert space L ( R , H ) , generated by { π ( x ) : x ∈ M} ∪ { λ ( s ) : s ∈ R } , where the operator π ( x ) is defined by( π ( x ) ξ )( t ) = σ ϕ − t ( x ) ξ ( t ) , ∀ ξ ∈ L ( R , H ) , ∀ t ∈ R , and the operator λ ( s ) is defined by( λ ( s ) ξ )( t ) = ξ ( t − s ) , ∀ ξ ∈ L ( R , H ) , ∀ t ∈ R . We will identify M and the subalgebra π ( M ) of N . The operators π ( x ) and λ ( t )satisfy λ ( t ) π ( x ) λ ( t ) ∗ = π ( σ ϕt ( x )) , ∀ t ∈ R , ∀ x ∈ M . Then σ ϕt ( x ) = λ ( t ) xλ ∗ ( t ) , x ∈ M , t ∈ R . We denote by { ˆ σ t } t ∈ R the dual action of R on N , this is a one parameter auto-morphism group of R on N , implemented by the unitary representation { W t } t ∈ R of R on L ( R , H : ˆ σ t ( x ) = W ( t ) xW ∗ ( t ) , ∀ x ∈ N , ∀ t ∈ R , (2.1)where W ( t )( ξ )( s ) = e − its ξ ( s ) , ∀ ξ ∈ L ( R , H ) , ∀ s, t ∈ R . Note that the dual action ˆ σ t is uniquely determined by the following conditions:for any x ∈ M and s ∈ R ,ˆ σ t ( x ) = x and ˆ σ t ( λ ( s )) = e − ist λ ( s ) , ∀ t ∈ R . Hence M = { x ∈ N : ˆ σ t ( x ) = x, ∀ t ∈ R } . Let τ be the unique normal semi-finite faithful trace on N satisfying τ ◦ ˆ σ t = e − t τ, ∀ t ∈ R . Also recall that the dual weight ˆ ϕ of our distinguished state ϕ has the Radon-Nikodym derivative D with respect to τ , which is the unique invertible positiveselfadjoint operator on L ( R , H ) , affiliated with N such thatˆ ϕ ( x ) = τ ( Dx ) , x ∈ N + . BEURLING-BLECHER-LABUSCHAGNE THEOREM FOR HAAGERUP NONCOMMUTATIVE L p SPACES3
Recall that the regular representation λ ( t ) above is given by λ ( t ) = D it , ∀ t ∈ R . Now we define Haagerup noncommutative L p -spaces. Let L ( N , τ ) denote thetopological ∗ -algebra of all operators on L ( R , H ) measurable with respect to( N , τ ). Then the Haagerup noncommutative L p -spaces, 0 < p ≤ ∞ , are definedby L p ( M , ϕ ) = { x ∈ L ( N , τ ) : ˆ σ t ( x ) = e − tp x, ∀ t ∈ R } . The spaces L p ( M , ϕ ) are closed selfadjoint linear subspaces of L ( N , τ ). It is nothard to show that L ∞ ( M , ϕ ) = M . Since for any ψ ∈ M + ∗ , the dual weight ˆ ψ has a Radon-Nikodym derivative withrespect to τ, denoted by D ψ :ˆ ψ ( x ) = τ ( D ψ x ) , x ∈ N + . Then D ψ ∈ L ( N , τ )and ˆ σ t ( D ψ ) = e − t D ψ , ∀ t ∈ R . So D ψ ∈ L ( M , ϕ ) + . It is well known that the map ψ D ψ on M + ∗ extends to a linear homeomorphismfrom M ∗ onto L ( M , ϕ ) (equipped with the vector space topology inherited from L ( N , τ )). This permits to transfer the norm M ∗ into a norm on L ( M , ϕ ),denoted by k·k . Moreover, L ( M , ϕ ) is equipped with a distinguished contractivepositive linear functional tr, defined by tr ( D ψ ) = ψ (1) , ψ ∈ M ∗ . Therefore, k x k = tr ( | x | ) for every x ∈ L ( M , ϕ ).Let 0 < p < ∞ and x ∈ L ( N , τ ). If x = u | x | is the polar decomposition of x ,then x ∈ L p ( M , ϕ ) iff u ∈ M and | x | ∈ L p ( M , ϕ ) iff u ∈ M and | x | p ∈ L ( M , ϕ ) . If we define k x k p = k| x | p k p , ∀ x ∈ L p ( M , ϕ ) , Then for 1 ≤ p < ∞ (resp. 0 < p < L p ( M , ϕ ) , k · k p )is a Banach space (resp. a quasi-Banach space), and k x k p = k x ∗ k p = k| x |k p , ∀ x ∈ L p ( M , ϕ ) . It is proved in [11] and [23] that L p ( M , ϕ ) is independent of ϕ up to isometry.Hence, we denote L p ( M , ϕ ) by L p ( M ).The usual Holder inequality also holds for the L p ( M ) spaces. It mean thatthe product of L ( N , τ ) , ( x, y ) xy , restricts to a contractive bilinear map L p ( M ) × L q ( M ) → L r ( M ), where r = p + q . In particular, if p + q = 1 then TURDEBEK N. BEKJAN AND MADI RAIKHAN the bilinear form ( x, y ) tr ( xy ) defines a duality bracket between L p ( M ) and L q ( M ), for which L q ( M ) coincides (isometrically) with the dual of L p ( M ) (if p = ∞ ). Moreover, the tr have the following property: tr ( xy ) = tr ( yx ) , ∀ x ∈ L p ( M ) , ∀ y ∈ L q ( M ) . Let 0 < p ≤ ∞ . For K ⊂ L p ( M ), we denote the closed linear span of K in L p ( M ) by [ K ] p (relative to the w*-topology for p = ∞ ) and the set { x ∗ : x ∈ K } by J ( K ).For 0 < p < ∞ , ≤ η ≤
1, we have that L p ( M ) = [ D − ηp M D ηp ] p . Proposition 2.1.
Let ≤ p, q, r < ∞ and q + r = p (resp. q − r = p ). If K is aclosed subspace of L q ( M ) , then [[ KD r ] p D − r ] q = K (resp. [[ KD − r ] p D r ] q = K ).Proof. We use same method as in the proof of Lemma 1.1 in [15]. Since D r ∈ L r ( M ), ˆ σ t ( D r ) = e − tr D r , ∀ t ∈ R . Hence, 1 = ˆ σ t ( D − r D r ) = e − tr D r ˆ σ t ( D − r ) , ∀ t ∈ R , so that ˆ σ t ( D − r ) = e tr D − r , ∀ t ∈ R . Moreover, for a ∈ L p ( M ), we haveˆ σ t ( aD − r ) = ˆ σ t ( a )ˆ σ t ( D − r ) = e − tp + tr aD − r = e − tq aD − r , ∀ t ∈ R . Thus aD − r ∈ L q ( M ) and L p ( M ) D − r ⊂ L q ( M ).Its clear that K ⊆ [[ KD r ] p D − r ] q . Let y ∈ L q ′ ( M ) such that tr ( yx ) = 0 for all x ∈ K , where q ′ is the conjugate index of q . If z ∈ [ KD r ] p , then we can find asequence ( x n ) ⊂ K , such that lim n →∞ k x n D r − z || p = 0. On the other hand, wehave that zD − r ∈ L q ( M ) , D − r y ∈ L p ′ ( M ), where p ′ is the conjugate index of p . Hence, tr ( yzD − r ) = tr ( zD − r y ) = lim n →∞ tr ( x n D r D − r y ) = lim n →∞ tr ( x n y ) = 0 . It follows that tr ( yw ) = 0 for all w ∈ [[ KD r ] p D − r ] q . Then by the Hahn-Banachtheorem, we obtain the desired result. The alternative claim follows analogously. (cid:3) Let D be a von Neumann subalgebra of M . Then there is an (unique) nor-mal faithful conditional expectation E of M with respect to D which leaves ϕ invariant. Definition 2.2.
A w*-closed subalgebra A of M is called a subdiagonal subal-gebra of M with respect to E (or to D ) if(1) A + J ( A ) is w*-dense in M ,(2) E ( xy ) = E ( x ) E ( y ) , ∀ x, y ∈ A , (3) A ∩ J ( A ) = D , BEURLING-BLECHER-LABUSCHAGNE THEOREM FOR HAAGERUP NONCOMMUTATIVE L p SPACES5
The algebra D is called the diagonal of A .If A is not properly contained in any other subalgebra of M which is a subdi-agonal with respect to E , We call A is a maximal subdiagonal subalgebra of M with respect to E . Let A = { x ∈ A : E ( x ) = 0 } Then by Theorem 2.2.1 in [1] , A is maximal if and only if A = { x ∈ M : E ( yxz ) = E ( yxz ) = 0 , ∀ y ∈ A , ∀ z ∈ A } . It follows from Theorem 2.4 in [12] and Theorem 1.1 in [24] that a subdiagonalsubalgebra A of M with respect to D is maximal if and only if σ ϕt ( A ) = A , ∀ t ∈ R . (2.2)In this paper A always denotes a maximal subdiagonal subalgebra in M withrespect to E . Definition 2.3.
For 0 < p < ∞ , we define the Haagerup noncommutative H p -space that H p ( A ) = [ A D p ] p , H p ( A ) = [ A D p ] p . If 1 ≤ p < ∞ , ≤ η ≤
1, then by Proposition 2.1 in [14], we have that H p ( A ) = [ D − ηp A D ηp ] p , H p ( A ) = [ D − ηp A D ηp ] p . (2.3)By Proposition 2.7 in [4], we know that A = { x ∈ M : tr ( xa ) = 0 , ∀ a ∈ H ( A ) } . (2.4)It is known that L p ( D ) = [ D − ηp D D ηp ] p , ∀ p ∈ [1 , ∞ ) , ∀ η ∈ [1 , . (2.5)For 1 ≤ p ≤ ∞ , the conditional expectation E extends to a contractive projectionfrom L p ( M ) onto L p ( D ). The extension will be denoted still by E . Let1 ≤ r, p, q ≤ ∞ , r = 1 p + 1 q . Then E ( xy ) = E ( x ) E ( y ) , ∀ x ∈ H p ( A ) , ∀ y ∈ H q ( A ) . Let M a be the family of analytic vectors in M . Recall that x ∈ M a if only ifthe function t σ t ( x ) extends to an analytic function from C to M . M a is aw*-dense ∗ -subalgebra of M (cf.[18]).The next result is known. For easy reference, we give its proof (see the proofof Theorem 2.5 in [13]) Lemma 2.4.
Let A a and D a be respectively the families of analytic vectors in A and D . If ≤ p < ∞ , then:(1) A a is a w*-dense in A , ( A a ) is a w*-dense in A and D a is a w*-densein D , where ( A a ) = { x ∈ A a : E ( x ) = 0 } ; TURDEBEK N. BEKJAN AND MADI RAIKHAN (2) D ± p A a = A a D ± p , D ± p ( A a ) = ( A a ) D ± p , D ± p D a = D a D ± p ; (3) A a D p is dense in H p ( A ) , ( A a ) D p is dense in H p ( A ) and D a D p is densein L p ( D ) .Proof. (1) Let x ∈ A . We define x n = r nπ Z R e − nt σ t ( x ) dt. By (2.2), x n ∈ A . Moreover by [[18], p. 58], x n ∈ A a and x n → x w*-weakly.Since σ ϕt ( A ) = A , σ ϕt ( D ) = D , ∀ t ∈ R (see [12], p.313), a similar argumentworks for A and D .(2) We prove only the first equivalence. The proofs of the two others are similar.Let x ∈ A a . Then D p x = [ D ± p xD ∓ p ] D ± p = [ σ ∓ ip ( x )] D ± p ∈ A a D ± p , whence xD ± p ⊆ A a D ± θp . The inverse inclusion can be proved in a similar way.(3) Let p ′ be the conjugate index of p . If y ∈ L p ′ ( M ) such that tr ( aD p y ) =0 , ∀ a ∈ A a , then by (1), tr ( aD p y ) = 0 , ∀ a ∈ A , since D p y ∈ L ( M ). Hence, by (2.3), tr ( xy ) = 0 , ∀ x ∈ H p ( A )By the Hahn-Banach theorem, A a D p is dense in H p ( A ). Similarly, we can provethe two others. (cid:3) Corollary 2.5. If ≤ p, q, r < ∞ and q + r = p , then [ H q ( A ) D r ] p = H p ( A ) . Proposition 2.6.
Let ≤ p, q, r < ∞ and q − r = p . Then [ H q ( A ) D − r ] p = H p ( A ) and [ L q ( D ) D − r ] p = L p ( D ) .Proof. From the proof of Proposition 2.1, it follows that [ H q ( A ) D − r ] p ⊂ L p ( M ).It is obvious that H p ( A ) ⊆ [ H q ( A ) D − r ] p . Let g ∈ L p ′ ( A ) such that tr ( gh ) = 0for all h ∈ H p ( A ), where p ′ is the conjugate index of p . If f ∈ H q ( A ), then byCorollary 2.5, we can find a sequence ( h n ) ⊂ H p ( A ), such that lim n →∞ k h n D r − f || q = 0. Note that f D − r ∈ L p ( M ) , D − r g ∈ L q ′ ( M ), where q ′ is the conjugateindex of q . Hence, tr ( gf D − r ) = tr ( f D − r g ) = lim n →∞ tr ( h n D r D − r g ) = lim n →∞ tr ( h n g ) = 0 . Consequently, tr ( gw ) = 0 for all w ∈ [ H q ( A ) D − r ] p . Using the Hahn-Banachtheorem, we obtain the desired result.The same reasoning applies to the second result. (cid:3) BEURLING-BLECHER-LABUSCHAGNE THEOREM FOR HAAGERUP NONCOMMUTATIVE L p SPACES7 A -invariant subspaces of L p ( M )We recall that a right (resp. left) A -invariant subspace of L p ( M ), is a closedsubspace K of L p ( M ) such that K A ⊂ K (resp. A K ⊂ K ).Let K be a right A -invariant subspace of L ( M ). Then W = K ⊖ [ K A ] isoften called the right wandering subspace of K . We say that K is type 1 if W generates K as an A -module (that is K = [ W A ] ) and K is type 2 if W = 0 (see[6]). Lemma 3.1.
Let ≤ p < ∞ , and let K be an A -invariant subspace of L p ( M ) .If ≤ q, r < ∞ and p − r = q (resp. p + r = q ), then [ KD − r ] q (resp. [ KD r ] q )is a right A -invariant subspace of L q ( M ) .Proof. It is clear that [ KD − r ] q is a closed subspace of L q ( M ). Using (2) ofLemma 2.4, we get that KD − r A a = K A a D − r ⊂ KD − r , and so [ KD − r ] q A a ⊂ [ KD − r A a ] q ⊂ [ KD − r ] q . On the other hand, if a ∈ A , thenfrom the proof of (1) in Lemma 2.4, we obtain a sequence ( a n ) ∈ A a such that a n → a w*-weakly. Hence, tr ( xD − r a n y ) → tr ( xD − r ay ) , ∀ x ∈ K, ∀ y ∈ L q ′ ( M ) , where q ′ is the conjugate index of q . Since the weak closure of KD − r A a is equalto [ KD − r A a ] q , xD − r a ∈ [ KD − r A a ] q . Thus [ KD − r A ] q ⊂ [ KD − r ] q . It follows that [ KD − r ] q A ⊂ [ KD − r ] q . The alter-native claim follows analogously. (cid:3) Using same method as in the proof of Lemma 3.1, we get the following result.
Lemma 3.2.
Let ≤ p < ∞ , and let K ⊂ L p ( M ) . If ≤ q, r < ∞ and p − r = q (resp. p + r = q ), then [ K A D − r ] q = [ KD − r A ] q , [ K A D − r ] q = [ KD − r A ] q and [ K D D − r ] q = [ KD − r D ] q (resp. [ K A D r ] q = [ KD r A ] q , [ K A D r ] q = [ KD r A ] q and [ K D D r ] q = [ KD r D ] q ). Definition 3.3.
Let 1 ≤ p < ∞ , and let K be a right A -invariant subspace of L p ( M ). If 1 ≤ p ≤ , p − r = (resp. 2 ≤ p < ∞ p + r = ) and W is the rightwandering subspace of [ KD − r ] (resp. [ KD r ] ), we define the right wanderingsubspace of K to be the L p -closure of [ W D r ] p (resp. [ W D − r ] p ).If K is a right A -invariant subspace of L p ( M ), we say that K is type 1 if theright wandering subspace of K generates K as an A -module, and K is type 2 if K = [ K A ] p .We can extend the result in [17] to the setting of the Haagerup noncommutative L p -spaces for 1 ≤ p < ∞ . Theorem 3.4.
Let ≤ p < ∞ , and let K is a right A -invariant subspace of L p ( M ) . Then: TURDEBEK N. BEKJAN AND MADI RAIKHAN (1) K may be written uniquely as a L p -column sum Z ⊕ col [ Y A ] p , where Z isa type 2 right A -invariant subspace of L p ( M ) , Y is the right wanderingsubspace of K such that Y = [ Y D ] p and J ( Y ) Y ⊂ L p ( D ) .(2) If K = { } then K is type 1 if and only if K = ⊕ coli u i H p ( A ) , for u i partialisometries with mutually orthogonal ranges and u ∗ i u i ∈ D .(3) If K = K ⊕ col K where K and K are types 2 and 1 respectively, thenthe right wandering subspace for K equals the right wandering subspacefor K .(4) The wandering quotient K/ [ K A ] p is isometrically D -isomorphic to theright wandering subspace of K.(5) The wandering subspace W of K is an L p ( D ) -module in the sense of Jungeand Sherman.Proof. (1) We prove the case 1 ≤ p ≤
2. The proof of the case 2 ≤ p < ∞ is similar. Let p − r = . Then by Lemma 3.1, K ′ = [ KD − r ] is a right A -invariant subspace of L ( M ). Using Theorem 2.3 and 2.8 in [17], we have that K ′ = Z ′ ⊕ col [ Y ′ A ] , where Z ′ is a type 2 right A -invariant subspace of L ( M ) and Y ′ is the right wandering subspace of K ′ with Y ′ = [ Y ′ D ] and J ( Y ′ ) Y ′ ⊂ L ( D ).Let Z = [ Z ′ D r ] p and Y = [ Y ′ D r ] p . By Lemma 3.1 and Definition 3.3, Z is aright A -invariant subspaces of L p ( M ) and Y is the right wandering subspace of K . For any x ∈ Z ′ , y ∈ [ Y ′ A ] , we have that x ∗ ya = 0, and so D r x ∗ yD r = 0.Hence, J ( Z )[ Y A ] p = { } . On the other hand, by Proposition 2.1, K = [ K ′ D r ] p .Therefore, K = Z ⊕ col [ Y A ] p . Since Z ′ = [ Z ′ A ] , Y ′ = [ Y ′ D ] , by Lemma 3.2, Z = [ Z ′ D r ] p = [[ Z ′ A ] D r ] p = [ Z ′ A D r ] p = [ Z ′ D r A ] p = [ Z A ] p and Y = [ Y ′ D r ] p = [[ Y ′ D ] D r ] p = [ Y ′ D D r ] p = [ Y ′ D r D ] p = [ Y D ] p . Suppose that Z is a type 2 right A -invariant subspace of L p ( M ) and Y is theright wandering subspace of K such that K = Z ⊕ col [ Y A ] p and Y = [ Y D ] p .Let Z ′ = [ Z D − r ] and Y ′ = [ Y D − r ] . By Lemma 3.2, we know that K ′ = Z ′ ⊕ col [ Y ′ A ] , Z ′ is a type 2 right A -invariant subspace of L ( M ) and Y ′ is theright wandering subspace of K ′ with Y ′ = [ Y ′ D ] . By by the uniqueness assertionin Theorem 2.3 of [17], Z ′ = Z ′ and Y ′ = Y ′ . Using Proposition 2.1, we obtainthat Z = [ Z ′ D r ] p = [ Z ′ D r ] p = [[ Z D − r ] D r ] p = Z and Y = [ Y ′ D r ] p = [ Y ′ D r ] p = [[ Y D − r ] D r ] p = Y . Since J ( Y ′ D r ) Y ′ D r = D r J ( Y ′ ) Y ′ D r ⊂ D r L ( D ) D r ⊂ L p ( D ), it follows that J ( Y ) Y ⊂ L p ( D ).(2) Let 1 ≤ p ≤ ≤ p < ∞ ). If K = { } and K is type 1, then K = [ W A ] p , where W is the right wandering space of K . By Definition 3.3 andProposition 2.1, we know that [ W D − r ] (resp. [ W D r ] ) is the right wandering BEURLING-BLECHER-LABUSCHAGNE THEOREM FOR HAAGERUP NONCOMMUTATIVE L p SPACES9 subspace of the right A -invariant subspace [ KD − r ] (resp. [ KD r ] ) of L ( M ).Applying Proposition 2.1 and Lemma 3.2, we obtain that[[ W D − r ] A ] = [ W D − r A ] = [ W A D − r ] = [[ W A ] p D − r ] = [ KD − r ] (resp . [[ W D r ] A ] = [ W D r A ] = [ W A D r ] = [[ W A ] p D r ] = [ KD r ] ) . Hence, [ KD − r ] ( resp. [ KD r ] ) is type 1. So, by Theorem 2.8 in [17], [ KD − r ] = ⊕ coli u i H ( A ) (resp. [ KD r ] = ⊕ coli u i H ( A )), for u i partial isometries with mutu-ally orthogonal ranges and u ∗ i u i ∈ D . Since u i [ H ( A ) D r ] p (resp. u i [ H ( A ) D − r ] p )is a closed subspace in L p ( M ) for any i , using Proposition 2.1 and 2.6, we get K = [[ KD − r ] D r ] p = ⊕ coli [ u i H ( A ) D r ] p = ⊕ coli u i [ H ( A ) D r ] p = ⊕ coli u i [ H p ( A )] p (resp . K = [[ KD r ] D − r ] p = ⊕ coli [ u i H ( A ) D − r ] p = ⊕ coli u i [ H p ( A )] p ) . If K = ⊕ coli u i H p ( A ), for u i as above, then[ KD − r ] = ⊕ coli u i H ( A ) (resp. [ KD r ] = ⊕ coli u i H ( A )) . So [ KD − r A ] = ⊕ coli u i H ( A ) (resp. [ KD r A ] = ⊕ coli u i H ( A )). From this it iseasy to argue that the right wandering subspace W of [ KD − r ] (resp. [ KD r ] )satisfies W = ⊕ coli u i L ( D ). By Definition 3.3 and Proposition 2.6, ⊕ coli u i L p ( D ) isthe right wandering subspace of K . Hence, [ ⊕ coli u i L p ( D ) A ] p = ⊕ coli u i H p ( A ) = K ,and so K is type 1.(3) If 1 ≤ p ≤ ≤ p < ∞ ), then [ KD − r ] (resp. [ KD r ] ) is aright A -invariant subspace of L ( M ) and [ KD − r ] = [ K D − r ] ⊕ col [ K D − r ] (resp. [ KD r ] = [ K D r ] ⊕ col [ K D r ] ). From the proof of (1) and (2), itfollows that [ K D − r ] and [ K D − r ] (resp. [ K D r ] and [ K D r ] ) are types 2and 1 respectively. By Proposition 2.7 in [17], the right wandering subspace for[ KD − r ] (resp. [ KD r ] ) equals the right wandering subspace for [ K D − r ] (resp.[ K D r ] ). This gives the result.(4) By (1), (2) and (3), we get that K = Z ⊕ coli u i H p ( A ), where Z is a type 2,and u i are partial isometries with mutually orthogonal ranges such that u ∗ i u i ∈ D and ⊕ coli u i L p ( D ) is the right wandering subspace of K . Using the properties of E , similar to the proof (2) of Theorem 4.5 in [6], we prove the desired result. Weomit the details.(5) Since J ( W ) W ⊂ L p ( D ), W is a right L p ( D )-module with inner product h ξ, η i = ξ ∗ η (see Definition 3.3 in [16]). (cid:3) Proposition 3.5.
Let K is a right A -invariant subspace of L ( M ) , and let W be the right wandering subspace of K . If W has a cyclic and separating vectorfor the D -action, then there is an isometry u ∈ M such that W = uL ( D ) .Proof. By an adaption of an argument from [16]) (see p.13) there exists an iso-metric D -module isomorphism ψ : L ( D ) → W . Let h = ψ ( D ) ∈ W . Then tr ( d ∗ h ∗ hd ) = k ψ ( D d ) k = tr ( d ∗ Dd ), for each d ∈ D . By (1) of Theorem 3.4, h ∗ h ∈ L ( D ), and so h ∗ h = D . Hence there exists an isometry u with initialprojection 1 such that h = uD . The modular action of ψ we will then have that ψ ( D d ) = ψ ( D ) d = uD d for any d ∈ D . Since L ( D ) = [ D D ], it follows that ψ ( L ( D )) = uL ( D ). Thus W = uL ( D ) and u ∗ u = 1. (cid:3) Similar to the above Proposition, we have the following result.
Proposition 3.6.
Let K is a left A -invariant subspace of L ( M ) , and let W be the left wandering subspace of K . If W has a cyclic and separating vectorfor the D -action, then there is a partial isometry v ∈ M such that vv ∗ = 1 and W = L ( D ) v . Outer elements of H p ( A ) Definition 4.1.
Let 0 < p ≤ ∞ . An operator h ∈ H p ( A ) is called left outer , right outer or bilaterally outer according to [ h A ] p = H p ( A ), [ A h ] p = H p ( A ) or[ A h A ] p = H p ( A ). Proposition 4.2.
Let ≤ p < ∞ , and let h ∈ H p ( A ) . The following areequivalent:(1) h is a bilaterally outer;(2) E ( h ) is a bilaterally outer in L p ( D ) and [ AhA ] p = [ A hA ] p = H p ( A ) ;(3) E ( h ) is a bilaterally outer in L p ( D ) and E ( h ) − h ∈ [ AhA ] p = [ A hA ] p .Proof. (i) ⇒ (ii). If h is a bilaterally outer, then for D p there exist two sequence( a n ) , ( b n ) ∈ A such that k a n hb n − D p k p → n → ∞ . (4.1)By continuity of E , we get kE ( a n ) E ( h ) E ( b n ) − D p k p → n → ∞ . Hence, by(2.5), we have that L p ( D ) = [ D p D ] p ⊂ [ DE ( h ) D ] p ⊂ L p ( D ) . So, E ( h ) is a bilaterally outer in L p ( D ). Using (2.3) and (4.1) we deduce that[ AhA ] p = [ A hA ] p = H p ( A ).(ii) ⇒ (iii) is trivial.(iii) ⇒ (i). It is clear that D p ∈ [ DE ( h ) D ] p ⊂ [ AE ( h ) A ] p and h ∈ [ A h A ] p .Hence, E ( h ) = ( E ( h ) − h ) + h ∈ [ A h A ] p . It follows that D p ∈ [ A h A ] p . By (2.3),we obtain that H p ( A ) = [ A h A ] p . (cid:3) Similar to Proposition 4.2, we have the following result.
Proposition 4.3.
Let ≤ p < ∞ , and let h ∈ H p ( A ) . The following areequivalent:(1) h is a left outer (resp. a right outer);(2) E ( h ) is a left outer (resp. a right outer) in L p ( D ) and [ hA ] p = H p ( A ) (resp. [ A h ] p = H p ( A ) );(3) E ( h ) is a left outer (resp., a right outer) in L p ( D ) and E ( h ) − h ∈ [ hA ] p (resp. E ( h ) − h ∈ [ A h ] p ). We will keep all previous notations throughout this section. We will say that h is outer if it is at the same time left and right outer. BEURLING-BLECHER-LABUSCHAGNE THEOREM FOR HAAGERUP NONCOMMUTATIVE L p SPACES11
Lemma 4.4.
Let < p < ∞ and h ∈ H p ( A ) . Suppose h is outer in H p ( A ) . If h = u | h | is the polar decomposition of h , then u is a unitary.Proof. Since h is a left outer, there exists a sequence a n ∈ A such that ha n → D p in norm in L p ( M ), so ha n → D p in measure. Let l ( h ) be the left supportprojection of h . Then l ( h ) ⊥ ha n → l ( h ) ⊥ D p in measure. On the other hand l ( h ) ⊥ ha n = 0 for all n , hence l ( h ) ⊥ D p = 0. Since D p is invertible, l ( h ) ⊥ = 0.So h must have dense range, i.e. uu ∗ = l ( h ) = 1. Similarly, from the fact that h is a right outer, we obtain that u ∗ u = r ( h ) = 1, where r ( h ) is the right supportprojection of h . Thus u is a unitary. (cid:3) Lemma 4.5.
Let ≤ p < ∞ and let h ∈ H p ( A ) be outer. If ≤ q, r < ∞ and p − r = q (resp. p + r = q ), then hD − r , D − r h ∈ H q ( A ) (resp. hD r , D r h ∈ H q ( A ) ) are outers.Proof. We only prove hD − r is an outer. A similar arguments works for D − r h .By Proposition 2.6, [ H p ( A ) D − r ] q = H q ( A ). We use same method as in theproof of (3) of Lemma 2.4 to obtain that [ h A a ] p = [ A a h ] p = H p ( A ). Hence,[[ h A a ] p D − r ] q = H q ( A ). On the other hand, similarly to the proof of Proposition2.6, we can prove that [[ h A a ] p D − r ] q = [ h A a D − r ] q . Therefore, H q ( A ) = [[ h A a ] p D − r ] q = [ h A a D − r ] q = [ hD − r A a ] q ⊂ [ hD − r A ] q ⊂ H q ( A ) . Thus hD − r is a left outer. Similarly we can show hD − r is a right outer. Thealternative claim follows analogously. (cid:3) Similar to this lemma, we have the following result.
Lemma 4.6.
Let ≤ p < ∞ and let d ∈ L p ( D ) be outer. If ≤ q, r < ∞ and p − r = q (resp. p + r = q ), then dD − r , D − r d ∈ L q ( D ) (resp. dD r , D − r d ∈ L q ( D ) ) are outers. Proposition 4.7.
Let ≤ p < ∞ , and let h ∈ H p ( A ) . If E ( h ) is outer in L p ( D ) , then there is a left outer g ∈ H p ( A ) and an isometry u ∈ A such that h = ug (resp. there is a right outer g ′ ∈ H p ( A ) and v ∈ A such that vv ∗ = 1 and h = g ′ v ).Proof. We prove the case 1 ≤ p ≤
2. The proof of the case 2 ≤ p < ∞ is similar.Let p − r = . By Lemma 4.6, E ( h ) D − r ∈ L ( D ) is outer. Let p ′ be the conjugateindex of p . Then for any d ∈ D , we have that tr ( E ( h ) D − r D d ) = tr ( E ( h ) D p ′ d ) = tr ( E ( hD p ′ d )) = tr ( E ( hD − r ) D d ) . By (2.5), we get tr ( E ( h ) D − r f ) = tr ( E ( hD − r ) f ) , ∀ f ∈ L ( D ) . Hence, E ( hD − r ) = E ( h ) D − r .We consider the orthogonal projection P : [ hD − r A ] → [ E ( hD − r ) D ] . Then P = E | [ hD − r A ] and [ E ( hD − r ) D ] = [ hD − r A ] ⊖ [ hD − r A ] . So E ( hD − r ) isa cyclic separating vector for the wandering subspace [ E ( hD − r ) D ] of [ hD − r A ] .By Proposition 3.5, there exists an isometry u ∈ M such that[ hD − r A ] = uH ( A ) . We may write hD − r = uf , for f ∈ H ( A ). Then[ f A ] = u ∗ u [ f A ] = u ∗ [ hD − r A ] = u ∗ uH ( A ) = H ( A ) , i.e., f is a left outer. On the other hand,0 = tr ( hD − r aD b ) = tr ( u ( f aD b )) , ∀ a ∈ A , ∀ b ∈ A . Since f is a left outer, by Proposition 4.3, [ f A ] = H ( A ). Hence, using (2.3)we obtain that [ f A D A ] = H ( A ). It follows that 0 = tr ( ua ) for any a ∈ H .By (2.4), u ∈ A . Let g = f D r . By Lemma 4.5, g is a left outer. This givesdesired result.The second result is proved similarly as the first one by using Lemma 4.6,Proposition 3.5 and Proposition 4.3. We omit the details. (cid:3) Lemma 4.8. If x ∈ L ( M ) and u ∈ M is a contraction such that k ux k = k x k ,then x = u ∗ ux .Proof. We have that x ∗ u ∗ ux ≤ x ∗ x and tr ( x ∗ u ∗ ux ) = k ux k = k x k = tr ( x ∗ x ).Hence, k x ∗ x − x ∗ u ∗ ux k = tr ( x ∗ x − x ∗ u ∗ ux ) = 0 , so that x ∗ x = x ∗ u ∗ ux . Thus k (1 − u ∗ u ) x k = k x ∗ (1 − u ∗ u ) x k = 0, therefore(1 − u ∗ u ) x = (1 − u ∗ u ) [(1 − u ∗ u ) x ] = 0, and x = u ∗ ux . (cid:3) The following result extends Theorem 4.4 in [8] to the Haagerup noncommu-tative H p -space case. Theorem 4.9.
Let ≤ p < ∞ , and let h ∈ H p ( A ) . The following are equivalent:(1) h is an outer;(2) E ( hD r ) is an outer in L ( D ) and kE ( hD r ) k = k P ( hD r ) k = k P ′ ( hD r ) k ,where P is the orthogonal projection from [ hD r A ] to [ hD r A ] ⊖ [ h A ] and P ′ is the orthogonal projection from [ A hD r ] to [ A hD r ] ⊖ [ A hD r ] and p + r = .Proof. (1) ⇒ (2). By Lemma 4.5, hD r is outer in H ( A ). Using Proposition 4.3,we obtain that E ( hD r ) is an outer in L ( D ). Since E is a contractive projectionfrom H ( A ) onto L ( D ) with kernel H p ( A ), we deduce that kE ( hD r ) k = inf h ∈ H ( A ) k hD r + h k . Using Proposition 4.3, we obtain that kE ( hD r ) k = inf h ∈ H ( A ) k hD r + h k = inf a ∈A k hD r + hD r a k = k P ( hD r ) k . BEURLING-BLECHER-LABUSCHAGNE THEOREM FOR HAAGERUP NONCOMMUTATIVE L p SPACES13
Similarly, we can prove kE ( hD r ) k = k P ′ ( hD r ) k .(2) ⇒ (1). By Proposition 4.7, hD r = ug where g ∈ H ( A ) is a left outer and u ∈ A is an isometry. Hence, kE ( hD r ) k = kE ( u ) E ( g ) k ≤ kE ( g ) k = inf a ∈ A k g + ga k = inf a ∈ A k u ∗ ( hD r + hD r a ) k ≤ inf a ∈ A k hD r + hD r a k = k P ( hD r ) k = kE ( hD r ) k . This gives kE ( u ) E ( g ) k = kE ( g ) k . Since the left support of E ( g ) is 1, usingLemma 4.8, we obtain that E ( u ) is an isometry. On the other hand, we have that DE ( hD r ) = DE ( u ) E ( g ) ⊂ DE ( g ). Hence, L ( D ) = [ DE ( hD r )] = [ DE ( u ) E ( g )] ⊂ [ DE ( g )] ⊂ L ( D ) , i.e., E ( g ) is a right outer. By Proposition 4.3, E ( g ) is an outer. By Lemma4.4, from E ( hD r ) = E ( u ) E ( g ) follows that E ( u ) is a unitary. Therefore, E (( u −E ( u )) ∗ ( u − E ( u ))) = 0. So u = E ( u ) ∈ D and hD r is a left outer. From the proofof Lemma 4.5, we know that h is a left outer.Using the second result of Proposition 4.7 as in the proof of the above, wededuce that h is a right outer. (cid:3) Acknowledgement.
T.N. Bekjan is partially supported by NSFC grant No.11771372,M. Raikhan is partially supported by project AP05131557 of the Science Com-mittee of Ministry of Education and Science of the Republic of Kazakhstan.
References [1]
W. B. Arveson , Analyticity in operator algebras,
Amer. J. Math. (1967) 578-642.[2] T. N. Bekjan, Q. Xu , Riesz and Szeg¨o type factorizations for noncommutative Hardyspaces,
J. Operator Theory (2009) 215-231.[3] T. N. Bekjan , Noncommutative symmetric Hardy spaces,
Integr. Equ. Oper. Theory 81 (2015) 191-212.[4]
T. N. Bekjan, Madi Raikhan , Interpolation of Haagerup noncommutative Hardyspaces, to apper
Banach J. Math. Anal. [5]
A. Beurling , On two problems concerning linear transformations in Hilbert space,
ActaMath. (1949) 239-255.[6] D. P. Blecher, L. E. Labuschagne , A Beurling theorem for noncommutative L p , J.Operator Theory (2008) 29-51.[7] D. P. Blecher, L. E. Labuschagne , Applications of the Fuglede-Kadison determinant:Szeg¨ o s theorem and outers for noncommutatuve H p , Trans. Amer. Math. Soc. (2008)6131-6147.[8]
D. P. Blecher, L. E. Labuschagne , Outers for noncommutative H p revisited, StudiaMath. (2013) 265-287.[9]
Y. Chen, D. Hadwin, J. Shen , A non-commutative Beurling’s theorem with respect tounitarily invariant norms,
J. Operator Theory (2016) 497-523.[10] A. Connes , Une classification des facteurs de type III,
Ann. Sci. ´Ecole Norm. Sup. (1973) 133-252.[11] U. Haagerup , ‘ L p - spaces associated wth an arbitrary von Neumann algegra’ in Alg`ebresd’op´erateurs et leurs applications en physique math´ematique ( Proc. Colloq., Marseille ,1977),
Colloq. Internat. CNRS (CNRS, Paris, 1979) 175-184.[12]
G. Ji, T. Ohwada, K-S. Saito , Certain structure of subdiagonal algebras,
J. OperatorTheory (1998) 309-317. [13] G. Ji , A noncommutative version of H p and characterizations of subdiagonal algebras, Integr. Equ. Oper. Theory (2012) 183-191.[14] G. Ji , Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonalalgebra,
Sci. China Math. (2014) 579-588.[15] M. Junge , Doob’s inequality for noncommutative martingales,
J. Reine. Angew. Math. (2002) 149-190.[16]
M. Junge,D. Sherman , Noncommutative L p -modules, J. Operator Theory (2005)3-34.[17] L.E. Labuschagne , Invariant subbspaces for H spaces of σ -finite algebras, Bull. LondonMath. Soc. (2017) 33–44.[18] G. K. Pedersen, M. Takesaki , The Radon-Nikodym theorem for von Neuman algebras,
Acta Math. (1973) 53-87.[19]
G. Pisier, Q. Xu ,Noncommutative L p -spaces, In Handbook of the geometry of Banachspaces , Vol. 2 (2003), 1459-1517.[20]
L. Sager , A Beurling-Blecher-Labuschagne theorem for noncommutative Hardy spacesassociated with semifinite von Neumann algebras,
Integr. Equ. Op. Theory (2016) 377-407.[21] L. Sager and W. Liu , A Beurling-Chen-Hadwin-Shen theorem for noncommutativeHardy spaces associated with semifinite von Neumann algebras with unitarily invariantnorms, arXiv:1801.01448[math.OA] .[22]
M. Takesaki , Duality for crossed products and the structure of von Neumann algebrasof type III,
Acta Math. (1973) 249-310.[23]
M. Terp , L p -spaces associated wth an arbitrary von Neumann algegras, Notes , Math.Institute, Copenhagen Univ., 1981.[24] Q. Xu , On the maximality of subdiagonal algebras,
J. Operator Theory (2005) 137-146. College of Mathematics and Systems Science, Xinjiang University, Urumqi830046, China.
E-mail address : [email protected] Faculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National Uni-versity, Astana 010008, Kazakhstan.
E-mail address ::