Integr. Equ. Oper. Theory 72 (2012), 131–149 DOI 10.1007/s00020-011-1920-1 Published online October 28, 2011 c Springer Basel AG 2011 

Integral Equations and Operator Theory

A Noncommutative Version of H p and Characterizations of Subdiagonal Algebras Guoxing Ji Abstract. Let M be a σ-finite von Neumann algebra and A a maximal subdiagonal algebra of M with respect to a faithful normal conditional expectation Φ. Based on Haagerup’s noncommutative Lp space Lp (M) associated with M, we give a noncommutative version of H p space relative to A. If h0 is the image of a faithful normal state ϕ in L1 (M) 1

such that ϕ ◦ Φ = ϕ, then it is shown that the closure of {Ah0p } in Lp (M) for 1 ≤ p < ∞ is independent of the choice of the state preserving Φ. Moreover, several characterizations for a subalgebra of the von Neumann algebra M to be a maximal subdiagonal algebra are given. Mathematics Subject Classification (2010). Primary 46L52, 47L75; Secondary 46K50, 46J15. Keywords. von Neumann algebra, subdiagonal algebra, noncommutative H p space.

1. Introduction The classical Hardy spaces H p (D), 1 ≤ p ≤ ∞ have played an important role in modern analysis. In 1960’s, the theory was generalized to the setting of abstract function algebras, such as weak∗ Dirichlet algebras in [26] by Srinivasan and Wang. Arveson in 1967 introduced the notion of subdiagonal algebras in von Neumann algebras to give a noncommutative analogue of weak∗ Dirichlet algebras in [1]. This concept unify the analysis of several classes of nonselfadjoint operator algebras and subsequently is studied by several authors. In particular, based on noncommutative Lp (1 ≤ p ≤ ∞) spaces, noncommutative H p spaces on a finite von Neumann algebra with a faithful normal trace are extensively studied (cf. [2–6,21,22,24] and references This research was supported by the National Natural Science Foundation of China (No. 10971123), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20090202110001) and the Fundamental Research Funds for the Central Universities (No. GK201001002).

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therein). Furthermore, very interesting characterizations for a subalgebra to be a finite subdiagonal algebra are considered. Arveson also considers subdiagonal algebras for general von Neumann algebras in [1]. Thus it would be interesting if there is some way to extend some of those to this context (cf. [2,6]). Haagerup [9] (cf. [28]) developed the noncommutative Lp space theory associated with a general von Neumann algebra. It then becomes a very important tool in the study of operator algebras. Based on this theory, we shall consider the noncommutative H p space associated with a maximal subdiagonal algebra in a σ-finite von Neumann algebra. Moreover, we extend some results on finite subdiagonal algebras to general ones by Haagerup’s reduction theorem. We firstly recall some notions. Let M be a σ-finite von Neumann algebra acting on a complex Hilbert H. We denote by M∗ the space of all σ-weakly continuous linear functionals on M. Let Φ be a faithful normal conditional expectation from M onto a von Neumann subalgebra D. A subalgebra A of M, containing D, is called a 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. Although subdiagonal algebras are not assumed to be σ-weakly closed in [1], the σ-weak closure of a subdiagonal algebra is again a subdiagonal algebra of M with respect to Φ (Remark 2.1.2 in [1]). Thus we assume that our subdiagonal algebras are always σ-weakly closed. 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 respect to Φ. Put A0 = {X ∈ A : Φ(X) = 0} and Am = {X ∈ M : Φ(AXB) = Φ(BXA) = 0, ∀A ∈ A, B ∈ A0 }. By Theorem 2.2.1 in [1], we recall that Am is a maximal subdiagonal algebra of M with respect to Φ containing A. If there is a faithful normal finite trace τ on M such that τ ◦ Φ = τ , we say that A is finite subdiagonal. An longstanding open question is whether a subdiagonal algebra must be maximal (cf. [1]). A positive answer for finite subdiagonal algebras is given by Exel [8] and a recent work on maximality for general case can be found in [29]. We next recall Haagerup’s noncommutative Lp space associated with a general von Neumann algebra M. Let ϕ be a faithful normal state on M and let σ ϕ = {σtϕ : t ∈ R} be the modular automorphism group of M associated with ϕ by Tomita-Takesaki theory. We consider the crossed product N = M σϕ R of M by R with respect to σ ϕ . Then we have that N is a von Neumann algebra on L2 (R, H) generated by the operators π(x), x ∈ M, and λ(s), s ∈ R defined by the equations ϕ (π(x)ξ)(t) = σ−t (x)ξ(t), ξ ∈ L2 (R, H), t ∈ R,

and (λ(s)ξ)(t) = ξ(t − s), ξ ∈ L2 (R, H), t ∈ R. We identify M with its image π(M) in N .

Vol. 72 (2012) Noncommutative H p Spaces and Subdiagonal Algebras 133 We denote by θ the dual action of R on N . Then {θs : s ∈ R} is an automorphisms group of N characterized by θs (X) = X, X ∈ M, θs (λ(t)) = eist λ(t), t ∈ R. Note that M = {X ∈ N : θs (X) = X, ∀s ∈ R}. N is a semifinite von Neumann algebra and there is the normal faithful semifinite trace τ on N satisfying τ ◦ θs = e−s τ,

∀s ∈ R.

Let L0 (N , τ ) denote the topological ∗-algebra of all τ -measurable operators on L2 (R, H) affiliated to N . According to Haagerup [9] (cf. [28]), the noncommutative Lp space Lp (M) for each 0 < p ≤ ∞ is defined as the set of all τ -measurable operators x in L0 (N , τ ) satisfying s

θs (x) = e− p x,

∀s ∈ R.

(1.1)

There is a linear bijection between the predual M∗ of M and L1 (M) : f → hf , ∀f ∈ M∗ . If we define tr(hf ) = f (I), then tr(|hf |) = tr(h|f | ) = |f |(I) = f

for all f ∈ M∗ and |tr(x)| ≤ tr(|x|) for all x ∈ L1 (M). It is known that Lp (M) for 1 ≤ p < ∞ is a Banach space 1 and for any x ∈ Lp (M), x p = (tr(|x|p )) p . As in [9], we define operators LA and RA on Lp (M)(1 ≤ p < ∞) by LA x = Ax and RA x = xA for all A ∈ M and x ∈ Lp (M). Note that L2 (M) is a Hilbert space with the inner product (a, b) = tr(b∗ a), ∀a, b ∈ L2 (M) and A → LA ( resp. A → RA ) is a faithful representation (resp. anti-representation) of M on L2 (M). We may identify M with L(M) = {LA : A ∈ M}. We also need to give a brief description of Haagerup’s reduction theorem (cf. [10,29]). Let G be the discrete subgroup ∪n≥1 2−n Z of R. We consider the discrete crossed product R = M σϕ G with respect to σ ϕ as above by replacing R by G and L2 (R, H) by 2 (G, H). If we again define π as above ϕ (x)ξ(t), ξ ∈ 2 (R, H), t ∈ G, then π on 2 (G, H), that is, (π(x)ξ)(t) = σ−t is a normal faithful representation of M on 2 (G, H) and we identify π(M) with M. Let ϕˆ be the dual weight of ϕ on R. Then ϕˆ is again a faithful normal state on R whose restriction on M is ϕ. Haagerup’s reduction theorem asserts that there is an increasing sequence {Rn }n≥1 of von Neumann subalgebras of R with the following properties: (i) each Rn is finite. (ii) ∪n≥1 Rn is σ-weakly dense in R. (iii) for each n ≥ 1 there is a faithful normal conditional expectation En from R onto Rn such that ϕˆ ◦ En = ϕ, ˆ En ◦ En+1 = En , n ≥ 1 and limn→∞ ψ ◦ En − ψ = 0 for all ψ ∈ R∗ . We refer the readers to [10] and [29] for more details.

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2. Noncommutative H p Spaces on von Neumann Algebras In this section we consider a noncommutative version of H p space determined by a maximal subdiagonal algebra in a σ-finite von Neumann algebra. To consider analytic crossed products and their invariant subspace structures, McAsey et al. [20] have studied a noncommutative version of H 2 . A related notion is considered in [13] by the author and Saito for factorization in a subdiagonal algebra. We note that the definition depends on the faithful normal state which preserves the conditional expectation. By use of Tomita-Takesaki theory, we may prove that it is in fact independent of the choice of state. We firstly extend a faithful normal conditional expectation to a positive contraction on noncommutative Lp for 1 ≤ p < ∞. Let Φ be a faithful normal conditional expectation from M onto a von Neumann subalgebra D. It is known that there is a faithful normal state ϕ ∈ M∗ such that ϕ ◦ Φ = ϕ. Let σ ϕ = {σtϕ : t ∈ R} be the modular automorphism group of M associated with ϕ. Then the conditional expectation Φ commutes with σtϕ , namely, Φ ◦ σtϕ = σtϕ ◦ Φ, ∀t ∈ R. We recall from Section 2 in [15] that if we consider the restriction φ = ϕ|D of ϕ on D, then the modular automorphism group associated with φ is the restriction of σtϕ on D, namely σtφ = σtϕ |D ,

∀t ∈ R.

It follows that the crossed products S = D σφ R is a von Neumann subalgebra of N = M σϕ R. We know that from the discussion of Sect. 2 in [15], the canonical normal semifinite faithful trace ν on S is the restriction of τ on S and the Radon-Nikodym derivatives of the dual weights of ϕ and φ with respect to τ and ν respectively are the same. In particular, the topological ∗-algebra L0 (S, ν) of ν measurable operators on L2 (R, H) is naturally identified with a subspace of L0 (N , τ ). Let h0 be the noncommutative Radon-Nikodym derivative of the dual weight of ϕ(resp. φ) with respect to τ (resp. ν). Then h0 is the image(hϕ ) of ϕ in L1 (M) and we have that the following representation of σtϕ (cf. [18,19]): −it σtϕ (X) = hit 0 Xh0 ,

∀t ∈ R,

∀X ∈ M. 1 p

(2.1) 1 p

For 1 ≤ p < ∞, we have that Lp (M) = [Mh0 ]p = [h0 M]p , where [S]p denotes the closed linear span of a subset S in Lp (M). It is known that the 1

space Lp (D) can be naturally isometrically identified with the space [Dh0p ]p = 1

1

1

1

[h0p D]p by identifying Dh0p in [Dh0p ]p with Dh0p in Lp (D) for any D ∈ D. 1

We refer readers to see [15] for detail. On the other hand, h02 ∈ L2 (M) is a cyclic and separating vector of M on L2 (M) and ϕ(X) = tr(Xh0 ) = 1

1

(Xh02 , h02 ), ∀X ∈ M. We know that the faithful normal conditional expectation Φ can extend to Lp (M)(1 ≤ p < ∞) in a natural way (cf. [15,16]). That is, for any 0 ≤ θ ≤ 1, we define 1−θ

θ

1−θ

θ

Φp,θ (h0 p Ah0p ) = h0 p Φ(A)h0p ,

∀A ∈ M.

Vol. 72 (2012) Noncommutative H p Spaces and Subdiagonal Algebras 135 The following result is from Lemmas 2.1, 2.2 and Proposition 2.3 in [15] (cf. Lemma 7.1 in [16]). Proposition 2.1. For each 1 ≤ p < ∞, Φp,θ extends to a contractive projec1

tion Φp which is independent of θ from Lp (M) onto [Dh0p ]p with the following properties. (i) tr ◦ Φ1 = tr. (ii) Φp (Dx) = DΦp (x), ∀D ∈ D, ∀x ∈ Lp (M) and Φp (Ay) = Φ(A)y, ∀A ∈ 1

M, ∀y ∈ [Dh0p ]p . (iii) Φp is positive as well as faithful, that is Φp (f ) ≥ 0 whenever f ≥ 0 and if f ≥ 0 and Φp (f ) = 0, then f = 0. By Proposition 2.1, we have following useful result. 1

Corollary 2.2. For any x ∈ [Dh0p ]p and x = U |x| is the polar decomposition, 1

we have |x| ∈ [Dh0p ]p and U ∈ D. 1

Proof. For any x ∈ [Dh0p ]p , let x = U |x| be the polar decomposition. Then |x| = U ∗ x and Φp (|x|) = Φ(U ∗ )x by Proposition 2.1(ii). Note that U is a partial isometry, we then have |Φp (|x|)|2 = (Φ(U ∗ )x)∗ Φ(U ∗ )x = x∗ Φ(U )Φ(U ∗ )x ≤ x∗ x = |x|2 . It follows that Φp (|x|) ≤ |x| from Lemma 2.3 in [25]. Note that Φp (|x| − 1

Φp (|x|)) = 0 since Φp is a projection. Then |x| = Φp (|x|) ∈ [Dh0p ]p from Proposition 2.1(iii). On the other hand, x = U |x| = Φ(U )|x| while Φ(U )∗ x = |x| from Proposition 2.1(ii). We easily have that |x| = Φ(U )∗ Φ(U )|x|. It is known that U ∗ U is a projection on the range of |x|. It now follows that U ∗ U ≤ Φ(U )∗ Φ(U ) ≤ Φ(U ∗ U ). Hence U ∗ U = Φ(U )∗ Φ(U ) = Φ(U ∗ U ). we have that U U ∗ = Φ(U )Φ(U )∗ = Φ(U U ∗ ) similarly. This means that Φ(U ) is a partial isometry with initial projection U ∗ U and final projection U U ∗ . Thus U = Φ(U ) ∈ D from the uniqueness of the polar decomposition.  Remark 2.3. We may get some elementary but useful facts on Lp (M) for 1 ≤ p < ∞. 1

1

(1) It easily follows that ker Φp = [ker(Φ)h0p ]p and Lp (M) = [Dh0p ]p ⊕ 1

1

[ker(Φ)h0p ]p . Moreover, for any X ∈ M, X ∈ D if and only if Xh0p ∈ 1

1

1

1

[Dh0p ]p . In fact, if Xh0p ∈ [Dh0p ]p , then we have that (X − Φ(X))h0p ∈ 1

1

1

[Dh0p ]p ∩ [ker(Φ)h0p ]p and thus is 0. Therefore X = Φ(X) since h0p is separating. The converse is trivial. ⊥ (2) [ker(Φ)h0 ]⊥ 1 = D, D⊥ = [ker(Φ)h0 ]1 , [ker(Φ)]⊥ = [Dh0 ]1 and [Dh0 ]1 = ⊥ ker(Φ) respectively, where S (resp. S⊥ ) denotes the annihilator (resp. pre-annihilator) of a subspace S of L1 (M) (resp. M) in M (resp. L1 (M)). In fact, it is trivial that [ker(Φ)h0 ]⊥ 1 ⊃ D. Now for any A ∈ M

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such that tr(ABh0 ) = (Bh02 , A∗ h02 ) = 0 for all B ∈ ker(Φ). Then 1

1

A∗ h02 ∈ [Dh02 ]2 and thus A∗ ∈ D from (1). Hence A ∈ D. Other proofs are similar. 1 p

We now consider noncommutative H p for 1 ≤ p < ∞. We note that

[Dh0 ]p ∩Lp (M)+ is a positive convex cone in Lp (M)+ . The following lemma might be known but I was unable to find a reference for it. For completeness, we give an elementary proof here. 1

Lemma 2.4. [Dh0 ]1 ∩ L1 (M)+ = {xp : x ∈ [Dh0p ]p ∩ Lp (M)+ }. Proof. We note that {Dh0 ∈ [Dh0 ]1 : Dh0 ≥ 0} is dense in [Dh0 ]1 ∩L1 (M)+ . In fact, for any x ∈ [Dh0 ]1 ∩ L1 (M)+ , there is a sequence {Dn h0 }∞ n=1 in Dh0 such that x = limn→∞ Dn h0 . Let Dn h0 = Un |Dn h0 | be the polar decomposition of Dn h0 . Then |Dn h0 | = Un∗ Dn h0 ∈ [Dh0 ]1 by Corollary 2.2 and limn→∞ |Dn h0 |−x 1 = limn→∞ Un∗ Dn h0 −x 1 = 0 from the main theorem in [17]. Let Dh0 ≥ 0 for some D ∈ D. Then Dh0 is an element of L1 (D) since Dh0 is also ν measurable and satisfying formula (1.1) for p = 1. It follows that there is a positive element x ∈ Lp (D)+ such that xp = Dh0 , which means that 1

p there is a sequence {An }∞ n=1 in D such that limn→∞ x − An h0 p = 0. Note that x is ν measurable and satisfying formula (1.1). Hence x is τ measurable 1

and satisfying formula (1.1). Then x ∈ Lp (M) and thus x ∈ [Dh0p ]p . It follows 1

1

that (Dh0 ) p = x ∈ [Dh0p ]p ∩ Lp (M)+ . Therefore {Dh0 } ∩ L1 (M)+ ⊆ {xp : 1

1

x ∈ [Dh0p ] ∩ Lp (M)+ }. On the other hand, {xp : x ∈ [Dh0p ]p ∩ Lp (M)+ } is 1

1

closed since x−y p ≤ xp −y p 1p for all x, y ∈ [Dh0p ]p ∩Lp (M)+ by the main 1

theorem of the Appendix in [11]. Then [Dh0 ]1 ∩L1 (M)+ ⊆ {xp : x ∈ [Dh0p ]p ∩ 1

1

Lp (M)+ }. Conversely, for any positive x = Dh0p ∈ {Dh0p }, x ∈ Lp (D) in a natural way. Then xp ∈ L1 (D) is ν measurable and limn→∞ xp − Bn h0 1 = 0 for a sequence {Bn }∞ n= in D. Note that Bn h0 ∈ [Dh0 ]1 for all n. Thus  xp ∈ [Dh0 ]1 . Theorem 2.5. Let M be a σ-finite von Neumann algebra and let A be a maximal subdiagonal algebra with respect to Φ such that ϕ ◦ Φ = ϕ for a faithful normal state. Then for any faithful normal state ψ such that ψ ◦ Φ = ψ, we have 1

1

1

1

1

1

[Dh0p ]p = [h0p D]p = [Dhψp ]p = [hψp D]p , 1

1

[Ah0p ]p = [h0p A]p = [Ahψp ]p = [hψp A]p and 1

1

1

1

[A0 h0p ]p = [h0p A0 ]p = [A0 hψp ]p = [hψp A0 ]p for 1 ≤ p < ∞, where hψ is the image of ψ in L1 (M)+ .

Vol. 72 (2012) Noncommutative H p Spaces and Subdiagonal Algebras 137 1

1

Proof. We know that [Dh0p ]p = [h0p D]p for 1 ≤ p < ∞. Let ψ be a faithful normal state on M such that ψ ◦ Φ = ψ and hψ is the image of ψ in L1 (M)+ . Then ψ(X) = tr(hψ X) = tr(hψ Φ(X)) for any X ∈ M. It follows that hψ ∈ ker(Φ)⊥ = [Dh0 ]1 from Remark 2.3, which implies that [Dhψ ]1 ⊆ [Dh0 ]1 . Therefore they must be equal by the symmetry. We simi1

1

larly have [Dhψ ]1 = [hψ D]1 . For 1 ≤ p < ∞, hψp ∈ [Dh0p ]p by Lemma 2.4 and 1

1

1

we then similarly have that [Dhψp ]p = [Dh0p ]p = [hψp D]p . On the other hand, we have A0 D = DA0 = A0 and AD = DA = 1

1

1

A respectively. Then we easily have that [Ah0p ]p = [Ahψp ]p , [A0 h0p ]p = 1

1

1

1

1

[A0 hψp ]p , [h0p A]p = [hψp A]p and [h0p A0 ]p = [hψp A0 ]p respectively. It is only 1

1

to show that [A0 h0p ]p = [h0p A0 ]p . Let T be the set of entire elements of M, that is, those elements X ∈ M for which the function t → σtϕ (X) can be extended to an M-valued entire function over C. For any X ∈ M and r > 0, we let   2 r Xr = e−rt σtϕ (X)dt. (2.2) π R

Then Xr ∈ T and



σzϕ (Xr )

=

r π



2

e−r(t−z) σtϕ (X)dt,

∀z ∈ C

(2.3)

R

−iz for any by the proof of Lemma VIII 2.3 in [27]. Note that σzϕ (X) = hiz 0 Xh0 z ∈ C from formula (2.1) for any entire element X. By Lemma VI 2.4 in [27], Xr converges σ-weakly to X as r → ∞. Note that A is maximal subdiagonal. Then both A and A0 are σtϕ -invariant from Theorem 2.4 in [12]. It follows that the set T ∩ A0 (resp. T ∩ A) of all entire elements in A0 (resp. A) is σweakly dense in A0 (resp. A) from formulae (2.2) and (2.3). However, for any 1

1

1

1

1

1

X ∈ T ∩ A0 , Xh0p = h0p σ ϕi (X), which implies that [A0 h0p ]p ⊆ [h0p A0 ]p . Thus 1 p

1 p

p

[A0 h0 ]p = [h0 A0 ]p by symmetry. Similarly we have [Ah0p ]p = [h0p A]p .



Definition 2.6. Let M be a σ-finite von Neumann algebra and let A be a maximal subdiagonal algebra with respect to Φ such that ϕ ◦ Φ = ϕ for a faithful normal state. For 1 ≤ p < ∞, we define the noncommutative H0p and H p respectively as follows: 1

H0p = [A0 h0p ]p

1

and

H p = [Ah0p ]p .

We now have that the noncommutative H0p and H p associated with a maximal subdiagonal algebra A with respect to a faithful normal conditional expectation Φ is independent of the choice of state which preserves Φ. For a general subdiagonal algebra A, we consider the noncommutative H p space associated with Am . Just as Theorem 2.2 in [13], H0p (resp. H p ) may be used to characterize maximal subdiagonal algebras.

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Theorem 2.7. Let A be a subdiagonal algebra with respect to Φ and Am its maximum. Then for 1 ≤ p < ∞, Am = {X = {X = {X = {X

∈ M : XH0p ∈ M : XH p ∈ M : H0p X ∈ M : H pX

⊆ H0p } ⊆ H p} ⊆ H0p } ⊆ H p }.

Proof. We note that Am is easily contained in any one of the sets described in right. Let 1 < q ≤ ∞ be the conjugate exponent of p. It is known that 1

for all x ∈ H0p and B ∈ Am , we have tr(xh0q B) = 0. Take any X ∈ M 1

1

such that XH0p ⊆ H0p . Then for any A ∈ (Am )0 , we have tr(XAh0p h0q B) = 1

1

1

1

tr(XAh02 h02 B) = (XAh02 , B ∗ h02 ) = 0. It follows that XH02 ⊆ H02 . Thus X ∈ Am by Theorem 2.2 in [13]. Hence Am = {X ∈ M : XH0p ⊆ H0p }. Other proofs are similar. 

3. Characterizations of Maximal Subdiagonal Algebras Recently, Blecher and Labuschagne have studied noncommutative H p theory on a finite von Neumann algebra and several very important characterizations for a tracial subalgebra to be a finite subdiagonal algebra are given (cf. [2–6]). They point out that it would be interesting if there is some way to extend some of their results on finite subdiagonal algebras to general ones (cf. [2,6]). In this section, we extend some of their results to general von Neumann algebras. Firstly, we give a version of tracial subalgebra in a σ-finite von Neumann algebra. Definition 3.1. Let A be a σ-weakly closed subalgebra of a σ-finite von Neumann algebra M and Φ a faithful normal conditional expectation from M onto the diagonal D = A ∩ A∗ of A. If Φ is multiplicative on A, then we say that A is an expectation algebra with respect to Φ. We note that if there is a faithful normal finite trace τ on M such that τ ◦ Φ = τ , then A is a tracial subalgebra (cf. [2]). Put A0 = ker Φ in A. Then A0 is a two sided ideal of A. If A + A∗ is σ-weakly dense in M, then A is a subdiagonal algebra with respect to Φ. In general, we may choose a faithful normal state ϕ on M such that ϕ ◦ Φ = ϕ. Recall that σ ϕ = {σtϕ : t ∈ R} is the modular automorphism group associated with ϕ. As in Sect. 2, we denote by h0 the image of ϕ in L1 (M). We say that A has the unique normal state extension property if for 1 any g ∈ L1 (M)+ , g(A) = ϕ(A) for all A ∈ A, then g = ϕ. If {(A + A∗ )h02 } is norm dense in L2 (M), then we say that A satisfies the L2 -density. The proof of the following lemma is parallel to Lemma 4.1 in [2] but a little different. We recall a notion to prove this lemma. Let ξ0 ∈ H be a cyclic and separating unit vector of M and ω0 denotes the vector state, that is ω0 (X) = (Xξ0 , ξ0 ) for all X ∈ M. Associated with {M, H, ξ0 } we have

Vol. 72 (2012) Noncommutative H p Spaces and Subdiagonal Algebras 139 the modular operator Δ (cf. [18,27]). We recall that the cone P α , α ∈ [0, 12 ] is the closure of the convex cone Δα M+ ξ0 in H (Definition 1.1 in [18]). Proposition 3.2. Let A be an expectation algebra with respect to Φ. Then A has the unique normal state extension property if and only if for any g ∈ L1 (M)+ and g(A) = 0 for all A ∈ A0 , then g ∈ [Dh0 ]1 . Proof. Sufficiency. Let g ∈ L1 (M)+ such that g(A) = ϕ(A) for all A ∈ A. Then g(A) = 0 for all A ∈ A0 and thus g ∈ [Dh0 ]1 by assumption. We also have g(D) = ϕ(D) for all D ∈ D. Since g = g ◦ Φ, g = h0 . Necessity. Suppose that A has the unique normal state extension property. Let g ∈ L1 (M)+ such that g(A) = 0 for all A ∈ A0 . Then g(A) = g(Φ(A)) for all A ∈ A. We may suppose that g ≥ h0 by replacing g by g + h0 if necessary. Then Φ1 (g) ≥ h0 from Proposition 2.1. Put 1

1

g0 = Φ1 (g). Then we have that h0 ≤ g0 . It now follows that h02 ≤ g02 1

1

from Lemma 2.3 in [25] and g02 ∈ [Dh02 ]2 by Lemma 2.4. It is elemen1

1

1

1

tary that [Dh02 ]2 = [h02 D]2 = [Dg02 ]2 = [g02 D]2 from Theorem 2.5 since 1

g0 ∈ [Dh0 ]1 is faithful. Then g02 is a cyclic and separating vector of D 1

1

on [Dh02 ]2 . We now consider D acting on [Dh02 ]2 as a von Neumann alge1 2

bra with the separating and cyclic vector g0 and two positive elements in 1

1

1

1

D∗ given by ω(D) = ϕ(D) = (Dh02 , h02 ) and ω0 (D) = (Dg02 , g02 ) for all D ∈ D. Then ω ≤ ω0 . It follows that there is unique C ∈ D such that 1 Cg02 ∈ P 0 (α = 0) and ω(X) = ω0 (C ∗ XC) for all X ∈ D from Corollary 1.3 in [18]. We claim that C is positive. In fact, taking any positive operator 1

1

sequence {Cn : n = 1, 2, · · · } ⊆ D+ such that limn→∞ Cn g02 − Cg02 2 = 0, 1

1

1

1

then limn→∞ Cn g02 D − Cg02 D 2 = 0, which implies that (Cg02 D, g02 D) = 1 2

1 2

1 2

1 2

limn→∞ (Cn g0 D, g0 D) = limn→∞ tr(D∗ g0 Cn g0 D) ≥ 0 for all D ∈ D. It follows that C is positive. Note that in this case, we have that h0 = Cg0 C. Setting f = CgC we have that f ∈ L1 (M)+ and f (A) = tr(CgCA) = tr(gCAC) = g(CAC) = g(Φ(CAC)) = g(CΦ(A)C) = tr(gCΦ(A)C) = tr(CgCΦ(A)) = tr(CΦ1 (g)CA) = tr(h0 A) = ϕ(A) for all A ∈ A. Then f = CgC = h0 = CΦ1 (g)C by the assumption. Note that C is injective since h0 is a non-singular τ -measurable positive operator  (cf. Lemma 2.1 in [18]). Therefore g = Φ1 (g) ∈ [Dh0 ]1 . 1

1

1

1

Put Am = {X ∈ M : X[A0 h02 ]2 ⊆ [A0 h02 ]2 , X[Ah02 ]2 ⊆ [Ah02 ]2 } and 1

1

1

1

AM = {X ∈ M : [h02 A0 ]2 X ⊆ [h02 A0 ]2 , [h02 A]2 X ⊆ [h02 A]2 }. It is trivial that both Am and AM are σ-weakly closed subalgebras containing A with diagonal D. If A is tracial, then Am is just the algebra A∞ defined in [2]. Proposition 3.3. Let A be an expectation algebra with respect to Φ. Then (i)

Both Am and AM are also expectation algebras with respect to Φ.

140 (ii)

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1

1

If A is {σtϕ : t ∈ R} invariant, then [A0 h02 ]2 = [h02 A0 ]2 (resp. [Ah02 ]2 = 1

[h02 A]2 ) and 1

1

1

1

Am = AM = {A ∈ M : A[Ah02 ]2 ⊆ [Ah02 ]2 } = {A ∈ M : Ah02 ∈ [Ah02 ]2 }. 1

1

1

Proof. (i) We know that [Ah02 ]2 = [Dh02 ]2 ⊕ [A0 h02 ]2 and DAm = Am D = Am . Then for any A ∈ Am and D ∈ D, we have 1 1 1 that (A − Φ(A))Dh02 , D(A − Φ(A))h02 ∈ [A0 h02 ]2 . Thus for any 1

1

1

1

A, B ∈ Am , ABh02 = Φ(AB)h02 ⊕ (AB − Φ(AB))h02 = Φ(A)Φ(B)h02 ⊕ 1

1

((A − Φ(A))Φ(B) + Φ(A)(B − Φ(B)))h02 . We thus have Φ(AB)h02 = 1

(ii)

1

Φ(A)Φ(B)h02 and therefore Φ(AB) = Φ(A)Φ(B) since h02 is separating. Hence Am is an expectation algebra with respect to Φ. So is AM by a similar way. If A is {σtϕ : t ∈ R} invariant, then A ∩ T is σ-weakly dense in A by use of formulae (2.2) and (2.3), where T is the set of entire elements of M used in the proof of Theorem 2.5. Similar to the proof of Theo1 1 1 1 rem 2.5, we have [A0 h02 ]2 = [h02 A0 ]2 and [Ah02 ]2 = [h02 A]2 . Then we 1

1

1

easily have that {A ∈ M : A[Ah02 ]2 ⊆ [Ah02 ]2 } = {A ∈ M : Ah02 ∈ 1

1

1

1

1

[Ah02 ]2 }. On the other hand, if Ah02 ∈ [Ah02 ]2 , then Ah02 B ∈ [Ah02 ]2 1

1

1

1

and (Ah02 B, h02 D) = (Ah02 , h02 DB ∗ ) = 0 for all B ∈ A0 and D ∈ D. 1

1

1

1

Hence Ah02 B ∈ [h02 A0 ]2 . It follows that A[A0 h02 ]2 ⊆ [A0 h02 ]2 and the desired result follows.  Next we consider factorization property in an expectation algebra. We recall that an invertible S ∈ M has a left(resp. right) partial factorization relative to A if there is an isometry(resp. a co-isometry) U ∈ M such that both U ∗ S and S −1 U are in A (cf. [23]). If U is unitary, then S is said to have a factorization relative to A. Pitts in [23] proved that if A is a nest subalgebra with an injective nest in M, then every invertible operator in M has a left partial factorization relative to A. We also proved in [13] that a maximal subdiagonal algebra A with respect to Φ in M has the same property. In fact, we may obtain that for any invertible S ∈ M, there is an isometry U ∈ M such that U ∗ S, S −1 U ∈ A and Φ(S −1 U )Φ(U ∗ S) = Φ(U ∗ S)Φ(S −1 U ) = I from the proof of Theorem 3.8 in [13]. Motivated by this fact, we give the following notion. Definition 3.4. Let A be an expectation algebra of M with respect to Φ. If for any invertible operator S ∈ M, there is an isometry U ∈ M such that both U ∗ S and S −1 U are in A and Φ(S −1 U )Φ(U ∗ S) = Φ(U ∗ S)Φ(S −1 U ) = I, then we say that A has the left partial factorization property. The following lemma is in fact formula (1.1) for p = 1 in [14]. Lemma 3.5. Let x, y ∈ L1 (M)+ such that x ≤ y. Then there is a contraction 1 1 C ∈ M such that x 2 = Cy 2 . Moreover, if x, y ∈ [Dh0 ]1 , then we may have C ∈ D.

Vol. 72 (2012) Noncommutative H p Spaces and Subdiagonal Algebras 141 Proposition 3.6. Let A be an expectation algebra with respect to Φ. If A is {σtϕ : t ∈ R} invariant and has the unique normal state extension property, then for any invertible operator S ∈ M, there are isometries U, V ∈ M such that S −1 U, V ∗ S ∈ Am and Φ(S −1 U )Φ(U ∗ S) = Φ(U ∗ S)Φ(S −1 U ) = Φ(V ∗ S)Φ(S −1 V ) = Φ(S −1 V )Φ(V ∗ S) = I. Moreover, if A satisfies the L2 -density, then Am has the left partial factorization property. 1

1

Proof. Since A is {σtϕ : t ∈ R} invariant, [A0 h02 ]2 = [h02 A0 ]2 from Proposition 1

1

1

3.3. Note that Sh02 ∈ S[A0 h02 ]2 . Then there are a unique ζ ∈ [A0 h02 ]2 and a 1

1

1

1

1

ξ ∈ S[Ah02 ]2 S[A0 h02 ]2 such that Sh02 = ξ ⊕Sζ. Since [Ah02 ]2 A0 ⊆ [A0 h02 ]2 , 1

it is trivial that [ξA0 ]2 ⊆ S[A0 h02 ]2 , which implies that (ξA, ξ) = tr(|ξ|2 A) = 1

0 for all A ∈ A0 . We then have that |ξ|2 ∈ [Dh0 ]1 and therefore |ξ| ∈ [Dh02 ]2 from Proposition 3.2 and Lemma 2.4. Let δ > 0 such that S ∗ S ≥ δ 2 I. Then 1 1 1 1 |ξ|2 = (h02 − ζ ∗ )S ∗ S(h02 − ζ) ≥ δ 2 (h02 − ζ ∗ )(h02 − ζ) and for any positive D ∈ D, we have 1

1

tr(|ξ|2 D) = tr((h02 − ζ ∗ )S ∗ S(h02 − ζ)D) 1 2

1 2

1

1

≥ δ 2 tr((h0 − ζ ∗ )(h0 − ζ)D) = δ 2 (h02 − ζ)D 2 22 1 2

1 2

1

1

≥ δ 2 Φ2 (h0 − ζ)D 2 ) 22 = δ 2 h0 D 2 22 = δ 2 tr(h0 D).

It follows that |ξ|2 ≥ δ 2 h0 . On the other hand, for any D ∈ D, we have that 1

SζD 22 ≤ SζD 22 + ξD 22 = Sh02 D 22 . Then ζD 22 ≤ S −1 2 SζD 22 ≤ 1

S −1

S

h02 D 22 . We thus have 1

1

tr(|ξ|2 D) = tr((h02 − ζ ∗ )S ∗ S(h02 − ζ)D) 1 2

1

≤ S 2 tr((h0 − ζ ∗ )(h02 − ζ)D) 1

1

1

1

= S 2 (h02 − ζ)D 2 22 = S 2 ( h02 D 22 + ζD 2 22 ) 1 2

1

≤ S 2 (1 + S −1

S ) h0 D 2 ) 22 = S 2 (1 + S −1

S )tr(h0 D).

This implies that |ξ|2 ≤ S 2 (1 + S −1 S )h0 . Thus, δ 2 h0 ≤ |ξ|2 ≤ S 2 (1 + S −1 S )h0 and 1

1

1

δh02 ≤ |ξ| ≤ S (1 + S −1 S ) 2 h02 by Lemma 2.3 in [25]. It is elementary that |ξ| is also a separating and 1

cyclic vector for D (resp. M) on [Dh02 ]2 (resp. L2 (M)). Thus there is an 1

operator C ∈ D such that h02 = C|ξ| = |ξ|C ∗ from Lemma 3.5 and C is in fact invertible. Let ξ = U |ξ| be the polar decomposition of ξ. Then U 1

is an isometry since |ξ| is right cyclic and ξ = U |ξ| = S(h02 − ζ). Then

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1

1

S −1 U h02 = S −1 U |ξ|C ∗ = (h02 − ζ)C ∗ ∈ [Ah02 ]2 . It follows that S −1 U ∈ Am from Proposition 3.3. On the other hand, 1

1

1

1

S −1 U h02 = Φ(S −1 U )h02 + (S −1 U − Φ(S −1 U ))h02 = h02 C ∗ − ζC ∗ . 1

1

1

Then Φ(S −1 U )h02 = h02 C ∗ = C|ξ|C ∗ = Ch02 . Thus Φ(S −1 U ) = C is invertible. On the other hand, we note that U [|ξ|A∗0 ]2 = [ξA∗0 ]2 , U [|ξ|D]2 = [ξD]2 and U [|ξ|A0 ]2 = [ξA0 ]2 respectively. This implies that the restriction of U ∗ ⊥ on [|ξ|(A∗ + A)]2 is unitary and therefore U ∗ [ξ(A∗ + A)]⊥ 2 ⊆ [|ξ|(A + A)]2 . ∗ ∗ ⊥ Since (Sζ, ξA ) = (SζA, ξ) = 0 for all A ∈ A, it is trivial that Sζ ∈ [ξA ]2 = ∗ ∗ ∗ ⊥ [ξA0 ]2 ⊕[ξ(A∗ +A)]⊥ 2 . Hence U Sζ ∈ U ([|ξ|A0 ]2 ⊕[|ξ|(A +A)]2 ) ⊆ [|ξ|A0 ]2 ⊕ 1

⊥ ∗ ∗ 2 [|ξ|(A∗ + A)]⊥ 2 ⊆ [|ξ|D]2 . It follows that Φ2 (U Sζ) = 0 and Φ2 (U Sh0 ) = 1 2

1 2

Φ(U ∗ S)h0 = |ξ| = C −1 h0 . Therefore Φ(U ∗ S) = C −1 = Φ(S −1 U )−1 . If we consider S ∗ −1 and A∗ , we can similarly have an isometry V ∈ M such that V ∗ S ∈ Am and Φ(V ∗ S)Φ(S −1 V ) = Φ(S −1 V )Φ(V ∗ S) = I. 1

∗ ⊥ 2 If A satisfies the L2 -density, then [|ξ|(A∗ +A)]⊥ 2 = [h0 (A +A)]2 = {0}. 1

Thus by the preceding proof, we have that U ∗ Sζ ∈ [|ξ|A0 ]2 = [h02 A0 ]2 . 1

1

Then U ∗ Sh02 = |ξ| + U ∗ Sζ ∈ [Ah02 ]2 . Thus U ∗ S, S −1 U ∈ Am by Proposition 3.3.  Proposition 3.7. Let A be an expectation algebra of M with the left partial factorization property. Then for any state f of M, we have that f ◦ Φ = f on M whenever f ◦ Φ = f on A. Proof. Let S ∈ M+ be invertible. Then there is an isometry U ∈ M 1 1 1 1 such that both U ∗ S 2 and S − 2 U are in A and Φ(S − 2 U )Φ(U ∗ S 2 ) = 1 1 Φ(U ∗ S 2 )Φ(S − 2 U ) = I. We thus have 1

1

1

1

f (Φ(S − 2 U )U ∗ S 2 ) = f (Φ(S − 2 U )Φ(U ∗ S 2 )) = f (I) = 1. By Cauchy–Schwarz and Kadison–Schwarz ineauqlity we deduce 1

1

1

1

1 ≤ f (Φ(S − 2 U )Φ(S − 2 U )∗ )f (S 2 U U ∗ S 2 ) 1 1 1 1 ≤ f (Φ(S − 2 U U ∗ S − 2 ))f (S 2 U U ∗ S 2 ) ≤ f (Φ(S −1 ))f (S). We can now follow the proof of Theorem 2.2 in [3]. That is, 1 ≤ f (Φ(e−tH ))f (etH ) = g(t) for any self-adjoint operator H ∈ M and all t ∈ R. Differentiating and noting that g  (0) = 0, we have that f (H) = f (Φ(H)).  We need two lemmas in order to give the main theorem of this section. Let K and S be two closed operators. If D(K) ⊆ D(S) and S|D(K) = K, then we say that S is an extension of K and denoted by K ⊆ S. We recall that a densely defined closed operator K on a Hilbert space H with domain D(K) is symmetric if K ⊆ K ∗ ; self-adjoint if K = K ∗ and positive if K is self-adjoint and (Kξ, ξ) ≥ 0 for all ξ ∈ D(K) (cf. [7]).

Vol. 72 (2012) Noncommutative H p Spaces and Subdiagonal Algebras 143 Lemma 3.8. Let K be a symmetric operator on H and P a positive extension of K. If E is a projection in B(H) such that E(D(K)) ⊆ D(K) and EK = KE, then E(D(P )) ⊆ D(P ) and EP = P E. Proof. Suppose that P is a positive extension of K, then K ⊆ P ⊆ K ∗ . Let L+ = ker(K ∗ − i) and L− = ker(K ∗ + i). Then there is a partial isometry W with initial space L+ and final space L− such that the domain of P is D(P ) = {x + y + W y : x ∈ D(K), y ∈ L+ } and P (x + y + W y) = Kx + iy − iW y,

∀x ∈ D(K), y ∈ L+

by Theorem 2.20 of Chapter X in [7]. Let H = EH ⊕(I −E)H. Then D(K) = E(D(K)) ⊕ (I − E)(D(K)) such that E(D(K))and (I − E)(D(K)) are dense in EH and (I − E)H and K = K1 ⊕ K2 for some closed symmetric operators K1 and K2 with domains E(D(K)) and (I −E)(D(K)) respectively. We then have that K ∗ = K1∗ ⊕K2∗ . Thus L+ = ker(K ∗ −i) = ker(K1∗ −i)⊕ker(K2∗ −i) = L1+ ⊕ L2+ and L− = ker(K ∗ + i) = ker(K1∗ + i) ⊕ ker(K2∗ + i) = L1− ⊕ L2− . We note that W is a unitary operator from L+ onto L− . Let   W11 W12 W = W21 W22 be the matrix form from L1+ ⊕ L2+ onto L1− ⊕ L2− . We claim that W12 = 0 and W21 = 0. For any x1 ∈ E(D(K)), y2 ∈ L2+ , we have ξ = x1 + y2 + W y2 = (x1 + W12 y2 ) ⊕ (y2 + W22 y2 ) ∈ D(P ) and P ξ = P (x1 + y2 + W y2 ) = (Kx1 −iW12 y2 )⊕(iy2 −iW22 y2 ). Since P is positive, Kx1 = P x1 and y2 2 =

W y2 2 = W12 y2 2 + W22 y2 2 , we have (P ξ, ξ) = (P x1 , x1 ) − 2Im(x1 , W12 y2 ) + 2Im(W22 y2 , y2 ) ≥ 0, where Imz is the imaginary part of a complex number z. It follows that 2|(x1 , W12 y2 )| ≤ (P x1 , x1 ) + 2Im(W12 y2 , y2 ),

∀x1 ∈ E(D(K)),

∀y2 ∈ L2+ .

Then W12 y2 = 0 for all y2 ∈ L2+ since E(D(K)) is dense in EH, that is, W12 = 0. We similarly have W21 = 0. Thus E(D(P )) ⊆ D(P ) and EP = P E.  On the other hand, we consider noncommutative L2 (R) associated with R defined in Section 1, that is, R is the crossed product of M σϕ G. We identify M with π(M). There is a unique faithful normal conditional expectation E from R onto M satisfying ϕˆ ◦ E = ϕˆ from formulae (2.5) and (2.6) in [29]. Since the dual weight ϕˆ of ϕ on R is a faithful normal state whose restriction on M is ϕ, we may still denote by h0 the image of ϕˆ in L1 (R) 1

and identify [Mh0 ]1 with L1 (M) as well as [Mh02 ]2 with L2 (M). Let A be a {σtϕ : t ∈ R} invariant expectation algebra with respect to Φ and let Aˆ be the σ-weakly closed subalgebra generated by A and {λ(s) : s ∈ G}. Note that Aˆ is the σ-weak closure of the set of all linear combinations of λ(s)X(resp. Xλ(s)), s ∈ G, X ∈ A. We again can extend Φ to a unique ˆ = D σϕ G by (3.2) ˆ from R onto D faithful normal conditional expectation Φ in [29], that is, ˆ Φ(λ(s)X) = λ(s)Φ(X), s ∈ G, X ∈ M.

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ˆ Then by proofs of Lemmas 3.1 and 3.2 in [29], We have Set An = Rn ∩ A. the following lemma without proof. ˆ and each An Lemma 3.9. Aˆ is an expectation algebra of R with respect to Φ ˆ is a tracial algebra of Rn with respect to Φ|Rn for all n. Theorem 3.10. Let A be an expectation algebra with respect to Φ satisfying the L2 -density. Then the following assertions are equivalent. (1) A is a maximal subdiagonal algebra with respect to Φ. (2) A has the left partial factorization property. (3) A is {σtϕ : t ∈ R} invariant and has the unique normal state extension property. Proof. The implication from (1) to (2) is Proposition 3.6 since A is σtϕ invariant for all t ∈ R by Theorem 2.4 in [12] and A = Am . (2) ⇒ (3) We recall that both Am and AM are expectation algebras containing A. We claim that A = Am = AM . In fact, for any X ∈ Am , we may assume that X is invertible in Am by considering λI + X for a positive constant λ > X if necessary. Then there is an isometry U ∈ M such that U ∗ X, X −1 U ∈ A ⊆ Am and Φ(U ∗ X)Φ(X −1 U ) = Φ(X −1 U )Φ(U ∗ X) = I. It follows that both U ∗ and U are in Am since X and X −1 are. Thus U ∈ D = Am ∩ A∗m . Note that Φ is multiplicative on Am . Then Φ(X) and Φ(U ∗ X) = U ∗ Φ(X) are invertible. Hence U is unitary in D. Therefore X = U U ∗ X ∈ A. That is, A = Am . Similarly we have A = AM . This implies that 1

1

[A0 h02 ]2 = [h02 A0 ]2

and

1

1

[A∗0 h02 ]2 = [h02 A∗0 ]2

(3.1)

since 1

1

[Dh02 ]2 = [h02 D]2 It follows that

and

1

1

1

L2 (M) = [A0 h02 ]2 ⊕ [Dh02 ]2 ⊕ [A∗0 h02 ]2 . (3.2)

and

⎧ ⎛ X11 ⎨ A0 = X ∈ M : X = ⎝ 0 ⎩ 0 ⎧ ⎛ D11 ⎨ D= D∈M:D=⎝ 0 ⎩ 0

0 D22 0

⎞⎫ X13 ⎬ X23 ⎠ , ⎭ X33 ⎞⎫ 0 ⎬ 0 ⎠ ⎭ D33

⎧ ⎛ X11 ⎨ A= X∈M:X=⎝ 0 ⎩ 0

X12 X22 0

⎞⎫ X13 ⎬ X23 ⎠ ⎭ X33

X12 0 0

according to the decomposition (3.2) of L2 (M).

1

Let Δ be the modular operator associated with {M, L2 (M), h02 }. Then 1

1

{(A + D + B ∗ )h02 : A, B ∈ A0 , D ∈ D} ⊆ D(Δ 2 ) (cf. P298 in [18]) and 1

1

1

Δ 2 Xh02 = h02 X,

∀X ∈ A0 + D + A∗0 .

(3.3)

Vol. 72 (2012) Noncommutative H p Spaces and Subdiagonal Algebras 145 1

1

Let K be the closure of the restriction of Δ 2 on {(A + D + B ∗ )h02 : A, B ∈ A0 , D ∈ D}. Then K is densely defined with domain D(K) such that 1 1 K ⊆ Δ 2 and therefore a symmetric operator. Thus Δ 2 is a positive extension of K. Denote by E1 , E2 and E3 the orthogonal projections from L2 (M) 1

1

1

onto [A0 h02 ]2 , [Dh02 ]2 and [A∗0 h02 ]2 respectively. Then it is easy to show that Ej (D(K)) ⊆ D(K) and Ej K = KEj for j = 1, 2, 3 by formulae (3.1), (3.2) 1 1 1 1 and (3.3). Thus Ej (D(Δ 2 )) ⊆ D(Δ 2 ) and Ej Δ 2 = Δ 2 Ej for j = 1, 2, 3 by Lemma 3.8. It follows that Δ = Δ1 ⊕ Δ2 ⊕ Δ3 for some closed positive operator Δj with domain D(Δj ) such that D(Δ) = D(Δ1 ) ⊕ D(Δ2 ) ⊕ D(Δ3 ). We thus easily have that σtϕ (A0 ) = A0 and σtϕ (A) = A for all t ∈ R from above matrix representations of A0 and A. Proposition 3.7 says that A has the unique normal state extension property. (3) ⇒ (1) It is known that Aˆ is an expectation algebra of R with respect ˆ ˆ R for all n by to Φ and each An is a tracial algebra of Rn with respect to Φ| n 1 1  2 ∗ 2 2 Lemma 3.9. Since L (R) = t∈G λ(t)[Mh0 ]2 and {(A + A )h0 } is dense in 1 L2 (M), {(Aˆ + Aˆ∗ )h02 } is dense in L2 (R), which means that Aˆ satisfies the L2 -density. We next show that Aˆ and An have the unique normal state extension property and the L2 -density for all n. As in §2, let En,2 be the projection 1

from L2 (R) onto [Rn h02 ]2 which is the extension of En . Since En (A) = An and 1 1 1 1 En,2 (Xh 2 ) = En (X)h 2 for any X ∈ R, {(An + A∗ )h 2 } = {En,2 (Aˆ + Aˆ∗ )h 2 } 0

1 2

n

0

0

0

2

is dense in [Rn h0 ]2 . Hence, An satisfies the L -density. On the other hand, suppose that f ∈ L1 (R)+ such that tr(f λ(t)A) = 1 tr(h0 λ(t)A) for all t ∈ G and A ∈ A. Then f 2 ∈ L2 (R)+ and therefore 1   1 f 2 = t∈G λ(t)ξt for some ξt ∈ [Mh02 ]2 with t∈G ξt 22 < ∞, which implies  1 1 that t∈G |ξt |2 ∈ L1 (M). We note that (f 2 A, f 2 ) = tr(f A) = tr(h0 A) = 1

1

(h02 A, h02 ) for all A ∈ A. Then 



tr(ξt∗ ξt A) = tr

t∈G

for all A ∈ A. It follows that



  |ξt |2

A

= tr(h0 A)

t∈G

 t∈G

|ξt |2 = h0 since A has the unique normal 1

2 state extension property. Thus ξt = At h 0 for some contractions At ∈ M ∗ from Lemma 3.5. It is elementary that t∈G At At = I. In fact, for any 1  1 1 1  ∗ 2 2 finite subset F ⊆ G, we have h02 ( t∈F A∗t At )h02 = t∈F h0 At At h0 = 1  1  2 ∗ 2 ∗ ∗ 2 t∈F |ξt | ≤ h0 , which implies that T h0 ( t∈F At At )h0 T ≤ T h0 T for 1 1 1  1  all T ∈ R. Then (( t∈F A∗t At )h02 T, h02 T ) = tr(T ∗ h02 ( t∈F A∗t At )h02 T ) ≤ 1 1  tr(T ∗ h0 T ) =(h02 T, h02 T ) for all T ∈ R. Thus t∈F A∗t At ≤ I for any subset F of G and t∈G A∗t At is σ-weakly convergent to a positive operator P ≤ I. 1 1   1 It is clear that P = I. Moreover, f 2 = t∈G λ(t)At h02 = t∈G h02 A∗t λ(−t). 1  2 Now for any subset F ⊆ G, we put xF = t∈F λ(t)At h0 . It is trivial 1 1 1 that f 2 − xF 22 + xF 22 = f 2 − x∗F 22 + x∗F 22 = f 2 22 = tr(f ) and

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limF f − x∗F xF 1 = 0. Then limF E1 (f ) − E1 (x∗F xF ) 1 = 0, where E1 is the projection from L1 (R) onto [Mh0 ]1 defined in Section 1 by the condi1  tional expectation E from R onto M. Note that x∗F xF = h02 ( s,t∈F A∗s λ(t − 1 1  1 ϕ s)At )h02 = h02 ( s,t∈F A∗s σ(t−s) (At ))λ(t − s)h02 . We have that E1 (x∗F xF ) = 1 1 1  1  ϕ h02 E( s,t∈F A∗s σ(t−s) (At )λ(t − s))h02 = h02 ( t∈F A∗t At )h02 by the definition of E. It follows that E1 (f ) = lim E1 (x∗F xF ) = F



1

1

h02 A∗t At h02 = h0 .

t∈G

Moreover, we have that tr((f − h0 )λ(t)A) = tr((f − h0 )Aλ(t)) = 0 for all t ∈ G and A ∈ A by the assumption. Then tr((f − h0 )λ(t)A) = 0 for all t ∈ G and A ∈ A + A∗ . Since σtϕˆ (λ(s)) = λ(s) for all s, t ∈ G from (2.3) in 1

1

[29], λ(s)h02 = h02 λ(s) for any s ∈ G. We similarly have that E1 (x∗F xF λ(−r)A) ⎛ ⎛ ⎞ ⎞  1 1 ϕ A∗s σ(t−s) (At )λ(t − s)⎠ h02 λ(−r)A⎠ = E1 ⎝h02 ⎝ s,t∈F

⎛

⎛



1 2

= E1 ⎝h0 ⎝

s,t∈F

⎛

⎛⎛

= ⎝h0 E ⎝⎝  1 2

= h0

s,t∈F



1 2

ϕ A∗s σ(t−s) (At )λ(t − s − r)⎠ h0 ⎠ A



1 2

⎞

⎞

⎞

⎞⎞ 1 2

ϕ A∗s σ(t−s) (At )λ(t − s − r)⎠⎠ h0 ⎠ A



A∗(t−r) σrϕ (At )

1

h02 A

t∈F

for all A ∈ M and r ∈ G. Now for any ξ ∈ L2 (M), we have 

 1 2

h0

t∈F

≤





 A∗(t−r) σrϕ (At ) ξ 1

t∈F 1

h02 A∗(t−r) σrϕ (At )ξ 1

t∈G

 ≤ =



1 2

h0 A∗(t−r) 22

t∈G

ξ 22 <



1

 12 



 12

λ(r)At λ(−r)ξ 22

t∈G

∞.

h02 A∗(t−r) σrϕ (At )ξ is convergent in L1 norm for any ξ ∈ 1  L2 (M) and therefore the series t∈G h02 A∗(t−r) σrϕ (At ) is convergent weakly It follows that

t∈G

Vol. 72 (2012) Noncommutative H p Spaces and Subdiagonal Algebras 147 to an element in L2 (M). Thus tr((f − h0 )λ(−r)A) = lim tr((x∗F xF − h0 )λ(−r)A) F

= lim tr ◦ E1 ((x∗F xF − h0 )λ(−r)A) F

= lim tr(E1 (x∗F xF λ(−r)A)) − tr ◦ E1 (h0 λ(−r)A) F     1 1 ∗ ϕ = tr h02 A(t−r) σr (At )h02 A t∈G

 = tr



 1 2

h0 A∗(t−r) σrϕ (At )

 1 2

(h0 A)

(3.4)

t∈G

for all nonzero r ∈ G and A ∈ M by Proposition 2.1(i). In particular,    1 1 ∗ ϕ 2 tr (h02 A)) = tr((f − h0 )λ(−r)A) = 0 h0 A(t−r) σr (At t∈G

for all nonzero r ∈ G and for all A ∈ A + A∗ . It follows that  1 h02 A∗(t−r) σrϕ (At ) = 0 t∈G

since A satisfies the L2 -density. Therefore      1  1 1 1 ∗ ∗ ϕ 2 2 2 h0 At−r λ(r)At h0 λ(−r) = h0 A(t−r) σr (At ) h02 = 0. t∈G

t∈G



1 2

1

We thus have that t∈G h0 A∗(t−r) λ(r)At h02 = 0. It follows from formula (3.4) again that tr((f − h0 )λ(−r)A) = 0 for all nonzero r ∈ G and A ∈ M. Since ker E is the σ-weakly closed linear span of {λ(r)A : r ∈ G − {0}, A ∈ M}, it follows that f − h0 ∈ (ker E)⊥ = [Mh0 ]1 by Remark 2.3. Thus f = h0 and Aˆ has the unique normal state extension property. We next claim that An has the unique normal state extension property for all n. let fn ∈ [Rn h0 ]1 be positive such that fn ((An )0 ) = 0. It is clear that fn ◦ En = fn . Since (An )0 = En (A0 ), we have that fn ◦ En (Aˆ0 ) = 0. Note that ˆ 0 ]1 Aˆ has the unique normal state extension property, then fn = fn ◦En ∈ [Dh from Proposition 3.2. It now follows that fn = (En )1 (fn ) ∈ [Dn h0 ]1 , where ˆ Then An has the unique normal state extension property for Dn = Rn ∩ D. all n from Proposition 3.2 again. Therefore An is a maximal finite subdiagˆ R for all n from Theorem 1.1 in [2], which onal algebra with respect to Φ| n ∗ implies that An + An is σ-weakly dense in Rn for all n. Thus Aˆ + Aˆ∗ is ˆ A + A∗ is σ-weakly dense in M by Haagerup’s dense in R. Since A = E(A), reduction theorem (cf. [10,29]) stated in Sect. 1. That is, A is a subdiagonal algebra with respect to Φ. Hence it is maximal by Theorem 1.1 in [29]. 

148

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IEOT

Acknowledgements The author would like to thank the referee for many helpful comments.

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