Unbounded Conditional Expectations for Partial O^∗-Algebras

Abstract

The main purpose of this paper is to generalize studies of unbounded conditional expectations for O^∗-algebras to those for partial O^∗-algebras.

1. Introduction

In probability theory, conditional expectations play a fundamental role. Conditional expectations for von Neumann algebra have been studied in noncommutative probability theory. In particular, Takesaki [1] characterized the existence of conditional expectation using Tomita's modular theory. Thus a conditional expectation does not necessarily exist for a general von Neumann algebra. The study of conditional expectations for O^∗-algebras was begun by Gudder and Hudson [2]. After that, in [3, 4] we have investigated an unbounded conditional expectation which is a positive linear map ℰ of an O^∗-algebra ℳ onto a given O^∗-subalgebra 𝒩 of ℳ. In this paper we will consider conditional expectations for partial O^∗-algebras. Suppose that ℳ is a self-adjoint partial O^∗-algebra containing identity 𝐼 on dense subspace 𝒟 of Hilbert space ℋ with a strongly cyclic vector 𝜉₀, and 𝒩 is a partial O^∗-subalgebra of ℳ such that (𝒩∩𝑅^w(ℳ))𝜉₀ is dense in ℋ_𝒩≡𝒩𝜉₀, where 𝑅^w(ℳ) is the set of all right multiplier of ℳ. The definitions of (self-adjoint) partial O^∗-algebra and a strongly cyclic vector are stated in Section 2. A map ℰ of ℳ onto 𝒩 is said to be a weak conditional-expectation of (ℳ,𝜉₀) with respect to, 𝒩 if it satisfies (𝐴𝑋𝜉₀∣𝑌𝜉₀)=(ℰ(𝐴)𝑋𝜉₀∣𝑌𝜉₀),forall𝐴∈ℳ,forall𝑋,𝑌∈𝒩∩𝑅^w(ℳ); but, the range ℰ(𝐴) of the weak conditional-expectation ℰ is not necessarily contained in 𝒩, and so we have considered a map ℰ of ℳ onto 𝒩 satisfying the following:

(i)the domain 𝐷(ℰ) of ℰ is a †-invariant subspace of ℳ containing 𝒩;(ii) ℰ is a projection; that is, it is hermitian (ℰ(𝐴)^†=ℰ(𝐴^†),forall𝐴∈𝐷(ℰ)) and ℰ(𝑋)=𝑋,forall𝑋∈𝒩;(iii)ℰ(𝐴□𝑋)=ℰ(𝐴)□𝑋,forall𝐴∈𝐷(ℰ),forall𝑋∈𝒩∩𝑅^w(ℳ), ℰ(𝑋□𝐴)=𝑋□ℰ(𝐴),forall𝐴∈𝐷(ℰ)∩𝑅^w(𝒩),forall𝑋∈𝒩;(iv) 𝜔_𝜉0(ℰ(𝐴))=𝜔_𝜉0(𝐴),forall𝐴∈𝐷(ℰ), where 𝜔_𝜉0 is a state on ℳ defined by 𝜔_𝜉0(𝐴)=(𝐴𝜉₀∣𝜉₀),𝐴∈ℳ; and call it an unbounded conditional expectation of (ℳ,𝜉₀) with respect to, 𝒩. In particular, if 𝐷(ℰ)=ℳ, then ℰ is said to be a conditional expectation of (ℳ,𝜉₀) with respect to, 𝒩.

Finally, we will investigate the scale of the domain of unbounded conditional expectations of partial GW^∗-algebra which is unbounded generalizations of von Neumann algebras.

2. Preliminaries

In this section we review the definitions and the basic theory of partial O^∗-algebras, partial GW^∗-algebras and partial EW^∗-algebras. For more details, refer to [5].

A partial∗-algebra is a complex vector space 𝔄 with an involution 𝑥→𝑥^∗ and a subset Γ⊂𝔄×𝔄 such that

(i) (𝑥,𝑦)∈Γ implies (𝑦^∗,𝑥^∗)∈Γ;(ii)(𝑥,𝑦₁),(𝑥,𝑦₂)∈Γ implies (𝑥,𝜆𝑦₁+𝜇𝑦₂)∈Γ, for all 𝜆,𝜇∈ℂ;(iii) whenever (𝑥,𝑦)∈Γ, there exists a product 𝑥⋅𝑦∈𝔄 with the usual properties of the multiplication: 𝑥⋅(𝑦+𝜆𝑧)=𝑥⋅𝑦+𝜆(𝑥⋅𝑧) and (𝑥⋅𝑦)^∗=𝑦^∗⋅𝑥^∗ for (𝑥,𝑦),(𝑥,𝑧)∈Γ and 𝜆∈ℂ. The element 𝑒 of the 𝔄 is called a unit if 𝑒^∗=𝑒,(𝑒,𝑥)∈Γ for all 𝑥∈𝔄, and 𝑒⋅𝑥=𝑥⋅𝑒=𝑥, for all 𝑥∈𝔄. Notice that the partial multiplication is not required to be associative. Whenever (𝑥,𝑦)∈Γ, 𝑥 is called a left multiplier of 𝑦 and 𝑦 is called a right multiplier of 𝑥, and we write 𝑥∈𝐿(𝑦) and 𝑦∈𝑅(𝑥). For a subset ℬ⊂𝔄, we write𝐿(ℬ)=_𝑥∈ℬ𝐿(𝑥),𝑅(ℬ)=_𝑥∈ℬ𝑅(𝑥). (2.1)

Let ℋ be a Hilbert space with inner product (⋅∣⋅) and 𝒟 a dense subspace of ℋ. We denote by ℒ^†(𝒟,ℋ) the set of all closable linear operators 𝑋 such that 𝒟(𝑋)=𝒟, 𝒟(𝑋^∗)⊇𝒟. The set ℒ^†(𝒟,ℋ) is a partial ∗-algebra with respect to the following operations: the usual sum 𝑋+𝑌, the scalar multiplication 𝜆𝑋, the involution 𝑋→𝑋^†(=𝑋^∗⌈𝒟), and the weak partial multiplication 𝑋□𝑌≡𝑋^†∗𝑌, defined whenever 𝑌 is a weak right multiplier of 𝑋 (𝑋∈𝐿^w(𝑌) or 𝑌∈𝑅^w(𝑋)), that is, if and only if 𝑌𝒟⊂𝒟(𝑋^†∗) and 𝑋^∗𝒟⊂𝒟(𝑌^∗). A partial ∗-subalgebra of ℒ^†(𝒟,ℋ) is called a partial O^∗-algebra on 𝒟.

Let ℳ be a partial O^∗-algebra on 𝒟. The locally convex topology on 𝒟 defined by the family {‖⋅‖_𝑋;𝑋∈ℳ} of seminorms ‖𝜉‖_𝑋=‖𝜉‖+‖𝑋𝜉‖,𝜉∈𝒟 is called the graph topology on 𝒟 and denoted by 𝑡_ℳ. The completion of 𝒟[𝑡_ℳ] is denoted by ^0𝑥0𝒟^02𝑑𝑐[𝑡_ℳ]. If the locally convex space 𝒟[𝑡_ℳ] is complete, then ℳ is called closed. We also define the following domains: 𝒟(ℳ)=⋂_𝑋∈ℳ𝒟(𝑋),𝒟^∗(ℳ)=⋂_𝑋∈ℳ𝒟(𝑋^∗𝒟),_∗∗(ℳ)=⋂_𝑋∈ℳ𝒟((𝑋^∗⌈𝒟^∗(ℳ))^∗), (2.2)and then𝒟⊂𝒟^{0𝑥002𝑑𝑐}(ℳ)⊂𝒟(ℳ)⊂𝒟^∗∗(ℳ)⊂𝒟^∗(ℳ). (2.3)The partial O^∗-algebra ℳ is called fully closed if 𝒟=𝒟(ℳ), self-adjoint if 𝒟=𝒟^∗(ℳ), essentially self-adjoint if 𝒟^∗(ℳ)=𝒟(ℳ), and algebraically self-adjoint if 𝒟^∗(ℳ)=𝒟^∗∗(ℳ).

We defined two weak commutants of ℳ. The weak bounded commutantℳ^′_w of ℳ is the set ℳ^′_w={𝐶∈ℬ(ℋ);(𝐶𝑋𝜉∣𝜂)=(𝐶𝜉∣𝑋^†𝜂)forevery𝑋∈ℳ,𝜉,𝜂∈𝒟}; (2.4)but the partial multiplication is not required to be associative, so we define the quasi-weak bounded commutantℳ^′_qw of ℳ as the set ℳ^′_qw={𝐶∈ℳ^′_w;(𝐶𝑋^†₁𝜉∣𝑋₂𝜂)=(𝐶𝜉∣(𝑋₁□𝑋₂)𝜂)∀𝑋₁∈𝐿(𝑋₂),𝜉,𝜂∈𝒟}. (2.5) In general, ℳ^′_qw⊊ℳ^′_w.

A ∗-representation of a partial ∗-algebra 𝔄 is a ∗-homomorphism of 𝔄 into ℒ^†(𝒟,ℋ), satisfying 𝜋(𝑒)=𝐼 whenever 𝑒∈𝔄, that is,

(i)𝜋 is linear;(ii)𝑥∈𝐿^w(𝑦) in 𝔄 implies 𝜋(𝑥)∈𝐿^w(𝜋(𝑦)) and 𝜋(𝑥)□𝜋(𝑦)=𝜋(𝑥𝑦);(iii)𝜋(𝑥^∗)=𝜋(𝑥)^†forevery𝑥∈𝔄.

Let 𝜋 be a ∗-representation of a partial ∗-algebra 𝔄 into ℒ^†(𝒟,ℋ). Then we define ^0𝑥0𝒟^02𝑑𝑐(𝜋):thecompletionof𝒟withrespecttothegraphtopology𝑡_𝜋(𝔄),_{0𝑥002𝑑𝑐}𝜋(𝑥)=𝜋(𝑥)⌈𝒟^{0𝑥002𝑑𝑐}(𝜋),𝑥∈𝔄;𝒟(𝜋)=_𝑥∈𝔄𝒟(𝜋(𝑥)),𝜋(𝑥)=𝒟𝜋(𝑥)⌈𝒟(𝜋),𝑥∈𝔄;_∗(𝜋)=_𝑥∈𝔄𝒟(𝜋(𝑥)^∗𝜋),_∗(𝑥)=𝜋(𝑥^∗)^∗⌈𝒟^∗(𝜋),𝑥∈𝔄. (2.6)

We say that 𝜋 is closed if 𝒟=𝒟^{0𝑥002𝑑𝑐}(𝜋); fully closed if 𝒟=𝒟(𝜋); essentially self-adjoint if 𝒟(𝜋)=𝒟^∗(𝜋); and self-adjoint if 𝒟=𝒟^∗(𝜋).

We introduce the weak and the quasi-weak commutants of a ∗-representaion 𝜋 of a partial ∗-algebra 𝔄 as follows: 𝜋(𝔄)^′_w={𝐶∈ℬ(ℋ);(𝐶𝜉∣𝜋(𝑥)𝜂)=(𝐶𝜋(𝑥^∗𝒞)𝜉∣𝜂),∀𝑥∈𝔄,𝜉,𝜂∈𝒟(𝜋)},_qw(𝜋)={𝐶∈𝜋(𝔄)^′_w;(𝐶𝜋(𝑥^∗₁)𝜉∣𝜋(𝑥₂)𝜂)=(𝐶𝜉∣𝜋(𝑥₁𝑥₂)𝜂),∀𝑥₁,𝑥₂∈𝔄suchthat𝑥₁∈𝐿(𝑥₂),andall𝜉,𝜂∈𝒟(𝜋)}, (2.7)respectively.

We define the notion of strongly cyclic vector for a partial O^∗-algebra ℳ on 𝒟 in ℋ. A vector 𝜉₀ in 𝒟 is said to be strongly cyclic if 𝑅^w(ℳ)𝜉₀ is dense in 𝒟[𝑡_ℳ], and 𝜉₀ is said to be separating if ℳ^′_w𝜉₀=ℋ, where 𝑅^w(ℳ)={𝑌∈ℳ;𝑋□𝑌iswell-defined,forall𝑋∈ℳ}.

We introduce the notion of partial GW^∗-algebras and partial EW^∗-algebras which are unbounded generalizations of von Neumann algebras. A fully closed partial O^∗-algebra ℳ on 𝒟 is called a partial 𝐺𝑊^∗-algebra if there exists a von Neumann algebra ℳ₀ on ℋ such that ℳ^′₀𝒟⊂𝒟 and ℳ=[ℳ₀⌈𝒟]^𝑠∗. A partial O^∗-algebra ℳ on 𝒟 is said to be a partial EW^∗-algebra if ℳ_𝑏≡{𝐴∈ℬ(ℋ);𝐴⌈𝒟∈ℳ} is a von Neumann algebra, ℳ_𝑏𝒟⊂𝒟 and ℳ_𝑏^′𝒟⊂𝒟.

3. Weak Conditional Expectations

In this section, let ℳ be a self-adjoint partial O^∗-algebra containing the identity 𝐼 on 𝒟 in ℋ with a strongly cyclic vector 𝜉₀ and let 𝒩 be a partial O^∗-subalgebra of ℳ such that

(N)(𝒩∩𝑅^w(ℳ))𝜉₀ is dense in ℋ_𝒩≡𝒩𝜉₀.

The following is easily shown.

Lemma 3.1. Put 𝒟(𝜋_𝒩)=(𝒩∩𝑅^w(ℳ))𝜉₀,𝜋_𝒩(𝑋)𝑌𝜉₀=(𝑋□𝑌)𝜉₀,∀𝑋∈𝒩,∀𝑌∈𝒩∩𝑅^w(ℳ). (3.1) Then 𝜋_𝒩 is a ∗-representations of 𝒩 in the Hilbert space ℋ_𝒩≡𝒟(𝜋_𝒩).

We denote by 𝑃_𝒩 the projection of ℋ onto ℋ_𝒩≡𝒟(𝜋_𝒩). This projection 𝑃_𝒩 plays an important role in this reserch. First we have the following.

Lemma 3.2. It holds that 𝑃_𝒩𝒟⊂𝒟^∗(𝜋_𝒩) and 𝜋^∗_𝒩(𝑋)𝑃_𝒩𝜉=𝑃_𝒩𝑋𝜉, forall𝑋∈𝒩andforall𝜉∈𝒟.

Proof. Take arbitrary 𝑋∈𝒩 and 𝜉∈𝒟. For any 𝑌∈𝒩∩𝑅^w(ℳ), we have (𝜋_𝒩(𝑋^†)𝑌𝜉₀∣𝑃_𝒩𝜉)=((𝑋^†□𝑌)𝜉₀∣𝑃_𝒩𝜉)=(𝑋^†𝑌𝜉₀∣𝜉)=(𝑌𝜉₀∣𝑋𝜉)=(𝑌𝜉₀∣𝑃_𝒩𝑋𝜉), (3.2)and so 𝑃_𝒩𝒟⊂𝒟^∗(𝜋_𝒩) and 𝜋^∗_𝒩(𝑋)𝑃_𝒩𝜉=𝑃_𝒩𝑋𝜉.

Definition 3.3. A map ℰ of ℳ into ℒ^†(𝒟(𝜋_𝒩),ℋ_𝒩) is said to be a weak conditional-expectation of (ℳ,𝜉₀) with respect to, 𝒩 if it satisfies (𝐴𝑋𝜉₀∣𝑌𝜉₀)=(ℰ(𝐴)𝑋𝜉₀∣𝑌𝜉₀),∀𝐴∈ℳ,∀𝑋,𝑌∈𝒩∩𝑅^w(ℳ). (3.3)

For weak conditional-expectation we have the following.

Theorem 3.4. There exists a unique weak conditional-expectation ℰ(⋅∣𝒩) of (ℳ,𝜉₀) with respect to, 𝒩, and ℰ(𝐴∣𝒩)=𝑃_𝒩𝐴⌈𝒟(𝜋_𝒩),∀𝐴∈ℳ. (3.4)The weak conditional-expectation ℰ(⋅∣𝒩) of (ℳ,𝜉₀) with respect to, 𝒩 satisfies the following:
(i)ℰ(⋅∣𝒩) is linear,(ii)ℰ(⋅∣𝒩) is hermitian, that is, ℰ(𝐴∣𝒩)^†=ℰ(𝐴^†∣𝒩),forall𝐴∈ℳ,(iii)ℰ(𝑋∣𝒩)=𝑋⌈𝒟(𝜋_𝒩),forall𝑋∈𝒩,(iv)ℰ(𝐴^†□𝐴∣𝒩)≥0,forall𝐴∈ℳs.t.𝐴^†□𝐴 is well-defined,(v)ℰ(𝐴∣𝒩)^†□ℰ(𝐴∣𝒩)≤ℰ(𝐴^†□𝐴∣𝒩),forall𝐴∈ℳs.t.𝐴^†□𝐴 and ℰ(𝐴∣𝒩)^†□ℰ(𝐴∣𝒩) are well-defined,(vi)ℰ(𝐴∣𝒩)□𝜋_𝒩(𝑋) is well-defined for any 𝐴∈ℳ and 𝑋∈𝒩∩𝑅^w(ℳ), and ℰ(𝐴∣𝒩)□𝜋_𝒩(𝑋)=ℰ(𝐴□𝑋∣𝒩),(vii)𝜋_𝒩(𝑋)□ℰ(𝐴∣𝒩) is well-defined for any 𝐴∈ℳ∩𝑅^w(𝒩) and forall𝑋∈𝒩, and 𝜋_𝒩(𝑋)□ℰ(𝐴∣𝒩)=ℰ(𝑋□𝐴∣𝒩),(viii)𝜔_𝜉0(ℰ(𝐴∣𝒩))=𝜔_𝜉0(𝐴),forall𝐴∈ℳ.

Proof. We put ℰ(𝐴∣𝒩)=𝑃_𝒩𝐴⌈𝒟(𝜋_𝒩),∀𝐴∈ℳ. (3.5)By Lemma 3.2, ℰ(𝐴∣𝒩) is a linear map of 𝒟(𝜋_𝒩) into 𝒟^∗(𝜋_𝒩) for any 𝐴∈ℳ, and furthermore we have ℰ(𝐴∣𝒩)^†=ℰ(𝐴^†∣𝒩),forall𝐴∈ℳ, so ℰ(⋅∣𝒩) is a map of ℳ into ℒ^†(𝒟(𝜋_𝒩),ℋ_𝒩).
Since (ℰ(𝐴∣𝒩)𝑋𝜉₀∣𝑌𝜉₀)=(𝑃_𝒩𝐴𝑋𝜉₀∣𝑌𝜉₀)=(𝐴𝑋𝜉₀∣𝑌𝜉₀) (3.6)for each 𝐴∈ℳ,𝑋,𝑌∈𝒩∩𝑅^w(ℳ), ℰ(⋅∣𝒩) is a weak conditional-expectation of (ℳ,𝜉₀) with respect to, 𝒩. It is easily shown that if ℰ is a weak conditional-expectation of (ℳ,𝜉₀) with respect to, 𝒩, ℰ(𝐴)=ℰ(𝐴∣𝒩) for each 𝐴∈ℳ. Thus the existence and uniqueness of weak conditional-expectations is shown. The statements (iii)–(viii) follow since ℰ(𝐴∣𝒩)=𝑃_𝒩𝐴⌈𝒟(𝜋_𝒩), forall𝐴∈ℳ. This completes the proof.

4. Unbounded Conditional Expectations for Partial O^∗-Algebras

Let ℳ be a self-adjoint partial O^∗-algebra containing 𝐼 on 𝒟 in ℋ and let 𝜉₀∈𝒟 be a strongly cyclic and separating vector for ℳ and suppose that 𝒩∋𝐼 is a partial O^∗-subalgebra of ℳ satisfying (N): (𝒩∩𝑅^w(ℳ))𝜉₀ is dense in ℋ_𝒩. We introduce unbounded conditional expectations of (ℳ,𝜉₀) with respect to, 𝒩.

Definition 4.1. A map ℰ of ℳ onto 𝒩 is said to be an unbounded conditional expectation of (ℳ,𝜉₀) with respect to, 𝒩 if
(i) the domain 𝐷(ℰ) of ℰ is a †-invariant subspace of ℳ containing 𝒩;(ii)ℰ is a projection; that is, it is hermitian (ℰ(𝐴)^†=ℰ(𝐴^†),forall𝐴∈𝐷(ℰ)) and ℰ(𝑋)=𝑋,forall𝑋∈𝒩;(iii)ℰ(𝐴□𝑋)=ℰ(𝐴)□𝑋,forall𝐴∈𝐷(ℰ),forall𝑋∈𝒩∩𝑅^w(ℳ), ℰ(𝑋□𝐴)=𝑋□ℰ(𝐴),forall𝐴∈𝐷(ℰ)∩𝑅^w(𝒩),forall𝑋∈𝒩;(iv)𝜔_𝜉0(ℰ(𝐴))=𝜔_𝜉0(𝐴),forall𝐴∈𝐷(ℰ). In particular, if 𝐷(ℰ)=ℳ, then ℰ is said to be a conditional expectation of (ℳ,𝜉₀) with respect to, 𝒩.

For unbounded conditional expectations we have the following.

Lemma 4.2. Let ℰ be an unbounded conditional expectation of (ℳ,𝜉₀) with respect to, 𝒩. Then, ℰ(𝐴)𝑋𝜉₀=𝑃_𝒩𝐴𝑋𝜉₀=ℰ(𝐴∣𝒩)𝑋𝜉₀,∀𝐴∈𝐷(ℰ),∀𝑋∈𝒩∩𝑅^w(ℳ). (4.1)

Proof. For all 𝐴∈𝐷(ℰ) and 𝑋,𝑌∈𝒩∩𝑅^w(ℳ), we have (ℰ(𝐴)𝑋𝜉₀∣𝑌𝜉₀)=(ℰ(𝐴□𝑋)𝜉₀∣𝑌𝜉₀)=(ℰ(𝑌^†□𝐴□𝑋)𝜉₀∣𝜉₀)=((𝑌^†□𝐴□𝑋)𝜉₀∣𝜉₀)=(𝐴𝑋𝜉₀∣𝑌𝜉₀)=(𝐴𝑋𝜉₀∣𝑃_𝒩𝑌𝜉₀)=(𝑃_𝒩𝐴𝑋𝜉₀∣𝑌𝜉₀). (4.2) Hence, ℰ(𝐴)𝑋𝜉₀=𝑃_𝒩𝐴𝑋𝜉₀=ℰ(𝐴∣𝒩)𝑋𝜉₀,forall𝐴∈𝐷(ℰ),forall𝑋∈𝒩∩𝑅^w(ℳ).

Let 𝔈 be the set of all unbounded conditional expectations of (ℳ,𝜉₀) with respect to, 𝒩. Then 𝔈 is an ordered set with the following order ⊂: ℰ₁⊂ℰ₂iff𝐷(ℰ₁)⊂𝐷(ℰ₂),ℰ₁(𝐴)=ℰ₂(𝐴),∀𝐴∈𝐷(ℰ₁). (4.3)

Theorem 4.3. There exists a maximal unbounded conditional expectation of (ℳ,𝜉₀) with respect to, 𝒩, and it is denoted by ℰ_𝒩.

Proof. We put 𝐷(ℰ₀)≡{𝐴∈ℳ;𝑃_𝒩𝐴⌈_(𝒩∩𝑅^w_(ℳ))𝜉0∈𝒩⌈_(𝒩∩𝑅^w_(ℳ))𝜉0}. (4.4)Then, for any 𝐴∈𝐷(ℰ₀), there exists a unique map ℰ₀ such that ℰ₀(𝐴)𝑋𝜉₀=𝑃_𝒩𝐴𝑋𝜉₀=ℰ(𝐴∣𝒩)𝑋𝜉₀,∀𝑋∈𝒩∩𝑅^w(ℳ). (4.5)It is easily shown that ℰ₀ is an unbounded conditional expectation of (ℳ,𝜉₀) with respect to, 𝒩. Furthermore, ℰ₀ is maximal in 𝔈. Indeed, let ℰ∈𝔈. Take an arbitrary 𝐴∈𝐷(ℰ). Then by Lemma 4.2 we haveℰ(𝐴)𝑋𝜉₀=𝑃_𝒩𝐴𝑋𝜉₀=ℰ(𝐴∣𝒩)𝑋𝜉₀,𝑋∈𝒩∩𝑅^w(ℳ), (4.6)which implies ℰ(𝐴)𝑋𝜉₀∈𝒩⌈_(𝒩∩𝑅^w_(ℳ))𝜉0. Hence ℰ⊂ℰ₀ and ℰ₀ is maximal in 𝔈. This completes the proof.

5. Existence of Conditional Expectations for Partial O^∗-Algebras

Let ℳ be a self-adjoint partial O^∗-algebra containing 𝐼 on 𝒟 in ℋ, 𝜉₀∈𝒟 be a strongly cyclic and separating vector for ℳ and 𝒩∋𝐼 a partial O^∗-subalgebra of ℳ such that

(N) (𝒩∩𝑅^w(ℳ))𝜉₀ is dense in ℋ_𝒩,(N₁)𝒩^′_w𝒟(𝒩)⊂𝒟(𝒩),(N₂)(𝒩∩𝑅^w(ℳ))𝜉₀ is essentially self-adjoint for 𝒩,(N₃)Δ^′′it_𝜉0(𝒩^′_w)^′Δ^′′−it_𝜉0=(𝒩^′_w)^′,forall𝑡∈ℝ, where Δ_𝜉^′′₀ is the modular operator for the full Hilbert algebra (ℳ^′_w)^′𝜉₀.

Lemma 5.1. It holds that 𝐷(ℰ_𝒩)={𝐴∈ℳ;𝑃_𝒩𝐴𝜉₀∈𝒩𝜉₀}.

Proof. We put 𝐷(ℰ)={𝐴∈ℳ;𝑃_𝒩𝐴𝜉₀∈𝒩𝜉₀}. (5.1)By Lemma 4.2, we have 𝑃_𝒩𝐴𝜉₀=ℰ_𝒩(𝐴)𝜉₀∈𝒩𝜉₀ (5.2)for each 𝐴∈𝐷(ℰ_𝒩). Hence, 𝐷(ℰ_𝒩)⊂𝐷(ℰ). We show the converse inclusion. Since 𝜉₀ is separating vector for ℳ, it follows that for any 𝐴∈𝐷(ℰ), there exists a unique element ℰ(𝐴) of 𝒩 such that 𝑃_𝒩𝐴𝜉₀=ℰ(𝐴)𝜉₀. Indeed, since ℰ_𝒩 is maximal in 𝔈, it is sufficient to show that ℰ is an unbounded conditional expectation of (ℳ,𝜉₀) with respect to, 𝒩. By assumption (N₁) and [5, Proposition 2.3.5], we have𝑋isaffiliatedwithvonNeumannalgebra(𝒩^′_w)^′foreach𝑋∈𝒩,𝒩^′_w=𝒩^′_qw. (5.3)(5.4) Since ℳ is self-adjoint and (𝒩∩𝑅^w(ℳ))𝜉₀ is dense in ℋ_𝒩, it follows that (𝒩∩𝑅^w(ℳ))𝜉₀ is a reducing subspace for 𝒩, that is, 𝒩(𝒩∩𝑅^w(ℳ))𝜉₀⊂(𝒩∩𝑅^w(ℳ))𝜉₀=𝒩𝜉₀, (5.5)which implies by assumption (N₂) and [5, Theorem 7.4.4] that 𝑃_𝒩∈𝑁^′_𝑤,𝑃_𝒩𝒟(𝒩)⊂𝒟(𝒩). (5.6) Furthermore, by (5.3) and (5.6), we have 𝒩𝜉₀=(𝒩^′_w)^′𝜉₀,thatis,𝒫_𝒩=𝒫_{(𝒩′w)′}. (5.7)Let 𝑆_𝜉0 and 𝑆^′′_𝜉0 be the closures of the maps: 𝑆_𝜉0𝐴𝜉₀=𝐴^†𝜉₀𝑆,𝐴∈ℳ,_𝜉′′0𝐵𝜉₀=𝐵^∗𝜉₀,𝐵∈(ℳ^′_w)^′. (5.8)By (5.3) we have 𝑆_𝜉0⊂𝑆_𝜉^′′₀. (5.9) Takesaki proved in [1] that assumtion (N₃ ) implies 𝑃_{(𝒩′w)′}𝑆_𝜉^′′₀⊂𝑆_𝜉^′′₀𝑃_{(𝒩′w)′} (5.10)and there exists a conditional expectation ℰ^′′ of the von Neumann algebra ((ℳ^′_w)^′,𝜉₀) with respect to, (𝒩^′_w)^′.
By (5.6), (5.9), and (5.10), we haveℰ(𝐴^†)𝜉₀=𝑃_𝒩𝐴^†𝜉₀=𝑃_𝒩𝑆_𝜉0𝐴𝜉₀=𝑃_𝒩𝑆_𝜉^′′₀𝐴𝜉₀=𝑆_𝜉^′′₀𝑃_𝒩𝐴𝜉₀=𝑆_𝜉^′′₀ℰ(𝐴)𝜉₀=𝑆_𝜉0ℰ(𝐴)𝜉₀=ℰ(𝐴)^†𝜉₀ (5.11)for each 𝐴∈𝐷(ℰ), which implies by the separateness of 𝜉₀ that ℰ is hermitian.
It is clear that ℰ(𝑋)=𝑋,forall𝑋∈𝒩. Take arbitrary 𝐴∈𝐷(ℰ) and 𝑋∈𝒩∩𝐿^w(ℳ). Since (𝑃_𝒩(𝑋□𝐴)𝜉₀∣𝑌𝜉₀)=(𝑃_𝒩𝐴𝜉₀∣𝑋^†𝑌𝜉₀)=(ℰ(𝐴)𝜉₀∣𝑋^†𝑌𝜉₀)=((𝑋□ℰ(𝐴))𝜉₀∣𝑌𝜉₀) (5.12) for each 𝑌∈𝒩∩𝑅^w(ℳ), it follows that 𝑋□𝐴∈𝐷(ℰ) and ℰ(𝑋□𝐴)=𝑋□ℰ(𝐴). Furthermore, since ℰ is hermitian, it follows that 𝐴□𝑋∈𝐷(ℰ) and ℰ(𝐴□𝑋)=ℰ(𝐴)□𝑋 for each 𝐴∈𝐷(ℰ) and 𝑋∈𝒩∩𝑅^w(ℳ). It is clear that 𝜔_𝜉0(ℰ(𝐴))=𝜔_𝜉0(𝐴) for each 𝐴∈𝐷(ℰ). Thus ℰ is an unbounded conditional expectation of (ℳ,𝜉₀) with respect to, 𝒩. This completes that proof.

By Lemma 5.1, we have the following.

Theorem 5.2. Let ℳ be a self-adjoint partial O^∗-algebra containing 𝐼 on 𝒟 in ℋ and let 𝜉₀∈𝒟 be a strongly cyclic and separating vector for ℳ and suppose that 𝒩∋𝐼 is a partial O^∗-subalgebra of ℳ satisfying (N), (N₁), (N ₂), and (N₃). Then there exists a conditional expectation of (ℳ,𝜉₀) with respect to, 𝒩 if and only if 𝑃_𝒩ℳ𝜉₀=𝒩𝜉₀.

It is important to investigate the scale of the domain of an unbounded conditional expectation. We consider the case of partial GW^∗-algebras.

Theorem 5.3. Let ℳ be a partial GW^∗-algebra on 𝒟 in ℋ and let 𝜉₀∈𝒟 be a strongly cyclic and separating vector for ℳ and suppose that 𝒩 be a partial GW^∗-subalgebra of ℳ satisfying (N), (N₁), (N ₂), and (N₃).
Then, 𝐷(ℰ_𝒩)⊃ linear span of {𝑋□𝐴;𝑋∈𝒩,𝐴∈(ℳ^′_w)^′s.t.𝑋□𝐴 and 𝑋□ℰ^′′(𝐴) are well defined}⊃linearspanof(ℳ^′_w)^′and𝒩.
In particular, if 𝒩_𝒫𝒩 is a partial GW^∗-algebra on 𝑃_𝒩𝒟, then ℰ_𝒩 is a conditional expectation of (ℳ,𝜉₀) with respect to, 𝒩.

Proof. Let 𝑋∈𝒩, and 𝐴∈(ℳ^′_w)^′s.t.𝑋□𝐴and𝑋□ℰ^′′(𝐴) are all defined. Then, it follows since 𝒩 is a partial GW^∗-subalgebra of ℳ that 𝑃_𝒩(𝑋□𝐴)𝜉₀=𝑃_𝒩𝑋^†∗𝐴𝜉₀=𝑋^†∗𝑃_𝒩𝐴𝜉₀=(𝑋□ℰ^′′(𝐴))𝜉₀∈𝒩𝜉₀, (5.13) which implies by Lemma 5.1 that 𝑋□𝐴∈𝐷(ℰ_𝒩) and 𝑃_𝒩(𝑋□𝐴)𝜉₀=(𝑋□ℰ^′′(𝐴))𝜉₀. Suppose that 𝒩_𝒫𝒩 is a partial GW^∗-algebra on 𝒫_𝒩𝒟.
By the result of Takesaki [1] there exists a unique conditional expectation ℰ^′′ of the von Neumann algebra (𝒩^′_w)^′ such that ℰ^′′(𝐴_𝛼)_𝑃𝒩=𝑃_𝒩𝐴𝑃_𝒩 for each 𝐴∈(ℳ^′_w)^′. Since ℳ is a partial GW^∗-algebra, for any 𝑋∈ℳ there is a net {𝐴_𝛼}∈(ℳ^′_w)^′ which converges strongly^∗ to 𝑋. Then ℰ^′′(𝐴_𝛼)_𝑃𝒩∈((𝒩^′_w)^′)_𝑃𝒩=((𝒩_𝑃𝒩)^′_w)^′, (5.14)and ℰ^′′(𝐴_𝛼)_𝑃𝒩 converges strongly^∗ to 𝑃_𝒩𝑋⌈𝑃_𝒩𝒟. Therefore, we have 𝑃_𝒩𝑋⌈𝑃_𝒩𝒟∈𝒩. Hence, 𝑋∈𝐷(ℰ_𝒩) and ℰ_𝒩 is a conditional expectation of (ℳ,𝜉₀) with respect to, 𝒩. This completes the proof.

Corollary 5.4. Let ℳ be a partial EW^∗-algebra on 𝒟 in ℋ and let 𝜉₀∈𝒟 be a strongly cyclic and separating vector for ℳ and suppose that 𝒩 be a partial EW^∗-subalgebra of ℳ satisfying (N₂) and (N₃). Then, 𝐷(ℰ_𝒩)⊃linearspanofℳ_𝑏𝒩and𝒩ℳ_𝑏. (5.15)

Proof. Since ℳ_𝑏⊂𝑅^w(ℳ), it follows that 𝒩∩𝑅^w(ℳ)⊃𝒩_𝑏, and so clearly (N) holds. Furthermore, (N₁) holds since 𝒩^′_w𝒟(𝒩)=𝒩_𝑏𝒟(𝒩)⊂𝒟(𝒩). This completes the proof.

We consider the case of the well-known Segal 𝐿^𝑝-space defined by 𝜏.

Example 5.5. Let ℳ₀ be a von Neumann algebra on a Hilbert space ℋ with a faithful finite trace 𝜏. We denote by 𝐿^𝑝(𝜏) the Banach space completion of ℳ₀ with respect to, the norm ‖𝐴‖_𝑝≡𝜏(|𝐴|^𝑝)^1/𝑝,𝐴∈ℳ₀. (5.16)Thenℳ₀≡𝐿^∞(𝜏)⊂𝐿^𝑝(𝜏)⊂𝐿²(𝜏)⊂𝐿^𝑞(𝜏)⊂𝐿¹(𝜏),1≤𝑞≤2≤𝑝<∞. (5.17)Let 2≤𝑝<∞. Here we define a ∗-representation 𝜋 of 𝐿^𝑝(𝜏) by 𝜋(𝑋)𝐴=𝑋𝐴,𝑋∈𝐿^𝑝(𝜏),𝐴∈𝐿^∞(𝜏). (5.18)Then ℳ≡𝜋(𝐿^𝑝(𝜏)) is a partial EW^∗-algebra on 𝐿^∞(𝜏) in 𝐿²(𝜏) with ℳ_𝑏=𝜋(𝐿^∞(𝜏)) which is integrable, that is, 𝜋(𝑋^†)=𝜋(𝑋)^∗ for each 𝑋∈𝐿^𝑝(𝜏). Furthremore, 𝜋(𝐿^𝑝(𝜏)) has a strongly cyclic and separating vector 𝜉₀≡𝜆_𝜏(𝐼), where 𝐼 is an identity operator on ℋ. Let 𝒩₀ be a von Neumann subalgebra of ℳ₀. We put 𝒩={𝜋(𝑋);𝑋∈𝐿^𝑝(𝜏),𝜋(𝑋)𝜆_𝜏(𝐼)∈𝐿^𝑝(𝜏⌈𝒩₀)},2≤𝑝≤∞. (5.19)Then 𝒩 is an integrable partial EW^∗-subalgebra of ℳ satisfying (N₂) and (N₃) and 𝑃_𝒩ℳ𝜉₀=𝒩𝜉₀. By Theorem 5.2, there exists a conditional expectation of (ℳ,𝜉₀).