# Unbounded Conditional Expectations for  PartialO∗-Algebras

Authors: Mayumi Takakura
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

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


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## Body

## 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∩Rw(ℳ))ξ0 is dense in ℋ𝒩≡𝒩ξ0¯, where Rw(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∩Rw(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 domainD(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∩Rw(ℳ), ℰ(X□A)=X□ℰ(A),forallA∈D(ℰ)∩Rw(𝒩),forallX∈𝒩;(iv)
ωξ0(ℰ(A))=ωξ0(A),forallA∈D(ℰ), where ωξ0 is a state on ℳ defined by ωξ0(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 partialGW*-algebra which
is unbounded generalizations of von Neumann algebras.


## 2. Preliminaries

In this section we review the definitions and the basic
theory of partialO*-algebras,
partial GW*-algebras  and
partial EW*-algebras. For
more details, refer to [5].Apartial*-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 write(2.1)L(ℬ)=⋂x∈ℬL(x),R(ℬ)=⋂x∈ℬR(x).Letℋ 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∈Lw(Y) or Y∈Rw(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 partialO*-algebra on 𝒟.Letℳ 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 tℳ. The completion of 𝒟[tℳ] is denoted by 𝒟˜[tℳ]. If the locally convex space 𝒟[tℳ] is complete,
then ℳ is called closed. We also define the following
domains: (2.2)𝒟^(ℳ)=⋂X∈ℳ𝒟(X¯),𝒟*(ℳ)=⋂X∈ℳ𝒟(X*),𝒟**(ℳ)=⋂X∈ℳ𝒟((X*⌈𝒟*(ℳ))*),and then(2.3)𝒟⊂𝒟˜(ℳ)⊂𝒟^(ℳ)⊂𝒟**(ℳ)⊂𝒟*(ℳ).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 ofℳ. The weak
bounded commutantℳw′ of ℳ is the
set (2.4)ℳw′={C∈ℬ(ℋ);(CXξ∣η)=(Cξ∣X†η)foreveryX∈ℳ,ξ,η∈𝒟};but the partial multiplication
is not required to be associative, so we define the quasi-weak bounded commutantℳqw′ of ℳ  as the
set (2.5)ℳqw′={C∈ℳw′;(CX1†ξ∣X2η)=(Cξ∣(X1□X2)η)∀X1∈L(X2),ξ,η∈𝒟}. In general, ℳqw′⊊ℳw′.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∈Lw(y) in 𝔄 implies π(x)∈Lw(π(y)) and π(x)□π(y)=π(xy);(iii)
π(x*)=π(x)†foreveryx∈𝔄.Letπ be a *-representation
of a partial *-algebra 𝔄 into ℒ†(𝒟,ℋ). Then we define  (2.6)𝒟˜(π):thecompletionof𝒟withrespecttothegraphtopologytπ(𝔄),π˜(x)=π(x)¯⌈𝒟˜(π),x∈𝔄;𝒟^(π)=⋂x∈𝔄𝒟(π(x)¯),π^(x)=π(x)¯⌈𝒟^(π),x∈𝔄;𝒟*(π)=⋂x∈𝔄𝒟(π(x)*),π*(x)=π(x*)*⌈𝒟*(π),x∈𝔄.We say thatπ is closed if 𝒟=𝒟˜(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: (2.7)π(𝔄)w′={C∈ℬ(ℋ);(Cξ∣π(x)η)=(Cπ(x*)ξ∣η),∀x∈𝔄,ξ,η∈𝒟(π)},𝒞qw(π)={C∈π(𝔄)w′;(Cπ(x1*)ξ∣π(x2)η)=(Cξ∣π(x1x2)η),∀x1,x2∈𝔄suchthatx1∈L(x2),andallξ,η∈𝒟(π)},respectively.We define the notion of strongly cyclic vector for a
partialO*-algebra ℳ on 𝒟 in ℋ. A vector ξ0 in 𝒟 is said to be strongly cyclic if Rw(ℳ)ξ0 is dense in 𝒟[tℳ], and ξ0 is said to be separating if ℳw'ξ0¯=ℋ, where Rw(ℳ)={Y∈ℳ;X□Yiswell-defined,forallX∈ℳ}.We introduce the notion of partialGW*-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 partialGW*-algebra if there exists a von Neumann algebra ℳ0 on ℋ such that ℳ0′𝒟⊂𝒟 and ℳ=[ℳ0⌈𝒟]s*. A partial O*-algebra ℳ on 𝒟 is said to be a
partial EW*-algebra if ℳb¯≡{A∈ℬ(ℋ);A⌈𝒟∈ℳ} is a von Neumann algebra, ℳb𝒟⊂𝒟 and ℳb¯'𝒟⊂𝒟.


## 3. Weak Conditional Expectations

In this section, letℳ 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∩Rw(ℳ))ξ0 is dense in ℋ𝒩≡𝒩ξ0¯.The following is easily shown.Lemma 3.1.
Put(3.1)𝒟(π𝒩)=(𝒩∩Rw(ℳ))ξ0,π𝒩(X)Yξ0=(X□Y)ξ0,∀X∈𝒩,∀Y∈𝒩∩Rw(ℳ).  Then π𝒩 is a *-representations
of 𝒩 in the Hilbert
space ℋ𝒩≡𝒟(π𝒩¯).We denote byP𝒩 the projection
of ℋ onto ℋ𝒩≡𝒟(π𝒩)¯. This projection P𝒩 plays an
important role in this reserch. First we have the following.Lemma 3.2.
It holds thatP𝒩𝒟⊂𝒟*(π𝒩)  and  π𝒩*(X)P𝒩ξ=P𝒩Xξ,  forallX∈𝒩andforallξ∈𝒟.Proof.
Take arbitraryX∈𝒩 and   ξ∈𝒟. For any Y∈𝒩∩Rw(ℳ), we have   (3.2)(π𝒩(X†)Yξ0∣P𝒩ξ)=((X†□Y)ξ0∣P𝒩ξ)=(X†Yξ0∣ξ)=(Yξ0∣Xξ)=(Yξ0∣P𝒩Xξ),and so P𝒩𝒟⊂𝒟*(π𝒩) and π𝒩*(X)P𝒩ξ=P𝒩Xξ.Definition 3.3.
A mapℰ  of  ℳ  into  ℒ†(𝒟(π𝒩),ℋ𝒩)
is said to be
a weak conditional-expectation of  (ℳ,ξ0)  with respect to,
𝒩
if it
satisfies (3.3)(AXξ0∣Yξ0)=(ℰ(A)Xξ0∣Yξ0),∀A∈ℳ,∀X,Y∈𝒩∩Rw(ℳ).For weak conditional-expectation we have
the following.Theorem 3.4.
There exists a unique weak
conditional-expectationℰ(⋅∣𝒩)  of  (ℳ,ξ0)  with respect to,
𝒩, and (3.4)ℰ(A∣𝒩)=P𝒩A⌈𝒟(π𝒩),∀A∈ℳ.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(π𝒩),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)□π𝒩(X)  is well-defined for any A∈ℳ and X∈𝒩∩Rw(ℳ), and ℰ(A∣𝒩)□π𝒩(X)=ℰ(A□X∣𝒩),(vii)
π𝒩(X)□ℰ(A∣𝒩)   is well-defined
for any A∈ℳ∩Rw(𝒩) and forallX∈𝒩, and π𝒩(X)□ℰ(A∣𝒩)=ℰ(X□A∣𝒩),(viii)
ωξ0(ℰ(A∣𝒩))=ωξ0(A),forallA∈ℳ.Proof.
We put(3.5)ℰ(A∣𝒩)=P𝒩A⌈𝒟(π𝒩),∀A∈ℳ.By Lemma 3.2, ℰ(A∣𝒩) is a linear map
of 𝒟(π𝒩) into 𝒟*(π𝒩) 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(π𝒩),ℋ𝒩).
Since(3.6)(ℰ(A∣𝒩)Xξ0∣Yξ0)=(P𝒩AXξ0∣Yξ0)=(AXξ0∣Yξ0)for each A∈ℳ,X,Y∈𝒩∩Rw(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)=P𝒩A⌈𝒟(π𝒩),  forallA∈ℳ. This completes the proof.


## 4. Unbounded Conditional Expectations for PartialO*-Algebras

Letℳ 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∩Rw(ℳ))ξ0 is dense in ℋ𝒩. We introduce unbounded conditional expectations of (ℳ,ξ0) with respect to,
𝒩.Definition 4.1.
A mapℰ  of  ℳ  onto  𝒩
is said to be
an unbounded conditional expectation of  (ℳ,ξ0)  with respect to,
𝒩  if
(i)
the domainD(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∩Rw(ℳ), ℰ(X□A)=X□ℰ(A),forallA∈D(ℰ)∩Rw(𝒩),forallX∈𝒩;(iv)
ωξ0(ℰ(A))=ωξ0(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.
Letℰ
be an unbounded
conditional expectation of  (ℳ,ξ0)  with respect to,
𝒩. Then, (4.1)ℰ(A)Xξ0=P𝒩AXξ0=ℰ(A∣𝒩)Xξ0,∀A∈D(ℰ),∀X∈𝒩∩Rw(ℳ).Proof.
For allA∈D(ℰ) and X,Y∈𝒩∩Rw(ℳ), we
have (4.2)(ℰ(A)Xξ0∣Yξ0)=(ℰ(A□X)ξ0∣Yξ0)=(ℰ(Y†□A□X)ξ0∣ξ0)=((Y†□A□X)ξ0∣ξ0)=(AXξ0∣Yξ0)=(AXξ0∣P𝒩Yξ0)=(P𝒩AXξ0∣Yξ0). Hence, ℰ(A)Xξ0=P𝒩AXξ0=ℰ(A∣𝒩)Xξ0,forallA∈D(ℰ),forallX∈𝒩∩Rw(ℳ).Let𝔈 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 ⊂: (4.3)ℰ1⊂ℰ2iffD(ℰ1)⊂D(ℰ2),ℰ1(A)=ℰ2(A),∀A∈D(ℰ1).Theorem 4.3.
There exists a maximal unbounded conditional
expectation of(ℳ,ξ0)  with respect to,
𝒩, and it is denoted by  ℰ𝒩.Proof.
We put(4.4)D(ℰ0)≡{A∈ℳ;P𝒩A⌈(𝒩∩Rw(ℳ))ξ0∈𝒩⌈(𝒩∩Rw(ℳ))ξ0}.Then, for any A∈D(ℰ0), there exists a
unique map ℰ0 such
that (4.5)ℰ0(A)Xξ0=P𝒩AXξ0=ℰ(A∣𝒩)Xξ0,∀X∈𝒩∩Rw(ℳ).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 have(4.6)ℰ(A)Xξ0=P𝒩AXξ0=ℰ(A∣𝒩)Xξ0,X∈𝒩∩Rw(ℳ),which implies ℰ(A)Xξ0∈𝒩⌈(𝒩∩Rw(ℳ))ξ0. Hence ℰ⊂ℰ0 and ℰ0 is maximal in 𝔈. This completes the proof.


## 5. Existence of Conditional Expectations for PartialO*-Algebras

Letℳ 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∩Rw(ℳ))ξ0 is dense in ℋ𝒩,(N1)
𝒩w′𝒟^(𝒩)⊂𝒟^(𝒩),(N2)
(𝒩∩Rw(ℳ))ξ0 is essentially
self-adjoint for 𝒩,(N3)
Δξ0′′it(𝒩w′)′Δξ0′′−it=(𝒩w′)′,forallt∈ℝ, where Δξ0′′ is the modular
operator for the full Hilbert algebra (ℳw′)′ξ0.Lemma 5.1.
It holds thatD(ℰ𝒩)={A∈ℳ;P𝒩Aξ0∈𝒩ξ0}.Proof.
We put(5.1)D(ℰ)={A∈ℳ;P𝒩Aξ0∈𝒩ξ0}.By  Lemma 4.2, we have (5.2)P𝒩Aξ0=ℰ𝒩(A)ξ0∈𝒩ξ0for each A∈D(ℰ𝒩). Hence, D(ℰ𝒩)⊂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 P𝒩Aξ0=ℰ(A)ξ0. Indeed, since ℰ𝒩 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 have(5.3)X¯isaffiliatedwithvonNeumannalgebra(𝒩w′)′foreachX∈𝒩,(5.4)𝒩w′=𝒩qw′. Since ℳ is self-adjoint
and (𝒩∩Rw(ℳ))ξ0 is dense in ℋ𝒩, it follows that (𝒩∩Rw(ℳ))ξ0 is a reducing
subspace for 𝒩, that is, (5.5)𝒩(𝒩∩Rw(ℳ))ξ0⊂(𝒩∩Rw(ℳ))ξ0¯=𝒩ξ0¯,which implies by assumption (N2) and
[5, Theorem 7.4.4]
that  (5.6)P𝒩∈Nw′,P𝒩𝒟^(𝒩)⊂𝒟^(𝒩).  Furthermore, by (5.3) and (5.6), we have (5.7)𝒩ξ0¯=(𝒩w′)′ξ0¯,thatis,𝒫𝒩=𝒫(𝒩w′)′.Let  Sξ0 and Sξ0′′ be the closures of the maps:  (5.8)Sξ0Aξ0=A†ξ0,A∈ℳ,Sξ0′′Bξ0=B*ξ0,B∈(ℳw′)′.By (5.3) we have (5.9)Sξ0⊂Sξ0′′. Takesaki proved
in [1]  that  assumtion (N3 ) implies (5.10)P(𝒩w′)′Sξ0′′⊂Sξ0′′P(𝒩w′)′and there
exists a conditional expectation ℰ′′ of the von
Neumann algebra ((ℳw′)′,ξ0) with respect to, (𝒩w′)′.
By (5.6), (5.9), and (5.10), we have(5.11)ℰ(A†)ξ0=P𝒩A†ξ0=P𝒩Sξ0Aξ0=P𝒩Sξ0′′Aξ0=Sξ0′′P𝒩Aξ0=Sξ0′′ℰ(A)ξ0=Sξ0ℰ(A)ξ0=ℰ(A)†ξ0for each A∈D(ℰ), which implies by the separateness of ξ0 that ℰ is hermitian.
It is clear thatℰ(X)=X,forallX∈𝒩. Take arbitrary A∈D(ℰ) and X∈𝒩∩Lw(ℳ). Since (5.12)(P𝒩(X□A)ξ0∣Yξ0)=(P𝒩Aξ0∣X†Yξ0)=(ℰ(A)ξ0∣X†Yξ0)=((X□ℰ(A))ξ0∣Yξ0) for each Y∈𝒩∩Rw(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∩Rw(ℳ). It is clear that ωξ0(ℰ(A))=ωξ0(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 Lemma5.1, we have the following.Theorem 5.2.
Letℳ
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), (N2), and (N3). Then there
exists a conditional expectation of  (ℳ,ξ0)  with respect to,
𝒩  if and only if  P𝒩ℳξ0=𝒩ξ0.It is important to investigate the scale of the domain
of an unbounded conditional expectation. We consider the case of partialGW*-algebras.Theorem 5.3.
Letℳ  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), (N2),  and (N3).
Then,D(ℰ𝒩)⊃ linear span of {X□A;X∈𝒩,A∈(ℳw′)′s.t.X□A and X□ℰ′′(A) are well defined}⊃linearspanof(ℳw′)′and𝒩.
In particular, if𝒩𝒫𝒩 is a partial GW*-algebra on P𝒩𝒟, then ℰ𝒩 is a
conditional expectation of (ℳ,ξ0) with respect to,
𝒩.Proof.
LetX∈𝒩, and A∈(ℳw′)′s.t.X□AandX□ℰ′′(A) are all defined. Then, it follows since 𝒩 is a partial GW*-subalgebra of ℳ that (5.13)P𝒩(X□A)ξ0=P𝒩X†*Aξ0=X†*P𝒩Aξ0=(X□ℰ′′(A))ξ0∈𝒩ξ0, which implies by Lemma 5.1 that X□A∈D(ℰ𝒩) and P𝒩(X□A)ξ0=(X□ℰ′′(A))ξ0. Suppose that 𝒩𝒫𝒩 is a partial GW*-algebra on 𝒫𝒩𝒟.
By the result of Takesaki [1] there exists a unique
conditional expectation ℰ′′ of the von
Neumann algebra (𝒩w′)′ such that ℰ′′(Aα)P𝒩=P𝒩AP𝒩 for each A∈(ℳw′)′. Since ℳ is a partial GW*-algebra, for
any X∈ℳ there is a net {Aα}∈(ℳ'w)′ which converges
strongly* to X. Then (5.14)ℰ′′(Aα)P𝒩∈((𝒩w′)′)P𝒩=((𝒩P𝒩)w′)′,and ℰ′′(Aα)P𝒩 converges
strongly* to P𝒩X⌈P𝒩𝒟. Therefore, we have P𝒩X⌈P𝒩𝒟∈𝒩. Hence, X∈D(ℰ𝒩) and ℰ𝒩 is a
conditional expectation of (ℳ,ξ0) with respect to,
𝒩. This completes the
proof.Corollary 5.4.
Letℳ
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, (5.15)D(ℰ𝒩)⊃linearspanofℳb𝒩and𝒩ℳb.Proof.
Sinceℳb⊂Rw(ℳ), it follows that 𝒩∩Rw(ℳ)⊃𝒩b, and so clearly (N) holds. Furthermore, (N1) holds since 𝒩w′𝒟^(𝒩)=𝒩b𝒟^(𝒩)⊂𝒟^(𝒩). This completes the proof.We consider the case of the well-known SegalLp-space defined
by τ.Example 5.5.
Letℳ0 be a von
Neumann algebra on a Hilbert space ℋ with a faithful
finite trace τ. We denote by Lp(τ) the Banach
space completion of ℳ0 with respect to,
the
norm (5.16)∥A∥p≡τ(|A|p)1/p,A∈ℳ0.Then(5.17)ℳ0≡L∞(τ)⊂Lp(τ)⊂L2(τ)⊂Lq(τ)⊂L1(τ),1≤q≤2≤p<∞.Let 2≤p<∞. Here we define a *-representation π of Lp(τ) by (5.18)π(X)A=XA,X∈Lp(τ),A∈L∞(τ).Then ℳ≡π(Lp(τ)) is a partial EW*-algebra on L∞(τ) in L2(τ) with ℳb=π(L∞(τ)) which is
integrable, that is, π(X†)¯=π(X)* for each X∈Lp(τ). Furthremore, π(Lp(τ)) has a strongly
cyclic and separating vector ξ0≡λτ(I), where I is an identity
operator on ℋ. Let 𝒩0 be a von
Neumann subalgebra of ℳ0. We put (5.19)𝒩={π(X);X∈Lp(τ),π(X)λτ(I)∈Lp(τ⌈𝒩0)},2≤p≤∞.Then 𝒩 is an
integrable partial EW*-subalgebra of ℳ satisfying (N2) and (N3) and P𝒩ℳξ0=𝒩ξ0. By Theorem 5.2, there exists a conditional
expectation of (ℳ,ξ0).



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*Source: 640948-2009-04-09.xml*