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∩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:

the domain D(ℰ) of ℰ is a †-invariant
subspace of ℳ containing 𝒩;

ℰ is a
projection; that is, it is hermitian
(ℰ(A)†=ℰ(A†),forallA∈D(ℰ)) and ℰ(X)=X,forallX∈𝒩;

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 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

(x,y)∈Γ implies (y*,x*)∈Γ;

(x,y1),(x,y2)∈Γ implies (x,λy1+μy2)∈Γ, for all λ,μ∈ℂ;

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).

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∈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 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 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: 𝒟^(ℳ)=⋂X∈ℳ𝒟(X¯),𝒟*(ℳ)=⋂X∈ℳ𝒟(X*),𝒟**(ℳ)=⋂X∈ℳ𝒟((X*⌈𝒟*(ℳ))*),and then𝒟⊂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 of ℳ. The weak
bounded commutantℳw′ of ℳ is the
set 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 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,

π is linear;

x∈Lw(y) in 𝔄 implies π(x)∈Lw(π(y)) and π(x)□π(y)=π(xy);

π(x*)=π(x)†foreveryx∈𝔄.

Let π be a *-representation
of a partial *-algebra 𝔄 into ℒ†(𝒟,ℋ). Then we define 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 that π 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: 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
partial O*-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 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 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, 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∩Rw(ℳ))ξ0 is dense in ℋ𝒩≡𝒩ξ0¯.

The following is easily shown.

Lemma 3.1.

Put 𝒟(π𝒩)=(𝒩∩R
w
(ℳ))ξ0,π𝒩(X)Yξ0=(X□Y)ξ0,∀X∈𝒩,∀Y∈𝒩∩R
w
(ℳ). Then π𝒩 is a *-representations
of 𝒩 in the Hilbert
space ℋ𝒩≡𝒟(π𝒩¯).

We denote by P𝒩 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 that P𝒩𝒟⊂𝒟*(π𝒩) and π𝒩*(X)P𝒩ξ=P𝒩Xξ, forallX∈𝒩andforallξ∈𝒟.

Proof.

Take arbitrary X∈𝒩 and ξ∈𝒟. For any Y∈𝒩∩Rw(ℳ), we have (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 map ℰ 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 (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-expectation ℰ(⋅∣𝒩) of (ℳ,ξ0) with respect to,
𝒩, and ℰ(A∣𝒩)=P𝒩A⌈𝒟(π𝒩),∀A∈ℳ.The weak conditional-expectation ℰ(⋅∣𝒩) of (ℳ,ξ0) with respect to,
𝒩 satisfies the
following:

ℰ(⋅∣𝒩) is linear,

ℰ(⋅∣𝒩) is hermitian,
that is, ℰ(A∣𝒩)†=ℰ(A†∣𝒩),forallA∈ℳ,

ℰ(X∣𝒩)=X⌈𝒟(π𝒩),forallX∈𝒩,

ℰ(A†□A∣𝒩)≥0,forallA∈ℳs.t.A†□A is well-defined,

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,

ℰ(A∣𝒩)□π𝒩(X) is well-defined for any A∈ℳ and X∈𝒩∩R
w
(ℳ), and ℰ(A∣𝒩)□π𝒩(X)=ℰ(A□X∣𝒩),

π𝒩(X)□ℰ(A∣𝒩) is well-defined
for any A∈ℳ∩R
w
(𝒩) and forallX∈𝒩, and π𝒩(X)□ℰ(A∣𝒩)=ℰ(X□A∣𝒩),

ωξ0(ℰ(A∣𝒩))=ωξ0(A),forallA∈ℳ.

Proof.

We put 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 (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 Partial <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M284"><mml:mrow><mml:msup><mml:mrow><mml:mtext>O</mml:mtext></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>-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∩Rw(ℳ))ξ0 is dense in ℋ𝒩. 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

the domain D(ℰ) of ℰ is a †-invariant
subspace of ℳ containing 𝒩;

ℰ is a
projection; that is, it is hermitian (ℰ(A)†=ℰ(A†),forallA∈D(ℰ)) and ℰ(X)=X,forallX∈𝒩;

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=P𝒩AXξ0=ℰ(A∣𝒩)Xξ0,∀A∈D(ℰ),∀X∈𝒩∩R
w
(ℳ).

Proof.

For all A∈D(ℰ) and X,Y∈𝒩∩Rw(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∣P𝒩Yξ0)=(P𝒩AXξ0∣Yξ0). Hence, ℰ(A)Xξ0=P𝒩AXξ0=ℰ(A∣𝒩)Xξ0,forallA∈D(ℰ),forallX∈𝒩∩Rw(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).

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 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 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 haveℰ(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 Partial <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M352"><mml:mrow><mml:msup><mml:mrow><mml:mtext>O</mml:mtext></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>-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∩Rw(ℳ))ξ0 is dense in ℋ𝒩,

𝒩w′𝒟^(𝒩)⊂𝒟^(𝒩),

(𝒩∩Rw(ℳ))ξ0 is essentially
self-adjoint for 𝒩,

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 that D(ℰ𝒩)={A∈ℳ;P𝒩Aξ0∈𝒩ξ0}.

Proof.

We put
D
(ℰ)={A∈ℳ;P𝒩Aξ0∈𝒩ξ0}.By Lemma 4.2, we have 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 (N_{1}) and [5, Proposition 2.3.5], we haveX¯isaffiliatedwithvonNeumannalgebra(𝒩w′)′foreachX∈𝒩,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, 𝒩(𝒩∩Rw(ℳ))ξ0⊂(𝒩∩Rw(ℳ))ξ0¯=𝒩ξ0¯,which implies by assumption (N_{2}) and
[5, Theorem 7.4.4]
that P𝒩∈Nw′,P𝒩𝒟^(𝒩)⊂𝒟^(𝒩). Furthermore, by (5.3) and (5.6), we have 𝒩ξ0¯=(𝒩w′)′ξ0¯,thatis,𝒫𝒩=𝒫(𝒩w′)′.Let Sξ0 and Sξ0′′ be the closures of the maps: Sξ0Aξ0=A†ξ0,A∈ℳ,Sξ0′′Bξ0=B*ξ0,B∈(ℳw′)′.By (5.3) we have Sξ0⊂Sξ0′′. Takesaki proved
in [1] that assumtion (N_{3} ) implies 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 haveℰ(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 that ℰ(X)=X,forallX∈𝒩. Take arbitrary A∈D(ℰ) and X∈𝒩∩Lw(ℳ). Since (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 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),
(N_{1}), (N_{ 2}), and (N_{3}). 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 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),
(N_{1}), (N_{ 2}), and (N_{3}).

Then, D(ℰ𝒩)⊃ 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 𝒩𝒫𝒩 is a partial GW*-algebra on P𝒩𝒟, then ℰ𝒩 is a
conditional expectation of (ℳ,ξ0) with respect to,
𝒩.

Proof.

Let X∈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 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 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.

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 (N_{2}) and (N_{3}).
Then, D(ℰ𝒩)⊃linearspanofℳb𝒩and𝒩ℳb.

Proof.

Since ℳb⊂Rw(ℳ), it follows that 𝒩∩Rw(ℳ)⊃𝒩b, and so clearly (N) holds. Furthermore, (N_{1}) holds since 𝒩w′𝒟^(𝒩)=𝒩b𝒟^(𝒩)⊂𝒟^(𝒩). This completes the proof.

We consider the case of the well-known Segal Lp-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 Lp(τ) the Banach
space completion of ℳ0 with respect to,
the
norm ∥A∥p≡τ(|A|p)1/p,A∈ℳ0.Thenℳ0≡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 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 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 (N_{2}) and (N_{3}) and P𝒩ℳξ0=𝒩ξ0. By Theorem 5.2, there exists a conditional
expectation of (ℳ,ξ0).

TakesakiM.Conditional expectations in von Neumann algebrasGudderS. P.HudsonR. L.A noncommutative probability theoryInoueA.TakakuraM.OgiH.Unbounded conditional expectations for O∗-algebrasInoueA.OgiH.TakakuraM.Conditional expectations for unbounded operator algebrasAntoineJ.-P.InoueA.TrapaniC.