Conditional Expectations for Unbounded Operator Algebras

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Volume 2007, Article ID 80152,22pages doi:10.1155/2007/80152

Research Article

Conditional Expectations for Unbounded Operator Algebras

Atsushi Inoue, Hidekazu Ogi, and Mayumi Takakura

Received 18 December 2006; Revised 20 March 2007; Accepted 19 May 2007 Recommended by Manfred H. Moller

Two conditional expectations in unbounded operator algebras (O^∗-algebras) are dis- cussed. One is a vector conditional expectation defined by a linear map of an O^∗-algebra into the Hilbert space on which the O^∗-algebra acts. This has the usual properties of conditional expectations. This was defined by Gudder and Hudson. Another is an unbounded conditional expectation which is a positive linear mapᏱof an O^∗-algebraᏹ onto a given O^∗-subalgebraᏺofᏹ. Here the domainD(Ᏹ) ofᏱdoes not equal toᏹin general, and so such a conditional expectation is called unbounded.

Copyright © 2007 Atsushi Inoue et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In probability theory, conditional expectations play a fundamental role. A noncommuta- tive analogue of conditional expectations in von Neumann algebras has been studied in [2–4]. A typical feature of probability in von Neumann algebras is that the observables permitted are usually bounded and some finiteness is imposed. But, unbounded observables occur naturally in quantum mechanics and quantum probability theory [1,5–8]

and so it is natural to consider conditional expectations in algebras of unbounded observables (O^∗-algebras). The first study of conditional expectations in O^∗-algebras was done by Gudder and Hudson [1]. Letᏹbe an O^∗-algebra on a dense subspaceᏰin a Hilbert spaceᏴwith a strongly cyclic and separating vectorξ0andᏺan O^∗-subalgebra ofᏹ. These notions are defined inSection 2. Gudder and Hudson have defined a conditional expectation given by (ᏺ,ξ0) by the mapA→PᏺAξ0ofᏹinto the closed subspace Ᏼᏺ≡ᏺξ0ofᏴ, which has the usual properties of a conditional expectation, wherePᏺ

is the projection ofᏴontoᏴᏺ. We call this the vector conditional expectation given by (ᏺ,ξ0). On the other hand, it is natural to consider when a conditional expectation of

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the O^∗-algebraᏹonto the O^∗-subalgebraᏺexists. Such a conditional expectation does not necessarily exist even for von Neumann algebras. In fact, Takesaki [2] has shown that there exists a conditional expectation of the von Neumann algebraᏹonto the von Neu- mann subalgebraᏺif and only ifΔ^it_ξ0ᏺΔ⁻_ξ0^it=ᏺfor allt∈R, whereΔξ0is the modular operator of the left Hilbert algebraᏹξ0. Here we consider a mapᏱ(· |ᏺ) :A→PᏺAᏺξ0

ofᏹinto the partial O^∗-algebraᏸ^†(ᏺξ0,Ᏼᏺ). We will show thatᏱ(· |ᏺ) has proper- ties similar to those of conditional expectations, so it will be called a weak conditional- expectation ofᏹwith respect to (ᏺ,ξ0). Unfortunately, the rangeᏱ(ᏹᏺ) of the weak conditional-expectationᏱ(· |ᏺ) is not necessarily contained inᏺ, and so we define an unbounded conditional-expectationᏱ ofᏹontoᏺwith respect toξ0as follows: Ᏹis a map ofᏹontoᏺsatisfying

(i) the domainD(Ᏹ) ofᏱis a†-invariant subspace ofᏹcontainingᏺsuch that ᏺD(Ᏹ)⊂D(Ᏹ);

(ii)Ᏹ(A)^†=Ᏹ(A^†), for allA∈D(Ᏹ) andᏱ(X)=X, for allX∈ᏺ; (iii)Ᏹ(AX)=Ᏹ(A)XandᏱ(XA)=XᏱ(A), for allA∈D(Ᏹ), for allX∈ᏺ;

(iv)ωξ0(Ᏹ(A))=ωξ0(A), for allA∈D(Ᏹ),

whereω_ξ0is a positive linear functional onᏹdefined byω_ξ0(A)=(Aξ0|ξ0),A∈ᏹ. By restriction of the weak conditional-expectationᏱ(· |ᏺ), we will show that there exists a maximal unbounded conditional expectationᏱᏺ of ᏹontoᏺwith respect to ξ0. Furthermore, we will investigate unbounded conditional-expectations in case thatᏹ andᏺare generalized von Neumann algebras which are unbounded generalization of von Neumann algebras and that the von Neumann algebra (ᏺw) (the usual commutant of the weak commutantᏺw ofᏺ) satisfies the Takesaki condition. As an application of vector conditional expectations we will establish the existence of coarse graining for absolutely continuous positive linear functionals.

2. Preliminaries

In this section we introduce the basic definitions and properties of (partial) O^∗-algebras.

We refer to [6–9] for O^∗-algebras and to [10] for partial O^∗-algebras.

LetᏴbe a Hilbert space with inner product (· | ·) andᏰa dense subspace ofᏴ. We denote byᏸ(Ᏸ,Ᏼ) the set of all linear operatorsXinᏴsuch thatᏰ(X) (the domain of X)=Ᏸ, and

ᏸ^†(Ᏸ,Ᏼ)=

X∈ᏸ(Ᏸ,Ᏼ); ᏰX^∗⊃Ᏸ, ᏸ^†(Ᏸ)=

X∈ᏸ^†(Ᏸ,Ᏼ); XᏰ⊂Ᏸ,X^∗Ᏸ⊂Ᏸ. (2.1) Thenᏸ(Ᏸ,Ᏼ) is a vector space with the usual operationsX+Y andλX, andᏸ^†(Ᏸ,Ᏼ) is equipped with the following operations and involution:

(i) the sumX+Y;

(ii) the scalar multiplicationλX;

(iii) the involutionX→X^†≡X^∗Ᏸ, that is, (X+λY)^†=X^†+λY^†,X^††=X;

(iv) the weak partial multiplicationXY=X^†∗Y, defined wheneverXis a left mul- tiplier ofY, (X∈L^w(Y) orY∈R^w(X)), that is, if and only ifYᏰ⊂Ᏸ(X^†∗) and X^†Ᏸ⊂Ᏸ(Y^∗).

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Thenᏸ^†(Ᏸ,Ᏼ) is a partial∗-algebra, that is, the following hold:

(i)X∈L^w(Y) if and only ifY^†∈L^w(X^†) and then (XY)^†=Y^†X^†;

(ii) ifX∈L^w(Y) andX∈L^w(Z), thenX∈L^w(λY+μZ) for allλ,μ∈CandX(λY+ μZ)=λ(XY) +μ(XZ).

ᏸ^†(Ᏸ) is a∗-algebra with the usual multiplicationXY (which coincides with the weak partial multiplicationXY) and the involutionX→X^†. A partial∗-subalgebra ofᏸ^†(Ᏸ, Ᏼ) is called a partial O^∗-algebra onᏰ, and a∗-subalgebra ofᏸ^†(Ᏸ) is called an O^∗- algebra onᏰ. Here we assume that a (partial) O^∗-algebra contains the identity operatorI.

In analogy with the notion of a closed symmetric (selfadjoint) operator, we define the notion of a closed O^∗-algebra (a selfadjoint O^∗-algebra). Letᏹbe an O^∗-algebra onᏰ. We define a natural graph topology onᏰ. This topologytᏹis a locally convex topology defined by a family{ · _X; X∈ᏹ}of seminorms ξ_X≡ ξ+Xξ, (ξ∈Ᏸ), and it is called the graph (or induced) topology on Ᏸ. If the locally convex space Ᏸ[tᏹ] is complete, thenᏹis said to be closed. We denote byᏰ(ᏹ) the completion of the locally convex spaceᏰ[tᏹ] and put

X=XᏰ(ᏹ), X∈ᏹ;

ᏹ= {X; X∈ᏹ}. (2.2)

Thenᏹis a closed O^∗-algebra onᏰ(ᏹ) in Ᏼwhich is the smallest closed extension of ᏹ, andᏰ(ᏹ)≡

X∈ᏹᏰ(X).ᏹis called the closure ofᏹ.

We next define the notion of selfadjointness ofᏹ. IfᏰ=Ᏸ^∗(ᏹ)≡

X∈ᏹᏰ(X^∗), then ᏹis said to be selfadjoint. IfᏰ(ᏹ)=Ᏸ^∗(ᏹ), thenᏹis said to be essentially selfadjoint.

It is clear that

Ᏸ⊂Ᏸ(ᏹ)⊂Ᏸ^∗(ᏹ),

X⊂X⊂X^∗, ∀X∈ᏹ. (2.3)

We define commutants and bicommutants ofᏹ. The weak commutantᏹw ofᏹis defined by

ᏹw=

C∈Ꮾ(Ᏼ); (CXξ|η)=

Cξ|X^†η,∀X∈ᏹ,∀ξ,η∈Ᏸ, (2.4) whereᏮ(Ᏼ) is a∗-algebra of all bounded linear operators onᏴ. Thenᏹw is a weakly closed∗-invariant subspace ofᏮ(Ᏼ) such that (ᏹ)w=ᏹw. IfᏹwᏰ⊂Ᏸ, thenᏹw is a von Neumann algebra; in particular, ifᏹis selfadjoint, thenᏹwᏰ⊂Ᏸ. The unbounded commutants and unbounded bicommutants ofᏹare defined by

ᏹ_δ=

S∈ᏸ(Ᏸ,Ᏼ); (SXξ|η)=

Sξ|X^†η,∀X∈ᏹ,∀ξ,η∈Ᏸ; ᏹ_σ=ᏹ_δ∩ᏸ^†(Ᏸ,Ᏼ);

ᏹc=ᏹ_σ∩ᏸ^†(Ᏸ);

ᏹwσ=

X∈ᏸ^†(Ᏸ,Ᏼ); (CXξ|η)=

Cξ|X^†η,∀C∈ᏹw,∀ξ,η∈Ᏸ; ᏹwc =ᏹwσ∩ᏸ^†(Ᏸ).

(2.5)

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Then the following hold.

(i)ᏹ_δis a subspace ofᏸ(Ᏸ,Ᏼ).

(ii)ᏹ_σ is a†-invariant subspace ofᏸ^†(Ᏸ,Ᏼ) and (ᏹ_σ)_b≡ {S∈ᏹ_σ;S∈Ꮾ(Ᏼ)} = ᏹwᏰ.

(iii)ᏹcis a subalgebra ofᏸ^†(Ᏸ).

(iv)ᏹwσ is a†-invariantτ_s^∗-closed subspace ofᏸ^†(Ᏸ,Ᏼ) containingᏹ, where the strong^∗topologyτs^∗ is defined by the family{p^∗_ξ(·);ξ∈Ᏸ}of seminorms

p_ξ^∗(X)≡ Xξ+X^†ξ, X∈ᏸ^†(Ᏸ,Ᏼ). (2.6) (iiv)ᏹwcis aτs^∗-closed O^∗-algebra onᏰsuch thatᏹ⊂ᏹwcand (ᏹwc)w=ᏹw. (iiiv) IfᏹwᏰ⊂Ᏸ, thenᏹwσis a partial O^∗-algebra onᏰsuch that

ᏹwσ=

X∈ᏸ^†(Ᏸ,Ᏼ); Xis aﬃliated withᏹw

= ᏹw

Ᏸ^τ^s^∗ theτ_s^∗-closure ofᏹw

Ᏸinᏸ^†(Ᏸ,Ᏼ), (2.7)

andᏹwcis an O^∗-algebra onᏰsuch that ᏹwc=

X∈ᏸ^†(Ᏸ);Xis aﬃliated withᏹw

= ᏹw

Ᏸ^τ^s^∗∩ᏸ^†(Ᏸ). (2.8)

We introduce the notions of generalized von Neumann algebras and extended W^∗- algebras which are unbounded generalizations of von Neumann algebras. IfᏹwᏰ⊂Ᏸ andᏹ=ᏹwc, thenᏹis said to be a generalized von Neumann algebra (or a GW^∗-algebra) on Ᏸ. A closed O^∗-algebraᏹon Ᏸis said to be an extended W^∗-algebra (simply, an EW^∗-algebra) if (I+X^†X)⁻¹exists inᏹb≡ {A∈ᏹ;A∈Ꮾ(Ᏼ)}for allX∈ᏹandᏹb≡ {A;A∈ᏹb}is a von Neumann algebra onᏴ.

We define the notion of strongly cyclic vectors for a closed O^∗-algebraᏹonᏰinᏴ. We denote by᏷the closure of a subset᏷ofᏴwith respect to the Hilbert space norm and denote byM^t^ᏹ the closure of a subsetMofᏰwith respect to the graph topology tᏹ. LetMbe anᏹ-invariant subspace of Ᏸ. ThenᏹM≡ {XM; X∈ᏹ}is an O^∗- algebra onMand its closureᏹM(≡(ᏹM)^∼) is a closed O^∗-algebra onM^t^ᏹ inM. If Mis essentially selfadjoint, that is,ᏹM is selfadjoint, then the projectionP_M ofᏴonto Mbelongs to ᏹw,P_MᏰ^∗(ᏹ)=M^t^ᏹ⊂Ᏸ andᏹM=ᏹPM≡ {X_P_M; X∈ᏹ}, where X_P_MP_Mξ=P_MXξ forX∈ᏹandξ∈Ᏸ. A vectorξ0inᏰis said to be strongly cyclic if ᏹξ0tᏹ

=Ᏸ, andξ0is said to be separating ifᏹwξ0=Ᏼ.

We define the notions of (unbounded)∗-representations of∗-algebras. LetᏭbe an

∗-algebra with identity 1. A (∗-)homomorphismπofᏭinto an O^∗-algebraᏸ^†(Ᏸ) with π(1)=Iis said to be a (∗-)representation ofᏭ. In this case,ᏰandᏴare denoted, re- spectively, byᏰ(π) andᏴπ.

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Letπ1andπ2be (∗-)representations ofᏭon the same Hilbert space. Ifπ1(x)⊂π2(x) for eachx∈Ꮽ, thenπ2is said to be an extension ofπ1and it is denoted byπ1⊂π2. Letπ be a (∗-)representation ofᏭ. We put

Ᏸ(π) =

x∈Ꮽ

Ᏸπ(x), π(x) =π(x)Ᏸ(π), x∈A. (2.9) Ifπ=π, then π is said to be closed;πis a closed (∗-)representation ofᏭwhich is the smallest closed extension ofπand it is called the closure ofπ. Letπbe a∗-representation ofᏭ. We put

Ᏸ^∗(π)=

x∈Ꮽ

Ᏸπ(x)^∗, π^∗(x)=πx^∗^∗Ᏸ^∗(π), x∈Ꮽ. (2.10) Thenπ^∗is a closed representation ofᏭsuch thatπ⊂π⊂π^∗and is called the adjoint of π. Ifπ=π^∗, thenπis said to be selfadjoint. We remark thatπis closed (resp., selfadjoint) if and only if the O^∗-algebraπ(Ꮽ) is closed (resp., selfadjoint).

3. Vector conditional expectations

Letᏹbe a closed O^∗-algebra onᏰinᏴ,ξ0∈Ᏸa strongly cyclic and separating vector forᏹ, andᏺan O^∗-subalgebra ofᏹ. Then

ᏺξ0⊂ᏺξ0tᏹ

⊂ᏺξ0tᏺ

⊂ᏺξ0

∩ ᏹξ0

tᏹ

=Ᏸ

⊂Ᏼ. (3.1)

Ifᏺis closed, thenᏺξ0⊂ᏺξ0 tᏹ

⊂ᏺξ0 tᏺ

⊂Ᏸ=ᏹξ0 tᏹ

. The following is easily shown.

Lemma 3.1. Put

Ᏸπᏺ

=ᏺξ0,

πᏺ(X)Yξ0=XYξ0, ∀X,Y∈ᏺ, Ᏸπ_ᏺ^ᏹ=ᏺξ0tᏹ

,

π_ᏺ^ᏹ(X)ξ=Xξ, ∀X∈ᏺ,∀ξ∈Ᏸπ_ᏺ^ᏹ.

(3.2)

Thenπᏺandπ_ᏺ^ᏹare faithful∗-representations ofᏺinᏴᏺ≡ᏺξ0such thatπᏺ⊂π_ᏺ^ᏹ⊂πᏺ, and

Ᏸπᏺ

⊂Ᏸπ_ᏺ^ᏹ⊂Ᏸπᏺ

, Ᏸ^∗πᏺ

=Ᏸ^∗π_ᏺ^ᏹ. (3.3) We denote byPᏺ the projection ofᏴontoᏴᏺ≡ᏺξ0. Then we have the following lemma.

Lemma 3.2. PᏺᏰ^∗(ᏹ)⊂Ᏸ^∗(πᏺ) andπ_ᏺ^∗(X)Pᏺξ=PᏺX^†∗ξ, for allX∈ᏺ and for all ξ∈Ᏸ^∗(ᏹ).

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Proof. Take arbitraryX∈ᏺandξ∈Ᏸ^∗(ᏹ). For anyY∈ᏺ, we have X^†Yξ0|Pᏺξ=

X^†Yξ0|ξ=

Yξ0|X^†∗ξ=

Yξ0|PᏺX^†∗ξ, (3.4) and soPᏺᏰ^∗(ᏹ)⊂Ᏸ^∗(πᏺ) andπ_ᏺ^∗(X)Pᏺξ=PᏺX^†∗ξ.

First we introduce the notion of a vector conditional expectation defined by Gudder and Hudson [1].

Definition 3.3. A mapEofᏹintoᏰ^∗(πᏺ) is said to be a vector conditional expectation of ᏹgiven by (ᏺ,ξ0) if the following hold.

(i)E(XA)=π_ᏺ^∗(X)E(A), for allA∈ᏹ, for allX∈ᏺ.

(ii)ω_ξ0(A)=(E(A)|ξ0), for allA∈ᏹ.

A mapEsatisfying the conditions ofDefinition 3.3was called a conditional expectation ofᏹgiven by (ᏺ,ξ0) by Gudder and Hudson [1]. They gave the following theorem.

We prove the theorem for the sake of completeness.

Theorem 3.4. A vector conditional expectationEofᏹgiven by (ᏺ,ξ0) exists uniquely, and

E(A)=PᏺAξ0, ∀A∈ᏹ. (3.5)

Denote byE(A|ᏺ) the unique vector conditional expectation ofᏹgiven by (ᏺ,ξ0), that is,

E(A|ᏺ)=PᏺAξ0, ∀A∈ᏹ. (3.6)

Proof. We put

E(A)=PᏺAξ0, A∈ᏹ. (3.7)

ByLemma 3.2Eis a map ofᏹintoᏰ^∗(πᏺ). It is clear thatEis linear. For anyA∈ᏹand X∈ᏺwe have, byLemma 3.2,

E(XA)=PᏺXAξ0=π_ᏺ^∗(X)PᏺAξ0=π_ᏺ^∗(X)E(A), ω_ξ0(XA)=

Aξ0|X^†ξ0

=

PᏺAξ0|X^†ξ0

=

π_ᏺ^∗(X)E(A)|ξ0

; (3.8)

in particular,

ω_ξ0(A)=

E(A)|ξ0

. (3.9)

HenceEis a vector conditional expectation ofᏹgiven by (ᏺ,ξ0).

We show the uniqueness of vector conditional expectations. LetE be any vector conditional expectation ofᏹgiven by (ᏺ,ξ0). For anyA∈ᏹandX∈ᏺwe have

E(A)|Xξ0

=

π_ᏺ^∗X^†E(A)|ξ0

=

EX^†A|ξ0

=ωξ0

X^†A

=

Aξ0|Xξ0

=

PᏺAξ0|Xξ0

, (3.10)

which implies that

E(A)=PᏺAξ0. (3.11)

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4. Unbounded conditional expectations for O^∗-algebras

We begin with the definition of unbounded conditional expectations of O^∗-algebras. In this section letᏹbe a closed O^∗-algebra onᏰinᏴwith a strongly cyclic and separating vectorξ0andᏺan O^∗-subalgebra ofᏹ.

Definition 4.1. A mapᏱofᏹontoᏺis said to be an unbounded conditional expectation ofᏹontoᏺwith respect toξ0if

(i) the domain D(Ᏹ) ofᏱis a†-invariant subspace ofᏹ containingᏺsuch that ᏺD(Ᏹ)⊂D(Ᏹ);

(ii)Ᏹis a projection; that is, it is hermitian (Ᏹ(A)^†=Ᏹ(A^†), for allA∈D(Ᏹ)) and Ᏹ(X)=X, for allX∈ᏺ;

(iii)Ᏹisᏺ-linear, that is,

Ᏹ(AX)=Ᏹ(A)X, Ᏹ(XA)=XᏱ(A), ∀A∈D(Ᏹ), ∀X∈ᏺ; (4.1) (iv)ω_ξ0(Ᏹ(A))=ω_ξ0(A), for allA∈D(Ᏹ).

In particular, ifD(Ᏹ)=ᏹ, thenᏱis said to be a conditional expectation ofᏹontoᏺ.

For unbounded conditional expectations we have the following lemma.

Lemma 4.2. LetᏱbe an unbounded conditional expectation ofᏹontoᏺwith respect toξ0. Then the following statements hold.

(1)Ᏹ(A)ξ0=E(A|ᏺ), for allA∈D(Ᏹ).

(2)Ᏹis anᏺ-Schwarz map, that is,

ᏱA^†Ᏹ(A)≤ᏱA^†A onᏰπ_ᏺ^ᏹwheneverA∈D(Ᏹ) s.t.A^†A∈D(Ᏹ). (4.2) Proof. (1) For allA∈D(Ᏹ) andX∈ᏺwe have

Ᏹ(A)ξ0|Xξ0

=

X^†Ᏹ(A)ξ0|ξ0

=

ᏱX^†Aξ0|ξ0

=ωξ0

X^†A=

Aξ0|Xξ0

=

PᏺAξ0|Xξ0

, (4.3)

which implies

Ᏹ(A)ξ0=PᏺAξ0=E(A|ᏺ). (4.4)

(2) Take an arbitraryA∈D(Ᏹ) s.t.A^†A∈D(Ᏹ). Then we have ᏱA^†Ᏹ(A)Xξ0|Xξ0

= Ᏹ(A)Xξ0 ²= Ᏹ(AX)ξ0 ²

= PᏺAXξ0 ²≤ AXξ0 ² by (1), ᏱA^†AXξ0|Xξ0

=

ᏱX^†A^†AXξ0|ξ0

=ωξ0

ᏱX^†A^†AX

=ω_ξ0

X^†A^†AX= AXξ0 ²,

(4.5)

for eachX∈ᏺ, which byᏰ(π_ᏺ^ᏹ)=ᏺξ0 tᏹ

implies that

ᏱA^†A≤Ᏹ(A)^†Ᏹ(A) onᏰπ_ᏺ^ᏹ. (4.6)

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LetEbe the set of all unbounded conditional expectations ofᏹontoᏺwith respect toξ0. ThenEis an ordered set with the following order⊂.

Ᏹ1⊂Ᏹ2 iﬀDᏱ1

⊂ᏰᏱ2

, Ᏹ1(A)=Ᏹ2(A), ∀A∈DᏱ1

. (4.7) InTheorem 4.6we will show that there exists a maximal unbounded conditional expectation ofᏹontoᏺwith respect toξ0.

Definition 4.3. A mapᏱofᏹinto the partial O^∗-algebraᏸ^†(Ᏸ(π^ᏹ_ᏺ),Ᏼᏺ) is said to be a weak conditional expectation ofᏹwith respect to (ᏺ,ξ0) if

(i)Ᏹis hermitian, that is,Ᏹ(A)^†=Ᏹ(A^†), for allA∈ᏹ; (ii)Ᏹ(ᏹ)Ᏸ(π_ᏺ^ᏹ)⊂Ᏸ^∗(π_ᏺ^ᏹ) and

π_ᏺ^ᏹ(X)Ᏹ(A)=Ᏹ(XA), ∀X∈ᏺ,∀A∈ᏹ; (4.8) (iii)ωξ0(A)=(Ᏹ(A)ξ0|ξ0), for allA∈ᏹ.

For weak conditional expectations we have the following.

Theorem 4.4. There exists a unique weak conditional-expectationᏱ(· |ᏺ) ofᏹwith re- spect to (ᏺ,ξ0), and

Ᏹ(A|ᏺ)=PᏺAᏰπ_ᏺ^ᏹ, ∀A∈ᏹ. (4.9) Proof. We first show the existence: we putᏱ(A|ᏺ)=PᏺAᏰ(π_ᏺ^ᏹ), A∈ᏹ. It follows fromLemma 3.2that for anyA∈ᏹ,Ᏹ(A|ᏺ) is a linear map ofᏰ(π_ᏺ^ᏹ) intoᏰ^∗(π_ᏺ^ᏹ), and furthermore

Ᏹ(A|ᏺ)ξ|η=

PᏺAξ|η=(Aξ|η)= ξ|A^†η

=

ξ|PᏺA^†η=

ξ|ᏱA^†|ᏺη (4.10) for eachξ,η∈Ᏸ(π_ᏺ^ᏹ), which implies thatᏱ(A|ᏺ)∈ᏸ^†(Ᏸ(π_ᏺ^ᏹ),Ᏼᏺ) andᏱ(A|ᏺ)^†= Ᏹ(A^†|ᏺ). ThusᏱ(· |ᏺ) satisfies the condition (i) inDefinition 4.3. Furthermore, we show that it satisfies the conditions (ii) and (iii) inDefinition 4.3.

(ii) Take arbitraryX∈ᏺandA∈ᏹ. SinceᏱ(A|ᏺ)∈ᏸ^†(Ᏸ(π_ᏺ^ᏹ),Ᏼᏺ) andᏱ(A| ᏺ)Ᏸ(π^ᏹ_ᏺ)⊂Ᏸ^∗(π_ᏺ^ᏹ) as shown above, it follows thatπ_ᏺ^ᏹ(X)Ᏹ(A|ᏺ) is well defined and

π_ᏺ^ᏹ(X)^†Ᏹ(A|ᏺ)^†ξ=π^ᏹ_ᏺ(X)^∗PᏺA^†ξ=PᏺX^†A^†ξ

=ᏱX^†A^†|ᏺξ (byLemma 3.2) (4.11) for eachA∈ᏹ,X∈ᏺ, andξ∈Ᏸ(π_ᏺ^ᏹ).

(iii) This follows from the equality ωξ0(A)=

Aξ0|ξ0

=

PᏺAξ0|ξ0

=

Ᏹ(A|ᏺ)ξ0|ξ0

(4.12)

for eachA∈ᏹ.

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We next show the uniqueness: letᏱbe any weak conditional expectation ofᏹwith respect to (ᏺ,ξ0). By (i) and (ii) inDefinition 4.3we have

Ᏹ(A)π_ᏺ^ᏹ(X)=Ᏹ(AX), ∀A∈ᏹ,∀X∈ᏺ, (4.13) which implies

Ᏹ(A)Xξ0|Yξ0

=

π_ᏺ^ᏹ(Y)^∗Ᏹ(AX)ξ0|ξ0

=

ᏱY^†AXξ0|ξ0

=ω_ξ0

Y^†AX

=

AXξ0|Yξ0

=

PᏺAXξ0|Yξ0

(4.14)

for eachA∈ᏹandX,Y∈ᏺ. Hence, we have

Ᏹ(A)Xξ0=PᏺAXξ0, ∀A∈ᏹ,∀X∈ᏺ, (4.15) which implies

Ᏹ(A)=PᏺAᏰπ_ᏺ^ᏹ, ∀A∈ᏹ. (4.16) The weak conditional expectation ofᏹwith respect to (ᏺ,ξ0) has the following properties.

Proposition 4.5. Ᏹ(· |ᏺ) is a map ofᏹinto the partial O^∗-algebraᏸ^†(Ᏸ(π_ᏺ^ᏹ),Ᏼᏺ) satisfying

(i)Ᏹ(A|ᏺ)Ᏸ(π_ᏺ^ᏹ)⊂Ᏸ^∗(π_ᏺ^ᏹ), for allA∈ᏹ;

(ii)Ᏹ(· |ᏺ) is linear;

(iii)Ᏹ(A|ᏺ)^†=Ᏹ(A^†|ᏺ), for allA∈ᏹ Ᏹ(X|ᏺ)=XᏰ(π_ᏺ^ᏹ), for allX∈ᏺ; (iv)Ᏹ(A^†A|ᏺ)≥0, for allA∈ᏹ;

(v)Ᏹ(A|ᏺ)^†Ᏹ(A|ᏺ)≤Ᏹ(A^†A|ᏺ) wheneverᏱ(A|ᏺ)^†∈L^w(Ᏹ(A|ᏺ));

(vi)Ᏹ(A|ᏺ)π_ᏺ^ᏹ(X) andπ^ᏹ_ᏺ(X)Ᏹ(A|ᏺ) are well defined for eachA∈ᏹandX∈ ᏺ, and

Proof. The statements (i), (ii), (iii), and (vi) follow fromTheorem 4.4.

(iv) This follows from the equality ᏱA^†A|ᏺξ|ξ=

PᏺA^†Aξ|ξ=

A^†Aξ|ξ= Aξ² (4.18) for eachA∈ᏹandξ∈Ᏸ(π_ᏺ^ᏹ).

(v) Take an arbitraryA∈ᏹs.t.Ᏹ(A|ᏺ)^†∈L^w(Ᏹ(A|ᏺ)). Then we have Ᏹ(A|ᏺ)^†Ᏹ(A|ᏺ)ξ|ξ

=

Ᏹ(A|ᏺ)^∗Ᏹ(A|ᏺ)ξ|ξ= Ᏹ(A|ᏺ)ξ²

= PᏺAξ²≤ Aξ²=

ᏱA^†A|ᏺξ|ξ (by (4.18))

(4.19)

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for eachξ∈Ᏸ(π_ᏺ^ᏹ), which implies that

Ᏹ(A|ᏺ)^†Ᏹ(A|ᏺ)≤ᏱA^†A|ᏺ. (4.20) (vii) This follows from

ω_ξ0(AX)=

AXξ0|ξ0

=

PᏺAXξ0|ξ0

=

Ᏹ(AX|ᏺ)ξ0|ξ0

(4.21)

for eachA∈ᏹandX∈ᏺ.

Here we put

DᏱᏺ

=

A∈ᏹ;Ᏹ(A|ᏺ)∈π_ᏺ^ᏹ(ᏺ). (4.22) Sinceπ_ᏺ^ᏹis faithful, for anyA∈D(Ᏹᏺ) there exists a unique elementX_Aofᏺsuch that Ᏹ(A|ᏺ)=π_ᏺ^ᏹ(X_A). Hence, the mapᏱᏺfromD(Ᏹᏺ) toᏺis defined by

Ᏹᏺ(A)=XA, A∈DᏱᏺ

. (4.23)

Then we have the following.

Theorem 4.6. Ᏹᏺis a maximal among unbounded conditional expectations ofᏹontoᏺ with respect toξ0.

Proof. We show that D(Ᏹᏺ) is a †-invariant subspace of ᏹ containing ᏺ such that ᏺD(Ᏹᏺ)⊂D(Ᏹᏺ). In fact, it is clear that D(Ᏹᏺ) is a subspace ofᏹcontainingᏺ. By Proposition 4.5(iii), D(Ᏹᏺ) is†-invariant, and it follows from Proposition 4.5(vi) that Ᏹ(XA|ᏺ)=π_ᏺ^ᏹ(X)Ᏹ(A|ᏺ)∈π_ᏺ^ᏹ(ᏺ) for eachX∈ᏺandA∈D(Ᏹᏺ), which implies thatᏺD(Ᏹᏺ)⊂D(Ᏹᏺ). It is easily shown thatᏱᏺis a projection. Since

π^ᏹ_ᏺᏱᏺ(AX)=Ᏹ(AX|ᏺ)=Ᏹ(A|ᏺ)π_ᏺ^ᏹ(X)=π_ᏺ^ᏹᏱᏺ(A)π_ᏺ^ᏹ(X)

=π_ᏺ^ᏹᏱᏺ(A)X, (byProposition 4.5(vi)) (4.24) for eachA∈D(Ᏹᏺ) andX∈ᏺ, it follows thatᏱᏺ(AX)=Ᏹᏺ(A)X. Similarly,Ᏹᏺ(XA)= XᏱᏺ(A). Hence,Ᏹᏺ isᏺ-linear. Furthermore, it follows fromProposition 4.5(vii) that ω_ξ0(Ᏹᏺ(A))=ω_ξ0(A) for eachA∈D(Ᏹᏺ). ThusᏱᏺis an unbounded conditional expectation ofᏹontoᏺwith respect toξ0. Finally we show thatᏱᏺis maximal. LetᏱbe any unbounded conditional expectation ofᏹontoᏺwith respect toξ0. Take an arbitrary A∈D(Ᏹ). Then it follows fromLemma 4.2(1) that

PᏺAXξ0=E(AX|ᏺ)=Ᏹ(A)Xξ0 (4.25) for eachX∈ᏺ, which implies that

PᏺAξ=Ᏹ(A)ξ, ∀ξ∈Ᏸπ_ᏺ^ᏹ. (4.26) Hence, we have

PᏺAᏰπ_ᏺ^ᏹ=Ᏹ(A)Ᏸπ_ᏺ^ᏹ∈π_ᏺ^ᏹ(ᏺ), (4.27) which impliesA∈D(Ᏹᏺ) andᏱᏺ(A)=Ᏹ(A). Thus,Ᏹ⊂Ᏹᏺ.

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5. Unbounded conditional expectations for special O^∗-algebras

In this section we consider conditional expectations for special O^∗-algebras (EW^∗- algebras, generalized von Neumann algebras). For conditional expectations for von Neu- mann algebras Takesaki [2] has obtained the following.

Lemma 5.1. Letᏹbe a von Neumann algebra on a Hilbert spaceᏴwith a separating and cyclic vectorξ0andᏺa von Neumann subalgebra ofᏹ. ThenᏱᏺis a conditional expectation ofᏹontoᏺwith respect toξ0if and only ifΔ^it_ξ₀ᏺΔ⁻_ξ₀^it=ᏺfor allt∈R, whereΔξ0is the modular operator of the left Hilbert algebraᏹξ0.

The following is our extension ofLemma 5.1to generalized von Neumann algebras.

Lemma 5.2. Letᏹbe a closed O^∗-algebra onᏰinᏴ,ξ0∈Ᏸa strongly cyclic and separating vector forᏹandᏺa closed O^∗-subalgebra ofᏹ. Suppose

(i)ᏺwᏰ⊂Ᏸ;

(ii)ᏺξ0is essentially selfadjoint forᏺ. Put

Ᏹ(A|ᏺ)=PᏺAPᏺᏰ, A∈ᏹwc. (5.1) ThenᏱ(· |ᏺ) is a linear map of the generalized von Neumann algebraᏹwc into the O^∗- algebraᏸ^†(PᏺᏰ) such that

(a)Ᏹ(A|ᏺ)^†=Ᏹ(A^†|ᏺ), for allA∈ᏹwc;Ᏹ(X|ᏺ)=XPᏺᏰ, for allX∈ᏺwc; (b)Ᏹ(A^†A|ᏺ)≥0, for allA∈ᏹwc;

(c)Ᏹ(A|ᏺ)^†Ᏹ(A|ᏺ)≤Ᏹ(A^†A|ᏺ), for allA∈ᏹwc;

(d)Ᏹ(A|ᏺ)X=Ᏹ(AX|ᏺ),XᏱ(A|ᏺ)=Ᏹ(XA|ᏺ), for allA∈ᏹwc, for allX∈ ᏺwc;

(e)ωξ0(AX)=(Ᏹ(AX|ᏺ)ξ0|ξ0), for allA∈ᏹwc, for allX∈ᏺwc. Furthermore, suppose

(iii)Δ^it_ξ₀(ᏺw)Δ⁻_ξ₀^it=(ᏺw), for allt∈R,

whereΔξ0 is the modular operator of the left Hilbert algebra (ᏹw)ξ0. Then,Ᏹ(A|ᏺ)∈ (ᏺPᏺ)_wc, for allA∈ᏹwc.

Proof. By (i) we haveᏹwᏰ⊂Ᏸ, and hence it follows from [6, Propositions 1.7.3, 1.7.5]

thatᏹwcis a generalized von Neumann algebra onᏰandᏺwcis a generalized von Neu- mann subalgebra ofᏹwc. Since theᏺ-invariant subspaceᏺξ0ofᏰis essentially selfadjoint, it follows from [7, Theorem 4.7] that

Pᏺ∈ᏺw, PᏺᏰ=ᏺξ0 tᏺ

⊂Ᏸ, (5.2)

ᏺξ0= ᏺw

ξ0. (5.3)

By (5.2), Ᏹ(· |ᏺ) is a linear map of ᏹwc intoᏸ^†(PᏺᏰ), and it is shown in a similar way to the proof ofProposition 4.5 thatᏱ(· |ᏺ) satisfies (a)–(e). Suppose (iii) holds.

We showᏱ(A|ᏺ)∈(ᏺPᏺ)_wc, for allA∈ᏹwc. By (5.3) we havePᏺ=P(ᏺw), and so by (iii) and by the Takesaki theorem [2] there exists a unique conditional expectationᏱof

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the von Neumann algebra (ᏹw) onto the von Neumann algebra (ᏺw) with respect to ξ0 such thatᏱ(A)Pᏺ=PᏺAPᏺ for eachA∈(ᏹw). Take an arbitraryA∈ᏹwc. Then there exists a net{A_α}in (ᏹw) which converges strongly^∗toA. From (5.2) it follows immediately that

ᏺw

Pᏺ= ᏺPᏺ

w, (5.4)

and by the basic theory of von Neumann algebras [11]

ᏺPᏺ

w

= ᏺw

Pᏺ

= ᏺw

Pᏺ. (5.5)

Hence we have

ᏱA_α_P_ᏺ∈ ᏺPᏺ

w

, ᏱA_α_P_ᏺ−−−−−→

τs^∗ PᏺAPᏺᏰ, (5.6)

which implies thatPᏺAPᏺᏰ∈((ᏺPᏺ)w)^τ^s^∗=(ᏺPᏺ)wc. Hence we have Ᏹ(A|ᏺ)=PᏺAPᏺᏰ∈

ᏺPᏺ

wc, A∈ᏹwc. (5.7) In a similar way to the proof ofTheorem 4.4one can show thatᏱ(· |ᏺ) is the unique weak conditional expectation of the generalized von Neumann algebraᏹwc with respect

to ((ᏺPᏺ)_wc,ξ0).

Now we put

DᏱᏺ

=

A∈ᏹwc;Ᏹ(A|ᏺ)∈ ᏺwc

Pᏺ

. (5.8)

Then, for any A∈D(Ᏹᏺ) there exists a unique element Ᏹᏺ(A) of ᏺwc such that Ᏹᏺ(A)PᏺᏰ=Ᏹ(A|ᏺ), and in a similar way to the proof ofTheorem 4.6we can show the following.

Lemma 5.3. Ᏹᏺis an unbounded conditional expectation of the generalized von Neumann algebraᏹwconto the generalized von Neumann algebraᏺwcwith respect toξ0which is an extension ofᏱᏺ.

By Lemmas5.2and5.3we have the following.

Theorem 5.4. Letᏹbe a generalized von Neumann algebra onᏰinᏴ,ξ0a strongly cyclic and separating vector forᏹandᏺa generalized von Neumann subalgebra ofᏹ. Suppose

(i)ᏺξ0is essentially selfadjoint forᏺ; (ii)Δ^it_ξ₀(ᏺw)Δ⁻_ξ₀^it=(ᏺw), for allt∈R,

whereΔξ0 is the modular operator of the left Hilbert algebra (ᏹw)ξ0. Then the following statements hold.

(1)Ᏹ(A|ᏺ)=Ᏹ(A|ᏺ)^∼∈(ᏺPᏺ)_wcfor eachA∈ᏹ. (2)Ᏹᏺ=Ᏹᏺ.