• 検索結果がありません。

# Conditional Expectations for Unbounded Operator Algebras

N/A
N/A
Protected

シェア "Conditional Expectations for Unbounded Operator Algebras"

Copied!
22
0
0

(1)

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

## 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 un- bounded 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 observ- ables 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 ob- servables (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 condi- tional expectation given by (ᏺ,ξ0) by the mapAP0ofᏹinto the closed subspace Ᏼᏺξ0ofᏴ, which has the usual properties of a conditional expectation, whereP

is the projection ofᏴontoᏴ. 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

(2)

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 ifΔitξ0ᏺΔξ0it=ᏺfor alltR, whereΔξ0is the modular operator of the left Hilbert algebraᏹξ0. Here we consider a mapᏱ(· |ᏺ) :APAᏺξ0

ofᏹinto the partial O-algebraᏸ(ᏺξ0,Ᏼ). 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);

whereωξ0is a positive linear functional onᏹdefined byωξ0(A)=(Aξ0|ξ0),Aᏹ. By restriction of the weak conditional-expectationᏱ(· |ᏺ), we will show that there exists a maximal unbounded conditional expectationᏱ 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,Ᏼ)=

Xᏸ(Ᏸ,Ᏼ); ᏰX, ᏸ(Ᏸ)=

X(Ᏸ,Ᏼ); XᏰᏰ,X. (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 involutionXXXᏰ, that is, (X+λY)=X+λY,X††=X;

(iv) the weak partial multiplicationXY=X†∗Y, defined wheneverXis a left mul- tiplier ofY, (XLw(Y) orYRw(X)), that is, if and only ifYᏰ(X†∗) and XᏰ(Y).

(3)

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

(i)XLw(Y) if and only ifYLw(X) and then (XY)=YX;

(ii) ifXLw(Y) andXLw(Z), thenXLw(λY+μZ) for allλ,μCandX(λY+ μZ)=λ(XY) +μ(XZ).

(Ᏸ) is a-algebra with the usual multiplicationXY (which coincides with the weak partial multiplicationXY) and the involutionXX. 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 topologytis a locally convex topology defined by a family{ · X; X}of seminorms ξXξ+, (ξᏰ), and it is called the graph (or induced) topology on Ᏸ. If the locally convex space Ᏸ[t] is complete, thenᏹis said to be closed. We denote byᏰ(ᏹ) the completion of the locally convex spaceᏰ[t] and put

X=XᏰ(ᏹ), Xᏹ;

= {X; X}. (2.2)

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

XᏰ(X).is called the closure ofᏹ.

We next define the notion of selfadjointness ofᏹ. IfᏰ=(ᏹ)

XᏰ(X), then ᏹis said to be selfadjoint. If(ᏹ)=(ᏹ), thenᏹis said to be essentially selfadjoint.

It is clear that

(ᏹ)(ᏹ),

XXX, Xᏹ. (2.3)

We define commutants and bicommutants ofᏹ. The weak commutantᏹw ofᏹis de- fined by

w=

CᏮ(Ᏼ); (CXξ|η)=

|Xη,Xᏹ,ξ, (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=w. IfᏹwᏰ, thenᏹw is a von Neumann algebra; in particular, ifᏹis selfadjoint, thenᏹwᏰ. The unbounded commutants and unbounded bicommutants ofᏹare defined by

δ=

Sᏸ(Ᏸ,Ᏼ); (SXξ|η)=

|Xη,Xᏹ,ξ,η; ᏹσ=δ(Ᏸ,Ᏼ);

c=σ(Ᏸ);

w σ=

X(Ᏸ,Ᏼ); (CXξ|η)=

|Xη,Cw,ξ,η; ᏹwc =w σ(Ᏸ).

(2.5)

(4)

Then the following hold.

(i)ᏹδis a subspace ofᏸ(Ᏸ,Ᏼ).

(ii)ᏹσ is a-invariant subspace ofᏸ(Ᏸ,Ᏼ) and (ᏹσ)b≡ {Sσ;SᏮ(Ᏼ)} =wᏰ.

(iii)ᏹcis a subalgebra ofᏸ(Ᏸ).

(iv)ᏹ wσ is a-invariantτs-closed subspace ofᏸ(Ᏸ,Ᏼ) containingᏹ, where the strongtopologyτs is defined by the family{pξ(·);ξ}of seminorms

pξ(X)+ Xξ , X(Ᏸ,Ᏼ). (2.6) (iiv)ᏹ wcis aτs-closed O-algebra onᏰsuch thatᏹ wcand (ᏹ wc)w=w. (iiiv) IfᏹwᏰ, thenᏹ wσis a partial O-algebra onᏰsuch that

w σ=

X(Ᏸ,Ᏼ); Xis aﬃliated withw

=w

τs theτs-closure ofw

Ᏸinᏸ(Ᏸ,Ᏼ), (2.7)

andᏹ wcis an O-algebra onᏰsuch that ᏹ wc=

X(Ᏸ);Xis aﬃliated withw

=w

τs(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 andᏹ= wc, 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+XX)1exists inᏹb≡ {Aᏹ;AᏮ(Ᏼ)}for allXᏹandᏹb {A;Ab}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 byMt the closure of a subsetMofᏰwith respect to the graph topology t. LetMbe anᏹ-invariant subspace of Ᏸ. ThenᏹM≡ {XM; X}is an O- algebra onMand its closureᏹM((ᏹM)) is a closed O-algebra onMt inM. If Mis essentially selfadjoint, that is,M is selfadjoint, then the projectionPM ofᏴonto Mbelongs to ᏹw,PM(ᏹ)=MtᏰ andᏹM=PM≡ {XPM; X}, where XPMPMξ=PM forXᏹandξᏰ. A vectorξ0inᏰis said to be strongly cyclic if ᏹξ0t

=Ᏸ, andξ0is said to be separating ifwξ0=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.

(5)

Letπ1andπ2be (-)representations ofᏭon the same Hilbert space. Ifπ1(x)π2(x) for eachxᏭ, thenπ2is said to be an extension ofπ1and it is denoted byπ1π2. Letπ be a (-)representation ofᏭ. We put

Ᏸ(π) =

x

π(x), π(x) =π(x)Ᏸ(π), xA. (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

(π)=

x

π(x), π(x)=πx(π), xᏭ. (2.10) Thenπis a closed representation ofᏭsuch thatπππand 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,ξ0Ᏸa strongly cyclic and separating vector forᏹ, andᏺan O-subalgebra ofᏹ. Then

ᏺξ0ᏺξ0t

ᏺξ0t

ᏺξ0

ᏹξ0

t

=

Ᏼ. (3.1)

Ifᏺis closed, thenᏺξ0ᏺξ0 t

ᏺξ0 t

=ᏹξ0 t

. The following is easily shown.

Lemma 3.1. Put

π

=ᏺξ0,

π(X)Yξ0=XYξ0, X,Yᏺ, Ᏸπ=ᏺξ0t

,

π(X)ξ=Xξ, Xᏺ,ξπ.

(3.2)

Thenπandπare faithful-representations ofinᏺξ0such thatπππ, and

π

ππ

, Ᏸπ

=π. (3.3) We denote byP the projection ofᏴontoᏴᏺξ0. Then we have the following lemma.

Lemma 3.2. P(ᏹ)) andπ(X)Pξ=PX†∗ξ, for allXand for all ξ(ᏹ).

(6)

Proof. Take arbitraryXᏺandξ(ᏹ). For anyYᏺ, we have X0|Pξ=

X0|ξ=

0|X†∗ξ=

0|PX†∗ξ, (3.4) and soP(ᏹ)) andπ(X)Pξ=PX†∗ξ.

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

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

(i)E(XA)=π(X)E(A), for allAᏹ, for allXᏺ.

(ii)ωξ0(A)=(E(A)|ξ0), for allAᏹ.

A mapEsatisfying the conditions ofDefinition 3.3was called a conditional expecta- tion 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 expectationEofgiven by (ᏺ,ξ0) exists uniquely, and

E(A)=P0, Aᏹ. (3.5)

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

E(A|ᏺ)=P0, Aᏹ. (3.6)

Proof. We put

E(A)=P0, Aᏹ. (3.7)

ByLemma 3.2Eis a map ofᏹintoᏰ). It is clear thatEis linear. For anyAᏹand Xᏺwe have, byLemma 3.2,

E(XA)=PXAξ0=π(X)P0=π(X)E(A), ωξ0(XA)=

0|Xξ0

=

P0|Xξ0

=

π(X)E(A)|ξ0

; (3.8)

in particular,

ωξ0(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 con- ditional expectation ofᏹgiven by (ᏺ,ξ0). For anyAᏹandXᏺwe have

E(A)|0

=

πXE(A)|ξ0

=

EXA|ξ0

=ωξ0

XA

=

0|0

=

P0|0

, (3.10)

which implies that

E(A)=P0. (3.11)

(7)

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 allAD(Ᏹ)) and Ᏹ(X)=X, for allXᏺ;

(iii)Ᏹisᏺ-linear, that is,

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. Letbe an unbounded conditional expectation ofontowith respect toξ0. Then the following statements hold.

(2)Ᏹis anᏺ-Schwarz map, that is,

Ᏹ(A)ξ0|0

=

XᏱ(A)ξ0|ξ0

=

XAξ0|ξ0

=ωξ0

XA=

0|0

=

P0|0

, (4.3)

which implies

Ᏹ(A)ξ0=P0=E(A|ᏺ). (4.4)

= Ᏹ(A)Xξ0 2= Ᏹ(AX)ξ0 2

= PAXξ0 2 AXξ0 2 by (1), AA0|0

=

XAAXξ0|ξ0

=ωξ0

XAAX

=ωξ0

XAAX= AXξ0 2,

(4.5)

for eachXᏺ, which byᏰ(π)=ᏺξ0 t

implies that

AAᏱ(A)Ᏹ(A) onᏰπ. (4.6)

(8)

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

12 iﬀD1

2

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

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

(i)Ᏹis hermitian, that is,Ᏹ(A)=Ᏹ(A), for allAᏹ; (ii)Ᏹ(ᏹ)Ᏸ(π)) and

π(X)Ᏹ(A)=Ᏹ(XA), Xᏺ,Aᏹ; (4.8) (iii)ωξ0(A)=(Ᏹ(A)ξ0|ξ0), for allAᏹ.

For weak conditional expectations we have the following.

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

Ᏹ(A|ᏺ)=PAπ, Aᏹ. (4.9) Proof. We first show the existence: we putᏱ(A|ᏺ)=PAᏰ(π), Aᏹ. It follows fromLemma 3.2that for anyAᏹ,Ᏹ(A|ᏺ) is a linear map ofᏰ(π) intoᏰ), and furthermore

Ᏹ(A|ᏺ)ξ|η=

P|η=(Aξ|η)= ξ|Aη

=

ξ|PAη=

ξ|A|η (4.10) for eachξᏰ(π), which implies thatᏱ(A|ᏺ)(Ᏸ(π),Ᏼ) 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 arbitraryXᏺandAᏹ. SinceᏱ(A|ᏺ)(Ᏸ(π),Ᏼ) andᏱ(A| ᏺ)Ᏸ(π)) as shown above, it follows thatπ(X)Ᏹ(A|ᏺ) is well defined and

π(X)Ᏹ(A|ᏺ)ξ=π(X)PAξ=PXAξ

=XA|ξ (byLemma 3.2) (4.11) for eachAᏹ,Xᏺ, andξᏰ(π).

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

0|ξ0

=

P0|ξ0

=

Ᏹ(A|ᏺ)ξ0|ξ0

(4.12)

for eachAᏹ.

(9)

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(X)=Ᏹ(AX), Aᏹ,Xᏺ, (4.13) which implies

Ᏹ(A)Xξ0|0

=

π(Y)Ᏹ(AX)ξ0|ξ0

=

YAXξ0|ξ0

=ωξ0

YAX

=

AXξ0|0

=

PAXξ0|0

(4.14)

for eachAᏹandX,Yᏺ. Hence, we have

Ᏹ(A)Xξ0=PAXξ0, Aᏹ,Xᏺ, (4.15) which implies

Ᏹ(A)=PAπ, Aᏹ. (4.16) The weak conditional expectation ofᏹwith respect to (ᏺ,ξ0) has the following prop- erties.

Proposition 4.5. Ᏹ(· |ᏺ) is a map ofᏹinto the partial O-algebra(Ᏸ(π),Ᏼ) satisfying

(i)Ᏹ(A|ᏺ)Ᏸ(π)), for allAᏹ;

(ii)Ᏹ(· |ᏺ) is linear;

(iii)Ᏹ(A|ᏺ)=Ᏹ(A|), for allAᏹ Ᏹ(X|ᏺ)=XᏰ(π), for allX; (iv)Ᏹ(AA|ᏺ)0, for allAᏹ;

(v)Ᏹ(A|ᏺ)Ᏹ(A|ᏺ)Ᏹ(AA|ᏺ) wheneverᏱ(A|ᏺ)Lw(Ᏹ(A|ᏺ));

(vi)Ᏹ(A|ᏺ)π(X) andπ(X)Ᏹ(A|) are well defined for eachAandX ᏺ, and

Ᏹ(A|ᏺ)π(X)=Ᏹ(AX|ᏺ), π(X)Ᏹ(A|ᏺ)=Ᏹ(XA|ᏺ); (4.17) (vii)ωξ0(AX)=(Ᏹ(AX|ᏺ)ξ0|ξ0) for eachAandXᏺ.

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

(iv) This follows from the equality AA|ξ|ξ=

PA|ξ=

A|ξ= 2 (4.18) for eachAᏹandξᏰ(π).

(v) Take an arbitraryAᏹs.t.Ᏹ(A|ᏺ)Lw(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)ξ 2

= P 22=

AA|ξ|ξ (by (4.18))

(4.19)

(10)

for eachξᏰ(π), which implies that

Ᏹ(A|ᏺ)Ᏹ(A|ᏺ)AA|. (4.20) (vii) This follows from

ωξ0(AX)=

AXξ0|ξ0

=

PAXξ0|ξ0

=

Ᏹ(AX|ᏺ)ξ0|ξ0

(4.21)

for eachAᏹandXᏺ.

Here we put

D

=

Aᏹ;Ᏹ(A|ᏺ)π(ᏺ). (4.22) Sinceπis faithful, for anyAD(Ᏹ) there exists a unique elementXAofᏺsuch that Ᏹ(A|ᏺ)=π(XA). Hence, the mapᏱfromD(Ᏹ) toᏺis defined by

. (4.23)

Then we have the following.

Theorem 4.6. Ᏹis a maximal among unbounded conditional expectations ofontowith respect toξ0.

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

π(AX)=Ᏹ(AX|ᏺ)=Ᏹ(A|ᏺ)π(X)=π(A)π(X)

=π(A)X, (byProposition 4.5(vi)) (4.24) for eachAD(Ᏹ) andXᏺ, it follows thatᏱ(AX)=(A)X. Similarly,Ᏹ(XA)= XᏱ(A). Hence,Ᏹ isᏺ-linear. Furthermore, it follows fromProposition 4.5(vii) that ωξ0(Ᏹ(A))=ωξ0(A) for eachAD(). ThusᏱis an unbounded conditional expec- tation ofᏹontoᏺwith respect toξ0. Finally we show thatᏱis maximal. LetᏱbe any unbounded conditional expectation ofᏹontoᏺwith respect toξ0. Take an arbitrary AD(Ᏹ). Then it follows fromLemma 4.2(1) that

PAXξ0=E(AX|ᏺ)=Ᏹ(A)Xξ0 (4.25) for eachXᏺ, which implies that

P=Ᏹ(A)ξ, ξπ. (4.26) Hence, we have

PAπ=Ᏹ(A)ππ(ᏺ), (4.27) which impliesAD() andᏱ(A)=Ᏹ(A). Thus,Ᏹ.

(11)

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. Letbe a von Neumann algebra on a Hilbert spacewith a separating and cyclic vectorξ0anda von Neumann subalgebra of. Thenis a conditional expectation ofontowith respect toξ0if and only ifΔitξ0ᏺΔξ0it=for alltR, whereΔξ0is the modular operator of the left Hilbert algebraᏹξ0.

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

Lemma 5.2. Letbe a closed O-algebra onin0a strongly cyclic and separating vector foranda closed O-subalgebra ofᏹ. Suppose

(i)ᏺwᏰ;

Ᏹ(A|ᏺ)=PAPᏰ, A wc. (5.1) ThenᏱ(· |) is a linear map of the generalized von Neumann algebrawc into the O- algebra(PᏰ) such that

(a)Ᏹ(A|ᏺ)=Ᏹ(A|ᏺ), for allA wc;Ᏹ(X|ᏺ)=XPᏰ, for allX wc; (b)Ᏹ(AA|ᏺ)0, for allA wc;

(c)Ᏹ(A|ᏺ)Ᏹ(A|ᏺ)Ᏹ(AA|ᏺ), for allA wc;

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

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

(iii)Δitξ0(ᏺwξ0it=(ᏺw), for alltR,

whereΔξ0 is the modular operator of the left Hilbert algebra (ᏹw)ξ0. Then,Ᏹ(A|ᏺ) (ᏺP) wc, for allAwc .

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

thatᏹ wcis a generalized von Neumann algebra onᏰandᏺ wcis a generalized von Neu- mann subalgebra ofᏹ wc. Since theᏺ-invariant subspaceᏺξ0ofᏰis essentially selfad- joint, it follows from [7, Theorem 4.7] that

Pw, P=ᏺξ0 t

Ᏸ, (5.2)

ᏺξ0=w

ξ0. (5.3)

By (5.2), Ᏹ(· |ᏺ) is a linear map of ᏹwc intoᏸ(PᏰ), 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)(ᏺP) wc, for allAwc . By (5.3) we haveP=P(w), and so by (iii) and by the Takesaki theorem [2] there exists a unique conditional expectationᏱ of

(12)

the von Neumann algebra (ᏹw) onto the von Neumann algebra (ᏺw) with respect to ξ0 such thatᏱ (A)P=PAP for eachA(ᏹw). Take an arbitraryA wc. Then there exists a net{Aα}in (ᏹw) which converges stronglytoA. From (5.2) it follows immediately that

w

P=P

w, (5.4)

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

P

w

=w

P

=w

P. (5.5)

Hence we have

AαPP

w

, Ᏹ AαP−−−−−→

τs PAPᏰ, (5.6)

which implies thatPAP((ᏺP)w)τs=(ᏺP) wc. Hence we have Ᏹ(A|ᏺ)=PAP

P

wc, A wc. (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 ((ᏺP) wc0).

Now we put

D

=

A wc;Ᏹ(A|ᏺ) wc

P

. (5.8)

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

Lemma 5.3. Ᏹis an unbounded conditional expectation of the generalized von Neumann algebra wconto the generalized von Neumann algebra wcwith respect toξ0which is an extension of.

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

Theorem 5.4. Letbe a generalized von Neumann algebra oninᏴ,ξ0a strongly cyclic and separating vector foranda generalized von Neumann subalgebra ofᏹ. Suppose

(i)ᏺξ0is essentially selfadjoint for; (ii)Δitξ0(ᏺwξ0it=(ᏺw), for alltR,

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

(1)Ᏹ(A|ᏺ)=Ᏹ(A|ᏺ)(ᏺP) wcfor eachA. (2)Ᏹ=.

A remarkable feature of irreducible affine isometric actions of a locally compact group G is that they remain irreducible under restriction to “most” lattices in G (see [Ner,

Related to this, we examine the modular theory for positive projections from a von Neumann algebra onto a Jordan image of another von Neumann alge- bra, and use such projections

In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type

Equivalent conditions are obtained for weak convergence of iterates of positive contrac- tions in the L 1 -spaces for general von Neumann algebra and general JBW algebras, as well

Furthermore, we will investigate unbounded conditional-expectations in case that ᏹ and ᏺ are generalized von Neumann algebras which are unbounded generalization of von Neumann

We will show that under different assumptions on the distribution of the state and the observation noise, the conditional chain (given the observations Y s which are not

Lions, “Existence and nonexistence results for semilinear elliptic prob- lems in unbounded domains,” Proceedings of the Royal Society of Edinburgh.. Section

In addition, as we are interested in graded division algebras arising from valued division algebras, we assume that the abelian group Γ (which contains Γ E ) is torsion free..