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 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 mapA→PᏺAξ0ofᏹinto the closed subspace
Ᏼᏺ≡ᏺξ0ofᏴ, which has the usual properties of a conditional expectation, wherePᏺ
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 ifΔitξ0ᏺΔ−itξ0 =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),
whereωξ0is a positive linear functional onᏹdefined byωξ0(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∈Lw(Y) orY∈Rw(X)), that is, if and only ifYᏰ⊂Ᏸ(X†∗) and
Thenᏸ†(Ᏸ,Ᏼ) is a partial∗-algebra, that is, the following hold:
(i)X∈Lw(Y) if and only ifY†∈Lw(X†) and then (XY)†=Y†X†;
(ii) ifX∈Lw(Y) andX∈Lw(Z), thenX∈Lw(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 ξX≡ 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) inᏴwhich 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
de-fined 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
ᏹδ=S∈ᏸ(Ᏸ,Ᏼ); (SXξ|η)=Sξ|X†η,∀X∈ᏹ,∀ξ,η∈Ᏸ; ᏹσ=ᏹδ∩ᏸ†(Ᏸ,Ᏼ);
ᏹc=ᏹσ∩ᏸ†(Ᏸ); ᏹ
wσ=
X∈ᏸ†(Ᏸ,Ᏼ); (CXξ|η)=Cξ|X†η,∀C∈ᏹ
w,∀ξ,η∈Ᏸ
; ᏹ
wc=ᏹw σ∩ᏸ†(Ᏸ).
Then the following hold.
(i)ᏹδis a subspace ofᏸ(Ᏸ,Ᏼ).
(ii)ᏹσ is a†-invariant subspace ofᏸ†(Ᏸ,Ᏼ) and (ᏹσ)b≡ {S∈ᏹσ;S∈Ꮾ(Ᏼ)} = ᏹwᏰ.
(iii)ᏹcis a subalgebra ofᏸ†(Ᏸ).
(iv)ᏹ wσ is a†-invariantτs∗-closed subspace ofᏸ†(Ᏸ,Ᏼ) containingᏹ, where the
strong∗topologyτs∗ is defined by the family{p∗
ξ(·);ξ∈Ᏸ}of seminorms
pξ∗(X)≡ Xξ+ X†ξ , X∈ᏸ†(Ᏸ,Ᏼ). (2.6)
(iiv)ᏹ wcis aτs∗-closed O∗-algebra onᏰsuch thatᏹ⊂ᏹ wcand (ᏹ wc)w=ᏹw.
(iiiv) IfᏹwᏰ⊂Ᏸ, thenᏹ wσis a partial O∗-algebra onᏰsuch that
ᏹ
wσ=
X∈ᏸ†(Ᏸ,Ᏼ); Xis affiliated withᏹ
w
=ᏹw
Ᏸτs∗
theτs∗-closure ofᏹwᏰinᏸ†(Ᏸ,Ᏼ),
(2.7)
andᏹ wcis an O∗-algebra onᏰsuch that
ᏹ
wc=
X∈ᏸ†(Ᏸ);Xis affiliated withᏹ
w
=ᏹw
Ᏸτs∗
∩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=ᏹ
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+X†X)−1exists inᏹ
b≡ {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 bythe closure of a subsetofᏴwith respect to the Hilbert space norm and denote byMtᏹ 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 onMtᏹ inM. If
Mis essentially selfadjoint, that is,ᏹM is selfadjoint, then the projectionPM ofᏴonto
Mbelongs to ᏹw,PMᏰ∗(ᏹ)=Mtᏹ⊂Ᏸ andᏹM=ᏹPM≡ {XPM; X∈ᏹ}, where XPMPMξ=PMXξ 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
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, thenπ 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⊂ᏺξ0t
ᏹ
⊂ᏺξ0t
ᏺ
⊂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,
Ᏸπᏺᏹ=ᏺξ0t
ᏹ ,
πᏺᏹ(X)ξ=Xξ, ∀X∈ᏺ,∀ξ∈Ᏸπᏺᏹ.
(3.2)
Thenπᏺandπᏺᏹare faithful∗-representations ofᏺinᏴᏺ≡ᏺξ0such thatπᏺ⊂πᏺᏹ⊂πᏺ, and
Ᏸπᏺ⊂Ᏸπᏺᏹ
⊂Ᏸπᏺ, Ᏸ∗πᏺ=Ᏸ∗πᏺᏹ
. (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
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)=πᏺ∗(X)E(A), for allA∈ᏹ, for allX∈ᏺ. (ii)ωξ0(A)=(E(A)|ξ0), for allA∈ᏹ.
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 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=πᏺ∗(X)PᏺAξ0=πᏺ∗(X)E(A), ωξ0(XA)=
Aξ0|X†ξ0
=PᏺAξ0|X†ξ0
=πᏺ∗(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 anyA∈ᏹandX∈ᏺwe have
E(A)|Xξ0
=πᏺ∗X†E(A)|ξ0
=EX†A|ξ0
=ωξ0
X†A =Aξ0|Xξ0=PᏺAξ0|Xξ0,
(3.10)
which implies that
E(A)=PᏺAξ0. (3.11)
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)ωξ0(Ᏹ(A))=ωξ0(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 onᏰπᏺᏹwheneverA∈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 =ωξ0
X†A†AX= AXξ0 2
,
(4.5)
for eachX∈ᏺ, which byᏰ(πᏺᏹ)=ᏺξ0
tᏹ
implies that
LetEbe the set of all unbounded conditional expectations ofᏹontoᏺwith respect toξ0. ThenEis an ordered set with the following order⊂.
Ᏹ1⊂Ᏹ2 iffD
Ᏹ1
⊂ᏰᏱ2
, Ᏹ1(A)=Ᏹ2(A), ∀A∈D
Ᏹ1
. (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(πᏹᏺ),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(πᏺᏹ)⊂Ᏸ∗(πᏹ
ᏺ) and
πᏺᏹ(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)=PᏺAᏰπᏺᏹ
, ∀A∈ᏹ. (4.9)
Proof. We first show the existence: we putᏱ(A|ᏺ)=PᏺAᏰ(πᏺᏹ), A∈ᏹ. It follows fromLemma 3.2that for anyA∈ᏹ,Ᏹ(A|ᏺ) is a linear map ofᏰ(πᏺᏹ) intoᏰ∗(πᏺᏹ), 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(πᏺᏹ), which implies thatᏱ(A|ᏺ)∈ᏸ†(Ᏸ(πᏺᏹ),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(πᏹ
ᏺ),Ᏼᏺ) andᏱ(A| ᏺ)Ᏸ(πᏹᏺ)⊂Ᏸ∗(πᏹ
ᏺ) as shown above, it follows thatπᏺᏹ(X)Ᏹ(A|ᏺ) is well defined and
πᏺᏹ(X)†Ᏹ(A|ᏺ)†ξ=πᏹᏺ(X)∗PᏺA†ξ=PᏺX†A†ξ
=ᏱX†A†|ᏺξ (byLemma 3.2) (4.11)
for eachA∈ᏹ,X∈ᏺ, andξ∈Ᏸ(πᏺᏹ). (iii) This follows from the equality
ωξ0(A)=
Aξ0|ξ0
=PᏺAξ0|ξ0
=Ᏹ(A|ᏺ)ξ0|ξ0
(4.12)
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)πᏺᏹ(X)=Ᏹ(AX), ∀A∈ᏹ,∀X∈ᏺ, (4.13)
which implies
Ᏹ(A)Xξ0|Yξ0
=πᏺᏹ(Y)∗Ᏹ(AX)ξ0|ξ0
=ᏱY†AXξ0|ξ0
=ωξ0
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)=PᏺAᏰπᏺᏹ
, ∀A∈ᏹ. (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∗-algebraᏸ†(Ᏸ(πᏺᏹ),Ᏼᏺ) satisfying
(i)Ᏹ(A|ᏺ)Ᏸ(πᏺᏹ)⊂Ᏸ∗(πᏹ
ᏺ), for allA∈ᏹ; (ii)Ᏹ(· |ᏺ) is linear;
(iii)Ᏹ(A|ᏺ)†=Ᏹ(A†|ᏺ), for allA∈ᏹ Ᏹ(X|ᏺ)=XᏰ(πᏺᏹ), 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)†∈Lw(Ᏹ(A|ᏺ));
(vi)Ᏹ(A|ᏺ)πᏺᏹ(X) andπᏺᏹ(X)Ᏹ(A|ᏺ) are well defined for eachA∈ᏹandX∈
ᏺ, and
Ᏹ(A|ᏺ)πᏺᏹ(X)=Ᏹ(AX|ᏺ), πᏺᏹ(X)Ᏹ(A|ᏺ)=Ᏹ(XA|ᏺ); (4.17)
(vii)ωξ0(AX)=(Ᏹ(AX|ᏺ)ξ0|ξ0) for eachA∈ᏹandX∈ᏺ.
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(πᏺᏹ).
(v) Take an arbitraryA∈ᏹs.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
= PᏺAξ 2≤ Aξ2=ᏱA†A|ᏺξ|ξ (by (4.18))
for eachξ∈Ᏸ(πᏺᏹ), which implies that
Ᏹ(A|ᏺ)†Ᏹ(A|ᏺ)≤ᏱA†A|ᏺ. (4.20)
(vii) This follows from
ωξ0(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)∈πᏺᏹ(ᏺ). (4.22)
Sinceπᏺᏹis faithful, for anyA∈D(Ᏹᏺ) there exists a unique elementXAofᏺsuch that Ᏹ(A|ᏺ)=πᏺᏹ(XA). Hence, the mapᏱᏺfromD(Ᏹᏺ) toᏺis defined by
Ᏹᏺ(A)=XA, A∈D
Ᏹᏺ. (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)=πᏺᏹ(X)Ᏹ(A|ᏺ)∈πᏺᏹ(ᏺ) for eachX∈ᏺandA∈D(Ᏹᏺ), which implies thatᏺD(Ᏹᏺ)⊂D(Ᏹᏺ). It is easily shown thatᏱᏺis a projection. Since
πᏹᏺᏱᏺ(AX)=Ᏹ(AX|ᏺ)=Ᏹ(A|ᏺ)πᏺᏹ(X)=πᏺᏹᏱᏺ(A)πᏺᏹ(X)
=πᏺᏹᏱᏺ(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
ωξ0(Ᏹᏺ(A))=ωξ0(A) for eachA∈D(Ᏹᏺ). ThusᏱᏺis an unbounded conditional
expec-tation 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∈Ᏸπᏺᏹ
. (4.26)
Hence, we have
PᏺAᏰπᏺᏹ=Ᏹ(A)Ᏸπᏺᏹ∈πᏺᏹ(ᏺ), (4.27)
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 ifΔitξ0ᏺΔ−itξ0 =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∈ᏹ
wc. (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∈ᏹ
wc;Ᏹ(X|ᏺ)=XPᏺᏰ, for allX∈ᏺ wc;
(b)Ᏹ(A†A|ᏺ)≥0, for allA∈ᏹ
wc;
(c)Ᏹ(A|ᏺ)†Ᏹ(A|ᏺ)≤Ᏹ(A†A|ᏺ), for allA∈ᏹ
wc;
(d)Ᏹ(A|ᏺ)X=Ᏹ(AX|ᏺ),XᏱ(A|ᏺ)=Ᏹ(XA|ᏺ), for allA∈ᏹ
wc, for allX∈
ᏺ
wc;
(e)ωξ0(AX)=(Ᏹ(AX|ᏺ)ξ0|ξ0), for allA∈ᏹ wc, for allX∈ᏺ wc. Furthermore, suppose
(iii)Δitξ0(ᏺw)Δξ−it0 =(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]
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
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
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∈ᏹ wc. Then
there exists a net{Aα}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
Ᏹ A α
Pᏺ∈
ᏺPᏺ
w
, Ᏹ A
α
Pᏺ−−−−−→τs∗ PᏺAPᏺᏰ,
(5.6)
which implies thatPᏺAPᏺᏰ∈((ᏺPᏺ)w)
τs∗
=(ᏺPᏺ) wc. Hence we have
Ᏹ(A|ᏺ)=PᏺAPᏺᏰ∈ᏺPᏺ
wc, A∈ᏹ 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 ((ᏺPᏺ) wc,ξ0).
Now we put
DᏱᏺ=A∈ᏹ
wc;Ᏹ(A|ᏺ)∈
ᏺ
wc
Pᏺ
. (5.8)
Then, for any A∈D(Ᏹᏺ) there exists a unique element Ᏹᏺ(A) of ᏺ wc 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 algebraᏹ wconto the generalized von Neumann algebraᏺ wcwith 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)Δitξ0(ᏺw)Δξ−it0 =(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.
Proof. (1) Sinceᏺξ0⊂Ᏸ(πᏺᏹ)⊂ᏺξ0tᏺ=PᏺᏰ, it follows thatᏱ(A|ᏺ)=Ᏹ(A|ᏺ)∼for
eachA∈ᏹ, andᏱ(A|ᏺ) is contained in (ᏺPᏺ) wcbyLemma 5.2.
(2) This follows from (1).
It is natural to consider the following question.
Question 1. Let (ᏹ,ξ0,ᏺ) be as inTheorem 5.4. DoesD(Ᏹᏺ) contain any elements ofᏹ?
For example, when isᏹb⊂D(Ᏹᏺ)?
For this question we have the following.
Proposition 5.5. Let (ᏹ,ξ0,ᏺ) be as inTheorem 5.4. Suppose thatᏺis an EW∗-algebra onᏰ, that is, (ᏺw) =ᏺb. Thenᏺᏹbᏺ⊂D(Ᏹᏺ).
Proof. ByLemma 5.1there exists a unique conditional expectationᏱ of the von Neu-mann algebra (ᏹw) onto the von Neumann algebra (ᏺw) such that
Ᏹ (A)P
ᏺ=PᏺAPᏺ, ∀A∈ᏹw
. (5.9)
Take an arbitraryA∈ᏹb. ThenᏱ (A)Ᏸ∈ᏺb; it follows that
PᏺAPᏺ=Ᏹ (A)Pᏺ∈ᏺbPᏺ, (5.10)
which impliesA∈D(Ᏹᏺ)=D(Ᏹᏺ). Thus,ᏹb⊂D(Ᏹᏺ), which impliesᏺᏹbᏺ⊂D(Ᏹᏺ).
Corollary 5.6. Let (ᏹ,ξ0,ᏺ) be as inTheorem 5.4. If one of the following conditions (i) and (ii) holds, thenᏺᏹbᏺ⊂D(Ᏹᏺ).
(i) (ᏺw) is commutative.
(ii)Ᏸ=Ᏸ∞(H )≡n∈NᏰ(H n), whereH is a selfadjoint operator inᏴaffiliated with
ᏺw.
Proof. Suppose (i) holds, that is, (ᏺw) is commutative. Then, (ᏺw)Ᏸ⊂ᏺwᏰ⊂Ᏸ.
Sup-pose (ii) holds. Then, (ᏺw)Ᏸ⊂Ᏸclearly. Henceᏺis an EW∗-algebra onᏰin either of
the cases (i) and (ii), and so it follows fromProposition 5.5thatᏺᏹbᏺ⊂D(Ᏹᏺ).
6. Absolute continuity and coarse graining
In this section we define the notions of absolutely continuous positive linear functionals and investigate them.
Letᏹbe an O∗-algebra onᏰinᏴwith a strongly cyclic vectorξ0. A linear functional Fonᏹis called hermitian ifF(A†)=F(A) for eachA∈ᏹand it is called positive (de-noted byF≥0) ifF(A†A)≥0 for eachA∈ᏹ. Sinceᏹcontains the identity operator, it follows that ifF≥0 then it is hermitian. The positive linear functionalωξ0 on ᏹis
defined by
ωξ0(A)=
Aξ0|ξ0
, A∈ᏹ. (6.1)
We define the notion ofωξ0-absolutely continuous linear functionals onᏹand investigate
Definition 6.1. LetFbe a linear functional onᏹ. If for anyA∈ᏹthere exists a constant
rA>0 such that
F(AX)2
≤rAωξ0
X†X, ∀X∈ᏹ, (6.2)
thenF is said to beωξ0-absolutely continuous and denoted byF < ωξ0. If there exists a
constantr >0 such that
FX†X≤rωξ0
X†X, ∀X∈ᏹ, (6.3)
thenFis said to beωξ0-dominated and denoted byF<dωξ0. Denote byᏹ∗(< ωξ0) (resp.,
ᏹ∗
h(< ωξ0), ᏹ∗+(< ωξ0)) the set of allωξ0-absolutely continuous (resp., hermitian,
posi-tive) linear functionals onᏹ; and denote byᏹ∗(<dωξ0) (reps.,ᏹ∗h(<dωξ0),ᏹ+∗(<dωξ0))
the set of allωξ0-dominated (resp., hermitian, positive) linear functionals onᏹ.
Theorem 6.2. LetFbe a linear functional onᏹ. (1) The following statements are equivalent.
(i)F∈ᏹ∗(< ω
ξ0) (resp.,ᏹ∗h(< ωξ0), ᏹ∗+(< ωξ0)).
(ii) There exists an elementξofᏰ∗(ᏹ) (resp.,Ᏸ∗(ᏹ)h,Ᏸ∗(ᏹ)+) such that
F(A)=Fξ(A)≡
Aξ0|ξ
, ∀A∈ᏹ, (6.4)
where
Ᏸ∗(ᏹ) h=
ξ∈Ᏸ∗(ᏹ); F
ξis hermitian
, Ᏸ∗(ᏹ)
+=
ξ∈Ᏸ∗(ᏹ); F
ξis positive
. (6.5)
(iii) There exists an element S of the unbounded commutant (ᏹᏹξ0)δ (resp.,
((ᏹᏹξ0)σ)h, ((ᏹᏹξ0)σ)+) of the O∗-algebraᏹᏹξ0onᏹξ0such that
F(A)=FS(A)≡
Aξ0|Sξ0
, ∀A∈ᏹ. (6.6)
The vectorξ in (ii) and the operatorSin (iii) are unique.Sis called the Radon-Nikodym derivative ofFwith respect toωξ0and denoted bydF/dωξ0.
(2)F∈ᏹ∗(<
dωξ0) (resp.,ᏹh∗(<dωξ0),ᏹ∗+(<dωξ0)) if and only ifdF/dωξ0∈ᏹw(resp.,
(ᏹw)h, (ᏹw)+).
Proof. (1) (i)⇒(ii). LetF∈ᏹ∗(< ω
ξ0). Then we have
F(A)2
≤rI Aξ0 2, ∀A∈ᏹ. (6.7)
Sinceξ0 is strongly cyclic, that is,ᏹξ0 is dense inᏰ[tᏹ], and the graph topologytᏹ is
finer than the topology defined by the Hilbert space norm · , it follows that ᏹξ0 is
dense inᏴ, which implies that the mapAξ0→F(A) can be extended to a continuous
linear functional onᏴand by the Riesz theorem there exists a unique elementξ ofᏴ such that
F(A)=Aξ0|ξ
Furthermore, sinceF < ωξ0, it follows that
X†Aξ
0|ξ=F
X†A≤rXωξ0
A†A=rX Aξ0 2 (6.9)
for allA,X∈ᏹ, which impliesξ∈Ᏸ∗(ᏹ). In particular, it is clear that ifF∈ᏹ∗ h(< ωξ0)
(resp.,ᏹ+∗(< ωξ0)) thenξ∈Ᏸ∗(ᏹ)h(resp.,ξ∈Ᏸ∗(ᏹ)+).
(ii)⇒(iii). We put
SAξ0=ξ, A∈ᏹ. (6.10)
Then since
Xξ0|SAYξ0=F(AY)†X=FY†A†X=A†Xξ0|SYξ0 (6.11)
for eachA,X,Y∈ᏹ, it follows thatS∈(ᏹᏹξ0)δandF=FS. It is clear thatSis uniquely determined. Furthermore, it is easily shown that ifFis hermitian (resp., positive) thenS
is hermitian (resp., positive). (iii)⇒(i). This is trivial.
(2) This is shown in a similar way to (1).
The equivalence of (i) and (ii) inTheorem 6.2follows from [1, Theorem 1].
The following schemes may serve as a sketch ofTheorem 6.2:
ξ∈Ᏸ∗(ᏹ)
resp.,Ᏸ∗(ᏹ)h,Ᏸ∗(ᏹ)+
bijection
bijection
Fξ∈ᏹ∗< ωξ0
resp.,ᏹ∗h< ωξ0
,ᏹ∗+
< ωξ0
bijection
dFξ
dωξ0
∈ᏹᏹξ0
δ
(resp.,ᏹᏹξ0
σ
h,
ᏹᏹξ0
σ
+
F∈ᏹ∗< dωξ0
resp.,ᏹ∗h<dωξ0
,ᏹ+∗
<dωξ0
bijection dωdFξ0
∈ᏹw
(resp.,ᏹw
h,
ᏹw
+
.
(6.12)
The Radon-Nikodym theorems for O∗-algebras can be found in [1,6,10,12,13]. Next we define and investigate the notion of coarse graining.
Letᏹbe a closed O∗-algebra onᏰinᏴwith a strongly cyclic (and separating) vector
ξ0andᏺan O∗-subalgebra ofᏹ.
Definition 6.3. A positive linear functionalFonᏹis said to be (ᏺ,ωξ0) coarse graining of
f, if the following conditions hold: (i) f ⊂F;
(ii)F < ωξ0;
(iii)F(A)=F(E(A|ᏺ))≡(E(A|ᏺ)|(dF/dωξ0)ξ0), for allA∈ᏹ.
We put
Fc(A)=
E(A|ᏺ)| df dωξ0ᏺ
ξ0
, A∈ᏹ. (6.13)
ThenFcis a linear functional onᏹsatisfying the following:
Fc⊃f . (6.14)
In fact, (6.14) follows from
Fc(X)=
Xξ0| df dωξ0ᏺ
ξ0
=f(X), ∀X∈ᏺ. (6.15)
Question 2. IsFc a (ᏺ,ωξ0) coarse graining of f? That is, doesFc satisfy the following
conditions?
Fcis positive, (6.16)
Fc< ωξ0. (6.17)
We remark that if (6.17) holds, then
Fc(A)=Fc
E(A|ᏺ), ∀A∈ᏹ. (6.18)
In fact, since
Aξ0| dFc dωξ0
ξ0
=Fc(A)=
E(A|ᏺ)| df dωξ0ᏺ
ξ0
=
Aξ0| df dωξ0ᏺ
ξ0
(6.19)
for allA∈ᏹ, it follows that (dFc/dωξ0)ξ0=(df /dωξ0ᏺ)ξ0, which implies thatFc(A)=
Fc(E(A|ᏺ)) for allA∈ᏹ.
Almost all the results ofTheorem 6.4,Proposition 6.6, andTheorem 6.12can be found in [1, Theorem 3], but it seems that they contain a few gaps. So, we introduce here these results and their proofs.
Theorem 6.4. The following statements are equivalent.
(i) f is (ᏺ,ωξ0) coarse grainable, that is, there exists a (ᏺ,ωξ0) coarse graining of f.
(ii) There exists a positive linear functionalFonᏹsuch thatF⊃ f,F < ωξ0and (dF/
dωξ0)ξ0∈ᏺξ0.
(iii) There exists a positive operatorSin (ᏹᏹξ0)σsuch thatSξ0=(df /dωξ0ᏺ)ξ0.
Proof. (i)⇒(ii). LetFbe a (ᏺ,ωξ0) coarse-graining of f. ThenFis a positive linear
func-tional onᏹsuch thatF⊃f andF < ωξ0. Furthermore, since
Aξ0| dF dωξ0
ξ0
=F(A)=FE(A|ᏺ)=
PᏺAξ0| dF dωξ0
ξ0
(6.20)
for allA∈ᏹ, we have
dF dωξ0
ξ0=Pᏺ dF
dωξ0
ξ0∈ᏺξ0. (6.21)
(ii)⇒(iii). We putS=dF/dωξ0. Then,S∈((ᏹᏹξ0)σ)+andSξ0∈ᏺξ0. Furthermore,
we have
Aξ0|Sξ0
=
PᏺAξ0| dF dωξ0
ξ0
=lim
n→∞
Xnξ0| dF dωξ0
ξ0 =lim n→∞F Xn =lim n→∞f
Xn=n→∞lim
Xnξ0| df dωξ0ᏺ
ξ0
=
Aξ0| df dωξ0ᏺ
ξ0
(6.22)
for allA∈ᏹ, where{Xn}is a sequence in ᏺ such that limn→∞Xnξ0=PᏺAξ0, which
implies thatSξ0=(df /dωξ0ᏺ)ξ0.
(iii)⇒(i). Since
Fc
B†A=
PᏺB†Aξ0| df dωξ0ᏺ
ξ0
=B†Aξ0|Sξ0
=Aξ0|SBξ0
(6.23)
for eachA,B∈ᏹ, it follows thatFc≥0 andFc< ωξ0. HenceFcis a (ᏺ,ωξ0) coarse graining
of f.
Finally we show thatFcis the unique (ᏺ,ωξ0) coarse graining of f. LetFbe a (ᏺ,ωξ0)
coarse graining of f. By the proof of (ii)⇒(iii) we have
F(A)=
Aξ0| dF dωξ0
ξ0
=
Aξ0| df dωξ0ᏺ
ξ0
=Fc(A) (6.24)
for allA∈ᏹ. This completes the proof.
Definition 6.5. ᏺis said to be positivity preserving if for anyA∈ᏹthere exists a sequence
{Xn}inᏺsuch that limn→∞Xn†Xnξ0=PᏺA†Aξ0.
It is clear that ifᏺis positivity preserving thenFcis positive. In fact, this follows from
Fc
A†A=EA†A|ᏺ| df dωξ0ᏺ
ξ0
=lim
n→∞
Xn†Xnξ0| df dωξ0ᏺ
ξ0
=lim
n→∞
Xnξ0| df dωξ0ᏺ
Xnξ0
≥0, ∀A∈ᏹ.
(6.25)
Proposition 6.6. Suppose thatᏺis positivity preserving. Then the following statements are equivalent.
(i) f is (ᏺ,ωξ0) coarse grainable.
(ii)Fcis a (ᏺ,ωξ0) coarse graining of f.
(iii)Fc< ωξ0.
(iv) (df /dωξ0ᏺ)ξ0∈Ᏸ∗(ᏹ).
Proof. (i)⇔(ii) This follows fromTheorem 6.4. (ii)⇒(iii). This is trivial.
(iii)⇒(ii). As shown above,Fcis automatically positive, and so (iii) implies (ii). (ii)⇒(iv). This follows from
df dωξ0ᏺ
ξ0= dFc dωξ0
ξ0∈Ᏸ∗(ᏹ). (6.26)
(iv)⇒(ii). Letξ≡(df /dωξ0ᏺ)ξ0. Then,ξ∈Ᏸ∗(ᏹ)=Ᏸ∗(ᏹᏹξ0), and so byTheorem
6.2
dFξ
dωξ0
∈ᏹᏹξ0
σ
+,
dFξ
dωξ0
ξ0= df dωξ0ᏺ
ξ0, (6.27)
which byTheorem 6.4implies thatFcis a (ᏺ,ωξ0) coarse graining of f.
Proposition 6.7. Suppose thatᏺ is positivity preserving and f <dωξ0ᏺ. Then Fc is a
(ᏺ,ωξ0) coarse graining of f andFc<dωξ0.
Proof. Since
FcA†A=
PᏺA†Aξ0| df dωξ0ᏺ
ξ0
=lim
n→∞
Xn†Xnξ0| df dωξ0ᏺ
ξ0
=lim
n→∞
Xnξ0| df dωξ0ᏺ
Xnξ0
≤ df dωξ0ᏺ
lim
n→∞ Xnξ0
2
= df dωξ0ᏺ
PᏺA†Aξ0|ξ0
≤ df dωξ0ᏺ
ωξ0
A†A
(6.28)
for allA∈ᏹ, we haveFc<dωξ0, which fromProposition 6.6implies thatFc is a (ᏺ,ωξ0)
coarse graining of f.
We characterize the coarse grainability of positive linear functionals onᏺby elements ofᏰ∗(ᏹ)+.
Proposition 6.8. Let f be a positive linear functional onᏺ. The following statements are equivalent.
(i) f is (ᏺ,ωξ0) coarse grainable.
Proof. (i)⇒(ii). LetFbe a (ᏺ,ωξ0) coarse graining off. Then,F < ωξ0impliesf < ωξ0ᏺ.
ByTheorem 6.4(ii) we have
ξ≡ dF dωξ0
ξ0∈Ᏸ∗(ᏹ)+, Pᏺξ=ξ, Fξ=F. (6.29)
(ii)⇒(i). It is easily shown thatFξis a (ᏺ,ωξ0) coarse graining of f.
ByProposition 6.8andTheorem 6.4we have the following:
ξ∈Ᏸ∗(ᏹ)
+; Pᏺξ=ξ∈ξ
bijection f;ᏺ,ω
ξ0
coarse-grainable positive linear functionals∈ fξ≡Fξᏺ
bijection
dFξ
dωξ0
∈S∈ᏹᏹξ0
σ
+;Sξ0∈ᏺξ0
.
(6.30)
Remark 6.9. Letξ∈Ᏸ∗(ᏹ)
+. ByTheorem 6.2,Fξis aωξ0-absolutely continuous positive
linear functional onᏹ, but it is not a (ᏺ,ωξ0) coarse graining of fξ≡Fξᏺin general. In
fact,
FξE(A|ᏺ)=PᏺAξ0|ξ
=Aξ0|Pᏺξ
=Fξ(A) in general. (6.31)
We consider whetherFPᏺξis a (ᏺ,ωξ0) coarse graining offξ.FPᏺξis a linear functional on ᏹsuch thatFPᏺξ⊃ fξ andFPᏺξ(E(A|ᏺ))=FPᏺξ(A), for allA∈ᏹ, but it is not positive and notωξ0-absolutely continuous in general. Hence we consider whenFPᏺξ is positive andωξ0-absolutely continuous.
Proposition 6.10. Suppose that ᏹis essentially selfadjoint, ᏺ is positivity preserving, and theᏺ-invariant subspaceᏺξ0is essentially selfadjoint. ThenPᏺᏰ+= {ξ∈Ᏸ+;Pᏺξ=
ξ} and the mapPᏺξ→ fPᏺξ = fξ is a bijection ofPᏺᏰ+ onto {f ∈ᏺ∗+; (ᏺ,ωξ0) coarse
grainable}, where Ᏸ+= {ξ∈Ᏸ; Fξ ≥0}. Hence, anyωξ0-absolutely continuous positive
linear functionalFonᏹis a (ᏺ,ωξ0) coarse graining ofFᏺ.
Proof. Sinceᏹis essentially selfadjoint andᏺξ0 is essentially selfadjoint, we havePᏺᏰ
⊂Ᏸ. Furthermore, sinceᏺis positivity preserving, we havePᏺᏰ+⊂Ᏸ+, which implies PᏺᏰ+= {ξ∈Ᏸ+;Pᏺξ=ξ}. Hence it follows fromProposition 6.8that the mapPᏺξ→ fξis a bijection. LetFbe anyωξ0-absolutely continuous positive linear functional onᏹ.
ByTheorem 6.2there exists an elementξ ofᏰ+such thatF=Fξ. By the aboveFPᏺξ is a (ᏺ,ωξ0) coarse graining of fξ=Fᏺ.
Definition 6.11. A mapIωξ0 ofᏹ∗+(< ωξ0) intoR+is said to be an information measure
with respect toωξ0if
(i)Iωξ0(F)≥0, for allF∈ᏹ∗+(< ωξ0) andIωξ0(ωξ0)=1;
(ii)Iωξ0(F1+F2)=Iωξ0(F1) +Iωξ0(F2), wheneverF1andF2are mutually singular, that
Theorem 6.12. (1) The information measureIωξ0 with respect toωξ0exists uniquely, and
Iωξ0(F)= (dF/dωξ0)ξ0
2for eachF∈ᏹ∗
+(< ωξ0).
(2) If f is a (ᏺ,ωξ0) coarse-grainable positive linear functional onᏺ, thenFcis theωξ0
-absolutely continuous extension of f with minimal information with respect toωξ0, that is,
Iωξ0(Fc)≤Iωξ0(F) for eachF∈ᏹ∗+(< ωξ0) s.t.Fᏺ= f.
Proof. (1) LetIωξ0 be an information measure with respect toωξ0. ByTheorem 6.2the
mapF→(dF/dωξ0)ξ0is a bijection ofᏹ+∗(< ωξ0) ontoᏰ∗(ᏹ)+, and so we may define a
mapϕofᏰ∗(ᏹ)+intoR+by
ϕ
dF dωξ0
ξ0
=Iωξ0(F), F∈ᏹ
∗
+
< ωξ0
. (6.32)
Take arbitraryF1,F2∈ᏹ+∗(< ωξ0) s.t. ((dF1/dωξ0)ξ0|(dF2/dωξ0)ξ0)=0. Then we have
ϕ
dF1 dωξ0
ξ0+ dF2 dωξ0
ξ0
=ϕ
dF1+F2 dωξ0
ξ0
=Iωξ0
F1+F2
=Iωξ0
F1
+Iωξ0
F2 =ϕ dF1 dωξ0
ξ0
+ϕ
dF2 dωξ0
ξ0
,
(6.33)
which by [14, Corollary 2.3] implies that
ϕ
dF dωξ0
ξ0
=r dF dωξ0
ξ0
2, ∀F∈ᏹ∗
+
< ωξ0
(6.34)
for somer >0. Sinceϕ(ξ0)=Iωξ0(ωξ0)=1, we haver=1 and
Iωξ0(F)=
dωdF
ξ0
ξ0
2, ∀F∈ᏹ∗
+
< ωξ0
. (6.35)
(2) Take an arbitraryF∈ᏹ∗
+(< ωξ0) s.t.Fᏺ=f. Then, byTheorem 6.4, we have
Xξ0| dFc dωξ0
ξ0
=
Xξ0| df dωξ0ᏺ
ξ0
=f(X)=
Xξ0| dF dωξ0
ξ0
(6.36)
for eachX∈ᏺ, which implies thatPᏺ(dF/dωξ0)ξ0=(dFc/dωξ0)ξ0. Hence we have
Iωξ0(F)=
dωdF
ξ0
ξ0
2≥ dFc dωξ0
ξ0
2=Iωξ0
Fc
. (6.37)
Letᏺandᏺ1 be O∗-subalgebras ofᏹsuch thatᏺ1⊂ᏺand f aωξ0ᏺ-absolutely
continuous positive linear functional onᏺ. Thenf1≡fᏺ1is aωξ0ᏺ1-absolutely
con-tinuous positive linear functional onᏺ1. We consider the following questions.
(1) Does the (ᏺ,ωξ0) coarse grainabilitiy of f imply the (ᏺ1,ωξ0) coarse grainability
of f1? Conversely, does the (ᏺ1,ωξ0) coarse grainability of f1 imply the (ᏺ,ωξ0) coarse
grainability of f?
Since
Fc(A)=
Aξ0| df dωξ0ᏺ
ξ0
, F1
c(A)=
Aξ0| df 1 dωξ0ᏺ1
ξ0
, A∈ᏹ, (6.38)
we have
Fc(X)= f(X)=f1(X)=
F1
c(X) (6.39)
for eachX∈ᏺ1, and so
df1 dωξ0ᏺ1
ξ0=Pᏺ1
df dωξ0ᏺ
ξ0. (6.40)
For the above question we have the following.
Proposition 6.13. The following statements are equivalent.
(i) f is (ᏺ,ωξ0) coarse grainable,f1is (ᏺ1,ωξ0) coarse grainable andFc=(F1)c.
(ii) f is (ᏺ,ωξ0) coarse grainable, f1 is (ᏺ1,ωξ0) coarse grainable, and Iωξ0(Fc)=
Iωξ0((F1)c).
(iii) f is (ᏺ,ωξ0) coarse grainable and (df /dωξ0ᏺ)ξ0∈ᏺ1ξ0.
(iv) f1is (ᏺ1,ωξ0) coarse grainable and (F1)c⊃f.
Proof. (i)⇒(ii). This is trivial.
(ii)⇒(iii). Since(dFc/dωξ0)ξ0 = (d(F1)c/dωξ0)ξ0, it follows that (df /dωξ0ᏺ)ξ0=
(dFc/dωξ0)ξ0∈ᏺ1ξ0.
(iii)⇒(iv). ByTheorem 6.4it is shown thatFcis a unique (ᏺ1,ωξ0) coarse graining of
f1, and so f1is (ᏺ1,ωξ0) coarse grainable and (F1)c=Fc⊃f.
(iv)⇒(i). ByTheorem 6.2it is shown that (F1)cis a (ᏺ,ωξ0) coarse graining of f, and
so f is (ᏺ,ωξ0) coarse grainable andFc=(F1)c.
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Atsushi Inoue: Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan
Email address:[email protected]
Hidekazu Ogi: Department of Functional Materials Engineering, Fukuoka Institute of Technology, Fukuoka 811-0295, Japan
Email address:[email protected]
Mayumi Takakura: Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan