International Journal of Mathematics and Mathematical Sciences

Volume 2009 (2009), Article ID 640948, 10 pages

http://dx.doi.org/10.1155/2009/640948

## Unbounded Conditional Expectations for Partial -Algebras

Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan

Received 18 September 2008; Accepted 26 January 2009

Academic Editor: Ingo Witt

Copyright © 2009 Mayumi Takakura. 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.

#### Abstract

The main purpose of this paper is to generalize studies of unbounded conditional expectations for -algebras to those for partial -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 -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 of an -algebra onto a given -subalgebra of . In this paper we will consider conditional
expectations for partial -algebras.
Suppose that is a
self-adjoint partial -algebra
containing identity on dense subspace of Hilbert
space with a strongly
cyclic vector , and is a partial -subalgebra of such that is dense in , where is the set of
all right multiplier of . The definitions of (self-adjoint) partial -algebra and a
strongly cyclic vector are stated in Section 2. A map of onto is said to be a *weak conditional-expectation* of with respect to, if it satisfies ; but, the range of the weak
conditional-expectation is not
necessarily contained in , and so we have considered a map of onto satisfying the following:

(i)the domain of is a -invariant
subspace of containing ;(ii) is a
projection; that is, it is hermitian
() and ;(iii), ;(iv) , where is a state on defined by ; and call it an *unbounded conditional expectation* of with respect to, . In particular, if , then is said to be a *conditional expectation* of with respect to, .

Finally, we will investigate the scale of the domain of unbounded conditional expectations of partial -algebra which is unbounded generalizations of von Neumann algebras.

#### 2. Preliminaries

In this section we review the definitions and the basic theory of partial -algebras, partial -algebras and partial -algebras. For more details, refer to [5].

A *partial**-algebra* is a complex vector space with an
involution and a subset such that

(i) implies ;(ii) implies , for all ;(iii) whenever , there exists a product with the usual
properties of the multiplication: and for and . The element of the is called a *unit* if for all , and , for all . Notice that the partial multiplication is not
required to be associative. Whenever , is called a *left
multiplier* of and is called a *right
multiplier* of , and we write and . For a subset , we write

Let be a Hilbert
space with inner product and a dense
subspace of . We denote by the set of all closable
linear operators such that , . The set is a partial -algebra with
respect to the following operations: the usual sum , the scalar multiplication , the involution , and the weak partial
multiplication , defined whenever is a weak right
multiplier of ( or ), that is, if and only if and . A partial -subalgebra of is called a *partial *-*algebra* on .

Let be a partial -algebra on . The locally convex topology on defined by the
family of seminorms is called the *graph topology* on and denoted by . The completion of is denoted by . If the locally convex space is complete,
then is called *closed*. We also define the following
domains: and thenThe partial -algebra is called *fully closed* if , *self-adjoint* if , *essentially self-adjoint* if , and *algebraically self-adjoint* if .

We defined two weak commutants of . The *weak
bounded commutant* of is the
set but the partial multiplication
is not required to be associative, so we define the *quasi-weak bounded commutant* of as the
set In general, .

A *-representation* of a partial -algebra is a -homomorphism
of into , satisfying whenever , that is,

(i) is linear;(ii) in implies and ;(iii).

Let be a -representation of a partial -algebra into . Then we define

We say that is *closed* if ; *fully closed* if ; *essentially self-adjoint* if ; and *self-adjoint* if .

We introduce the weak and the quasi-weak commutants of a -representaion of a partial -algebra as follows: respectively.

We define the notion of strongly cyclic vector for a
partial -algebra on in . A vector in is said to be *strongly cyclic* if is dense in , and is said to be *separating* if , where .

We introduce the notion of partial -algebras and
partial -algebras which
are unbounded generalizations of von Neumann algebras. A fully closed partial -algebra on is called a *partial **-algebra* if there exists a von Neumann algebra on such that and . A partial -algebra on is said to be a
partial -algebra if is a von Neumann algebra, and .

#### 3. Weak Conditional Expectations

In this section, let be a self-adjoint partial -algebra containing the identity on in with a strongly cyclic vector and let be a partial -subalgebra of such that

(N) is dense in .

The following is easily shown.

Lemma 3.1. *Put ** Then is a -representations
of in the Hilbert
space .*

We denote by the projection of onto . This projection plays an important role in this reserch. First we have the following.

Lemma 3.2. *It holds that and , *

*Proof. * Take arbitrary and . For any , we have and so and .

*Definition 3.3. *A map of into
is said to be
a weak conditional-expectation of with respect to,
if it
satisfies

For weak conditional-expectation we have the following.

Theorem 3.4. *There exists a unique weak
conditional-expectation of with respect to,
, and **The weak conditional-expectation of with respect to,
satisfies the
following:*

(i)* is linear,*(ii)* is hermitian,
that is, ,*(iii)*,*(iv)* is well-defined,*(v)* and are well-defined,*(vi)* is well-defined for any and , and ,*(vii)* is well-defined
for any and , and ,*(viii)*.*

*Proof. * We put By Lemma 3.2, is a linear map
of into for any , and furthermore we have , so is a map of into .

Since for each , is a weak
conditional-expectation of with respect to,
. It is easily shown that if is a weak conditional-expectation
of with respect to,
, for each . Thus the existence and uniqueness of weak
conditional-expectations is shown. The statements (iii)–(viii) follow since , . This completes the proof.

#### 4. Unbounded Conditional Expectations for Partial -Algebras

Let be a self-adjoint partial -algebra containing on in and let be a strongly cyclic and separating vector for and suppose that is a partial -subalgebra of satisfying (N): is dense in . We introduce unbounded conditional expectations of with respect to, .

*Definition 4.1. *A map of onto
is said to be
an unbounded conditional expectation of with respect to,
if

(i) the domain of is a -invariant
subspace of containing ;(ii) is a
projection; that is, it is hermitian () and ;(iii), ;(iv). In particular, if , then is said to be a *conditional expectation* of with respect to,
.

For unbounded conditional expectations we have the following.

Lemma 4.2. *Let
be an unbounded
conditional expectation of with respect to,
. Then, *

*Proof. * For all and , we
have Hence, .

Let be the set of all unbounded conditional expectations of with respect to, . Then is an ordered set with the following order :

Theorem 4.3. *There exists a maximal unbounded conditional
expectation of with respect to,
, and it is denoted by . *

*Proof. * We put Then, for any , there exists a
unique map such
that It is easily shown that is an unbounded
conditional expectation of with respect to,
. Furthermore, is maximal in . Indeed, let . Take an arbitrary . Then by Lemma 4.2
we havewhich implies . Hence and is maximal in . This completes the proof.

#### 5. Existence of Conditional Expectations for Partial -Algebras

Let be a self-adjoint partial -algebra containing on in , be a strongly cyclic and separating vector for and a partial -subalgebra of such that

(N) is dense in ,(N_{1}),(N_{2}) is essentially
self-adjoint for ,(N_{3}), where is the modular
operator for the full Hilbert algebra .

Lemma 5.1. *It holds that . *

*Proof. * We put By Lemma 4.2, we have for each . Hence, . We show the converse inclusion. Since is separating
vector for , it follows that for any , there exists a
unique element of such that . Indeed, since is maximal in , it is sufficient to show that is an unbounded
conditional expectation of with respect to,
. By assumption (N_{1}) and [5, Proposition 2.3.5], we have Since is self-adjoint
and is dense in , it follows that is a reducing
subspace for , that is, which implies by assumption (N_{2}) and
[5, Theorem 7.4.4]
that Furthermore, by (5.3) and (5.6), we have Let and be the closures of the maps: By (5.3) we have Takesaki proved
in [1] that assumtion (N_{3} ) implies and there
exists a conditional expectation of the von
Neumann algebra with respect to, .

By (5.6), (5.9), and (5.10), we havefor each , which implies by the separateness of that is hermitian.

It is clear that . Take arbitrary and . Since for each , it follows that and . Furthermore, since is hermitian,
it follows that and for each and . It is clear that for each . Thus is an unbounded
conditional expectation of with respect to,
. This completes that proof.

By Lemma 5.1, we have the following.

Theorem 5.2. *Let
be a
self-adjoint partial -algebra
containing on in and let
be a strongly
cyclic and separating vector for
and suppose
that is a partial -subalgebra of satisfying (N),
(N _{1}), (N_{ 2}), and (N_{3}). Then there
exists a conditional expectation of with respect to,
if and only if . *

It is important to investigate the scale of the domain of an unbounded conditional expectation. We consider the case of partial -algebras.

Theorem 5.3. *Let be a partial -algebra on in and let be a strongly
cyclic and separating vector for
and suppose
that
be a partial -subalgebra of satisfying (N),
(N _{1}), (N_{ 2}), and (N_{3}). *

*Then, linear span of and are well defined*

*In particular, if is a partial -algebra on , then is a conditional expectation of with respect to, .*

*Proof. * Let , and are all defined. Then, it follows since is a partial -subalgebra of that which implies by Lemma 5.1 that and . Suppose that is a partial -algebra on .

By the result of Takesaki [1] there exists a unique
conditional expectation of the von
Neumann algebra such that for each . Since is a partial -algebra, for
any there is a net which converges
to . Then and converges
to . Therefore, we have . Hence, and is a
conditional expectation of with respect to,
. This completes the
proof.

Corollary 5.4. *Let
be a partial -algebra on in and let be a strongly
cyclic and separating vector for
and suppose
that
be a partial -subalgebra of satisfying (N _{2}) and (N_{3}).
Then, *

*Proof. * Since , it follows that , and so clearly (N) holds. Furthermore, (N_{1}) holds since . This completes the proof.

We consider the case of the well-known Segal -space defined by .

*Example 5.5. * Let be a von
Neumann algebra on a Hilbert space with a faithful
finite trace . We denote by the Banach
space completion of with respect to,
the
norm ThenLet . Here we define a -representation of by Then is a partial -algebra on in with which is
integrable, that is, for each . Furthremore, has a strongly
cyclic and separating vector , where is an identity
operator on . Let be a von
Neumann subalgebra of . We put Then is an
integrable partial -subalgebra of satisfying (N_{2}) and (N_{3}) and . By Theorem 5.2, there exists a conditional
expectation of .

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