#### 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 .