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 ,(N₁),(N₂) is essentially self-adjoint for ,(N₃), 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 (N₁) 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 (N₂) 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 (N₃ ) 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.

Theor