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International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 640948, 10 pages
http://dx.doi.org/10.1155/2009/640948
Research Article

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 ,(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.

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), (N1), (N 2), and (N3). 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), (N1), (N 2), and (N3).
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 (N2) and (N3). Then,

Proof. Since , it follows that , and so clearly (N) holds. Furthermore, (N1) 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 (N2) and (N3) and . By Theorem 5.2, there exists a conditional expectation of .

References

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