International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 80152, 22 pages
doi:10.1155/2007/80152
Research Article

Conditional Expectations for Unbounded Operator Algebras

Atsushi Inoue,1 Hidekazu Ogi,2 and Mayumi Takakura1

1Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan
2Department of Functional Materials Engineering, Fukuoka Institute of Technology, Fukuoka 811-0295, Japan

Received 18 December 2006; Revised 20 March 2007; Accepted 19 May 2007

Academic Editor: Manfred H. Moller

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.

Abstract

Two conditional expectations in unbounded operator algebras (O∗-algebras) are discussed. 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 unbounded conditional expectation which is a positive linear map ℰ of an O∗-algebra ℳ onto a given O∗-subalgebra 𝒩 of ℳ. Here the domain D(ℰ) of ℰ does not equal to ℳ in general, and so such a conditional expectation is called unbounded.