International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 80152, 22 pages
doi:10.1155/2007/80152
Conditional expectations for unbounded operator algebras
Atsushi Inoue1
, Hidekazu Ogi2
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
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.