Conditional Expectations for Unbounded Operator Algebras

摘要

Two conditional expectations in unbounded operator algebras (O?-algebras) are discussed. One is a vector conditional expectation defined by a linear map of anO?-algebra into the Hilbert space on which theO?-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?of anO?-algebra?onto a givenO?-subalgebra𝒩of?.Here the domainD(?)of?does not equal to?in general, and so such a conditional expectation is called unbounded.