An operator algebra conditional expectation onto a unital 
-subalgebra 
is a unital completely
positive map 
that fixes 
and satisfies 
for 
and 
. It is an idempotent projection onto 
. For von Neumann algebras,
normality means preservation of increasing bounded suprema of positive
elements, and faithfulness means 
implies 
. An inclusion admitting a faithful normal conditional expectation
is called expected. The diagonal map 
is an example. In commutative probability algebras,
this construction recovers conditional expectation onto a sigma-algebra.
Operator Algebra Conditional Expectation
See also
Completely Positive Map, Conditional Expectation, Maximal Abelian Subalgebra, von Neumann AlgebraExplore with Wolfram|Alpha
References
Anantharaman, C. and Popa, S. An Introduction to
Factors. https://www.math.ucla.edu/~popa/Books/IIun.pdf.
Cite this as:
Weisstein, Eric W. "Operator Algebra Conditional Expectation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OperatorAlgebraConditionalExpectation.html