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