An operator algebra state on a unital 
-algebra 
is a positive linear
functional 
with 
.
It is faithful when 
implies 
. For a von Neumann algebra,
a state is normal when it preserves suprema of bounded increasing nets of positive
elements. Equivalently, it belongs to the canonical predual.
For example, 
on 
is a state whenever 
is positive semidefinite and 
. It is faithful precisely when 
is positive definite. Faithful normal states define modular
automorphism groups and bicentralizers.
Operator Algebra State
See also
Modular Automorphism Group, Positive Linear Functional, Predual, Quantum State, Tracial StateExplore 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 State." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OperatorAlgebraState.html