extension and restriction of states
0.1 Restriction of States
- Given a state of , its restriction (http://planetmath.org/RestrictionOfAFunction) to is also a state of .
Remark - Note that the requirement that the -algebras and have a (common) identity element is necessary.
For example, let be a compact space and the -algebra of continuous functions . Pick a point and consider the -subalgebra of continuous functions which vanish at . Notice that this subalgebra never has the same identity element of (the constant function that equals ). In fact, this subalgebra may not have an identity at all.
0.2 Extension of States
Let be a -algebra and a -subalgebra (not necessarily unital).
|Title||extension and restriction of states|
|Date of creation||2013-03-22 18:09:35|
|Last modified on||2013-03-22 18:09:35|
|Last modified by||asteroid (17536)|