extension and restriction of states



0.1 Restriction of States

Let 𝒜 be a C*-algebra (http://planetmath.org/CAlgebra) and 𝒜 a C*-subalgebra, both having the same identity elementMathworldPlanetmath.

- Given a state ϕ of 𝒜, its restriction (http://planetmath.org/RestrictionOfAFunction) ϕ| to is also a state of .

Remark - Note that the requirement that the C*-algebras 𝒜 and have a (common) identity element is necessary.

For example, let X be a compact space and C(X) the C*-algebra of continuous functionsMathworldPlanetmathPlanetmath X. Pick a point x0X and consider the C*-subalgebra of continuous functions X which vanish at x0. Notice that this subalgebra never has the same identity element of C(X) (the constant function that equals 1). In fact, this subalgebra may not have an identityPlanetmathPlanetmathPlanetmath at all.

Now the evaluation mapping at x0, i.e. the function evx0:C(X)


is a state of C(X). Of course, its restriction to the subalgebra in question is the zero mapping, therefore not being a state.

0.2 Extension of States

Let 𝒜 be a C*-algebra and 𝒜 a C*-subalgebra (not necessarily unital).

Theorem 1 - Every state ϕ of admits an extensionPlanetmathPlanetmathPlanetmath to a state ϕ~ of 𝒜. Moreover, every pure state ϕ of admits an extension to a pure state ϕ~ of 𝒜.

Theorem 2 - The set of extensions of a state ϕ of is a compactPlanetmathPlanetmath and convex subset of S𝒜, the of 𝒜 endowed with the weak-* topologyMathworldPlanetmath.

Title extension and restriction of states
Canonical name ExtensionAndRestrictionOfStates
Date of creation 2013-03-22 18:09:35
Last modified on 2013-03-22 18:09:35
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 10
Author asteroid (17536)
Entry type Theorem
Classification msc 46L30
Related topic State