extension and restriction of states

restriction

0.1 Restriction of States

Let $𝒜$ be a $C^{*}$-algebra (http://planetmath.org/CAlgebra) and $ℬ\subset 𝒜$ a $C^{*}$-subalgebra, both having the same identity element.

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- Given a state $\varphi$ of $𝒜$, its restriction (http://planetmath.org/RestrictionOfAFunction) ${\varphi |}_{ℬ}$ to $ℬ$ is also a state of $ℬ$.

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

For example, let $X$ be a compact space and $C(X)$ the $C^{*}$-algebra of continuous functions $X\to ℂ$. Pick a point $x_0\in X$ and consider the $C^{*}$-subalgebra of continuous functions $X\to ℂ$ which vanish at $x_0$. 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 identity at all.

Now the evaluation mapping at $x_0$, i.e. the function ${\mathrm{ev}}_{x_0}:C(X)\to ℂ$

${\mathrm{ev}}_{x_0}(f):=f(x_0)$

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 $ℬ\subset 𝒜$ a $C^{*}$-subalgebra (not necessarily unital).

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Theorem 1 - Every state $\varphi$ of $ℬ$ admits an extension to a state $\tilde{\varphi }$ of $𝒜$. Moreover, every pure state $\varphi$ of $ℬ$ admits an extension to a pure state $\tilde{\varphi }$ of $𝒜$.

Theorem 2 - The set of extensions of a state $\varphi$ of $ℬ$ is a compact and convex subset of $S_{𝒜}$, the of $𝒜$ endowed with the weak-* topology.