extension and restriction of states

0.1 Restriction of States

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

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

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

For example, let $X$ be a compact space and $C(X)$ the $C^{*}$-algebra of continuous functions $X\to\mathbb{C}$. Pick a point $x_{0}\in X$ and consider the $C^{*}$-subalgebra of continuous functions $X\to\mathbb{C}$ 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 $\mathrm{ev}_{x_{0}}:C(X)\to\mathbb{C}$

 $\displaystyle\mathrm{ev}_{x_{0}}(f):=f(x_{0})$

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

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

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

Title extension and restriction of states ExtensionAndRestrictionOfStates 2013-03-22 18:09:35 2013-03-22 18:09:35 asteroid (17536) asteroid (17536) 10 asteroid (17536) Theorem msc 46L30 State