State on Unital C*-Subalgebra extends to whole C*-Algebra

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Theorem

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.

Let $B$ be a unital $\text C^\ast$-subalgebra of $A$.

Let $f_1 : B \to \C$ be a state.


Then there exists a state $f : A \to \C$ extending $f_1$.


Proof

Let $\le_A$ be the canonical preordering of $A$.

Let $B_{\mathbf {SA} }$ and $A_{\mathbf {SA} }$ be the set of Hermitian elements of $B$ and $A$ respectively.

From Hermitian Elements of *-Algebra form Real Vector Subspace, these are vector spaces over $\R$.

From Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum, for each $a \in A_{\mathbf {SA} }$ we have:

$a \le_A \norm a {\mathbf 1}_A$

where $\norm a {\mathbf 1}_A \in B_{\mathbf {SA} }$.

Hence $B_{\mathbf {SA} }$ is cofinal in the preordered vector space $\struct {A_{\mathbf {SA} }, \le_A}$.

From Positive Linear Functional on C*-Algebra preserves Star and Complex Number equals Conjugate iff Wholly Real, $f_1$ is real-valued on $B_{\mathbf {SA} }$.

Hence from Extension Theorem for Positive Linear Functional defined on Cofinal Linear Subspace, there exists a positive linear functional $f_0 : A_{\mathbf {SA} } \to \R$ extending $f_1 \restriction {B_{\mathbf {SA} } }$.

For general $a \in A$, define:

$\map f a = \map {f_0} {\map \Re a} + i \map {f_0} {\map \Im a}$

For $b \in B$, we have:

\(\ds \map f b\) \(=\) \(\ds \map {f_0} {\map \Re b} + i \map {f_0} {\map \Im b}\)
\(\ds \) \(=\) \(\ds \map {f_1} {\map \Re b} + i \map {f_1} {\map \Im b}\) $f_0 \restriction B_{\mathbf{SA} } = f_1 \restriction B_{\mathbf{SA} }$
\(\ds \) \(=\) \(\ds \map {f_1} {\map \Re b + i \map \Im b}\) $f_1$ is linear
\(\ds \) \(=\) \(\ds \map {f_1} b\) Element of *-Algebra Uniquely Decomposes into Hermitian Elements

Hence $f$ extends $f_1$.


We show that $f$ is as desired.

We first show that $f$ is linear.

Firstly, for $a \in A$ we have:

\(\ds \map f {i a}\) \(=\) \(\ds \map {f_0} {\map \Re {i a} } + i \map {f_0} {\map \Im {ia} }\)
\(\ds \) \(=\) \(\ds \map {f_0} {-\map \Im a} + i \map {f_0} {\map \Re a}\) Real Part of Imaginary Unit times Element of *-Algebra, Imaginary Part of Imaginary Unit times Element of *-Algebra
\(\ds \) \(=\) \(\ds i \map {f_0} {\map \Re a} - \map {f_0} {\map \Im a}\) since $f_0$ is linear
\(\ds \) \(=\) \(\ds i \paren {\map {f_0} {\map \Re a} + i \map {f_0} {\map \Im a} }\)
\(\ds \) \(=\) \(\ds i \map f a\)

Now let $a \in A$ and $\lambda \in \R$, we have that:

\(\ds \map f {\lambda a}\) \(=\) \(\ds \map {f_0} {\map \Re {\lambda a} } + i \map {f_0} {\map \Im {\lambda a} }\)
\(\ds \) \(=\) \(\ds \map {f_0} {\lambda \map \Re a} + i \map {f_0} {\lambda \map \Im a}\) Real Part of Element of *-Algebra is Real Linear and Imaginary Part of Element of *-Algebra is Real Linear
\(\ds \) \(=\) \(\ds \lambda \paren {\map {f_0} {\map \Re a} + i \map {f_0} {\map \Im a} }\) $f_0$ is linear
\(\ds \) \(=\) \(\ds \lambda \map f a\)

We also have, for $a, b \in A$:

\(\ds \map f {a + b}\) \(=\) \(\ds \map {f_0} {\map \Re {a + b} } + i \map {f_0} {\map \Im {a + b} }\)
\(\ds \) \(=\) \(\ds \map {f_0} {\map \Re a + \map \Re b} + i \map {f_0} {\map \Im a + \map \Im b}\) Real Part of Element of *-Algebra is Real Linear and Imaginary Part of Element of *-Algebra is Real Linear
\(\ds \) \(=\) \(\ds \paren {\map {f_0} {\map \Re a} + i \map {f_0} {\map \Im a} } + \paren {\map {f_0} {\map \Re b} + i \map {f_0} {\map \Im b} }\) $f_0$ is linear
\(\ds \) \(=\) \(\ds \map f a + \map f b\)

Hence for $a, b \in A$ and $\lambda \in \C$ we have:

\(\ds \map f {a + \lambda b}\) \(=\) \(\ds \map f a + \map f {\paren {\map \Re \lambda + i \map \Im \lambda} b}\)
\(\ds \) \(=\) \(\ds \map f a + \map \Re \lambda \map f b + \map \Im \lambda \map f {i b}\)
\(\ds \) \(=\) \(\ds \map f a + \map \Re \lambda \map f b + i \map \Im \lambda \map f b\)
\(\ds \) \(=\) \(\ds \map f a + \paren {\map \Re \lambda + i \map \Im \lambda} \map f b\)
\(\ds \) \(=\) \(\ds \map f a + \lambda \map f b\)


For $a \in A$ positive, we have:

$\map f a = \map {f_0} a \ge 0$

Hence $f$ is a positive linear functional.

From Norm of Positive Linear Functional on Unital C*-Algebra, we have:

\(\ds \norm f_{A^\ast}\) \(=\) \(\ds \map f { {\mathbf 1}_A}\)
\(\ds \) \(=\) \(\ds \map {f_1} { {\mathbf 1}_A}\) Identity Element in Unital *-Algebra is Hermitian
\(\ds \) \(=\) \(\ds \map {f_0} { {\mathbf 1}_B}\) $B$ is unital
\(\ds \) \(=\) \(\ds \norm {f_0}_{B^\ast}\) Norm of Positive Linear Functional on Unital C*-Algebra
\(\ds \) \(=\) \(\ds 1\) Norm of Positive Linear Functional on Unital C*-Algebra

Hence $f$ is a state that extends $f_0$ as required.

$\blacksquare$


Sources

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