47 lines
1.7 KiB
TeX
47 lines
1.7 KiB
TeX
\section{*-Homomorphisms}
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\label{section:c-star-homomorphism}
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\begin{definition}[*-Homomorphism]
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\label{definition:c-star-homomorphism}
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Let $A, B$ be involutive algebras over $\complex$ and $\phi: A \to B$, then $\phi$ is a \textbf{*-homomorphism} if:
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\begin{enumerate}[label=(SH\arabic*)]
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\item For each $x, y \in A$ and $\lambda \in \complex$, $\phi(\lambda x + y) = \lambda \phi(x) + \phi(y)$.
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\item For each $x, y \in A$, $\phi(xy) = \phi(x)\phi(y)$.
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\item For each $x \in A$, $\phi(x^*) = \phi(x)^*$.
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\end{enumerate}
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If $A$ and $B$ are unital, then $\phi$ is \textbf{unital} if:
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\begin{enumerate}
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\item[(U)] $\phi(1_A) = 1_B$.
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\end{enumerate}
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\end{definition}
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\begin{proposition}
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\label{proposition:star-homomorphism-contractive}
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Let $A, B$ be unital $C^*$-algebras and $\phi: A \to B$ be a unital *-homomorphism, then for each $x \in A$,
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\begin{enumerate}
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\item $\sigma_B(\phi(x)) \subset \sigma_A(x)$.
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\item $\norm{\phi(x)}_B \le \norm{x}_A$.
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\end{enumerate}
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\end{proposition}
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\begin{proof}
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(1): Since $\phi$ is unital, $\phi(G(A)) \subset G(B)$, so $\sigma_B(\phi(x)) \subset \sigma_A(x)$.
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(2): By (1) and \autoref{corollary:c-star-unique-norm},
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\begin{align*}
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\norm{\phi(x)}_B^2 &= \sup\bracsn{|\lambda|\ | \lambda \in \sigma_B(\phi(x^*x))} \\
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&\ge \sup\bracsn{|\lambda|\ | \lambda \in \sigma_A(x^*x)} = \norm{x}_A^2
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\end{align*}
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\end{proof}
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\begin{theorem}
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\label{theorem:continuity-of-homomorphism-c-star}
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Let $A, B$ be unital $C^*$-algebras and $\Phi: A \to B$ be a unital *-homomorphism, then $\Phi(A)$ is closed.
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\end{theorem}
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\begin{proof}[Proof, {{\cite[Theorem 11.1]{Zhu}}}. ]
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\end{proof}
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