58 lines
2.2 KiB
TeX
58 lines
2.2 KiB
TeX
\section{Involutions}
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\label{section:involutions}
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\begin{definition}[Involution]
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\label{definition:involution}
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Let $A$ be an associative algebra over $\complex$, and $*: A \to A$ be a $\real$-linear map, then $*$ is an \textbf{involution} if:
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\begin{enumerate}
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\item[(C1)] For each $\lambda \in \complex$, $(\lambda x)^* = \ol \lambda x^*$.
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\item[(C2)] For each $x \in A$, $x^{**} = x$.
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\item[(I)] For every $x, y \in A$, $(xy)^* = y^*x^*$.
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\end{enumerate}
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The space $A$ equipped with an involution is an \textbf{involutive algebra} over $\complex$.
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\end{definition}
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\begin{definition}[$C^*$-Algebra]
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\label{definition:c-star-algebra}
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Let $A$ be an involutive Banach algebra over $\complex$, then $A$ is a \textbf{$C^*$-algebra} if for every $x \in A$, $\normn{x^*x}_A = \norm{x}_A^2$.
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\end{definition}
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\begin{proposition}
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\label{proposition:c-star-algebra-gymnastics}
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Let $A$ be a $C^*$ algebra, then:
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\begin{enumerate}
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\item For each $x \in A$, $\norm{x}_A = \normn{x^*}_A\norm{x}_A$.
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\end{enumerate}
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If $A$ is unital, then
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\begin{enumerate}[start=1]
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\item For each $\lambda \in \complex$, $\lambda^* = \ol \lambda$.
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\item For any $x \in A$, $x \in G(A)$ if and only if $x^* \in G(A)$.
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\item For every $x \in A$, $\sigma_A(x^*) = \bracsn{\ol \lambda| \lambda \in \sigma_A(x)}$.
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\item For each $x \in A$, $[x]_{sp} = [x^*]_{sp}$.
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\end{enumerate}
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\end{proposition}
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\begin{proof}
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(1): For each $x \in A$, $\norm{x}_A^2 = \normn{x^*x}_A \le \norm{x}_A \normn{x^*}_A$.
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(2): For every $x \in A$, $1^*x^* = (x1)^* = x^* = (1x)^* = x^*1^*$, so $1^* = 1$ by uniqueness of the inverse.
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(3): For any $x \in A$, $(x^{-1})^*x^* = (x^{-1}x)^* = 1 = (xx^{-1})^* = x^*(x^{-1})^*$.
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\end{proof}
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\begin{definition}[*-Homomorphism]
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\label{definition:star-homomorphism}
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Let $A, B$ be $C^*$-algebras and $\phi: A \to B$, then $\phi$ is a \textbf{*-homomorphism} if:
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\begin{enumerate}
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\item $\phi$ is a homomorphism of Banach algebras.
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\item For every $x \in A$, $\phi(x^*) = \phi(x)^*$.
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\end{enumerate}
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If in addition, $\phi(1) = 1$, then $\phi$ is a \textbf{unital *-homomorphism}.
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\end{definition}
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