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\begin{definition}[$C_0(X)$]
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\label{definition:vanishing-infinity-algebra}
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Let $X$ be a LCH space, then $C_0(X; \complex)$ equipped with pointwise operations and the uniform norm is a $C^*$-algebra.
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Let $X$ be an LCH space, then $C_0(X; \complex)$ equipped with pointwise operations and the uniform norm is a $C^*$-algebra.
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\end{definition}
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\begin{theorem}
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\label{theorem:vanishing-infinity-multiplicative-functional}
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Let $X$ be a LCH space, then the mapping
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Let $X$ be an LCH space, then the mapping
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\[
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E: X \to \Omega(C_0(X)) \quad E(x)(f) = f(x)
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\]
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