Let $C_0(\mathbb{R})$ be the $C^{\ast}$-algebra of continuous complex-valued functions vanishing at infinity, with involution given by $f^{\ast}(x) = \overline{f(x)}$. How can I prove that this commutative $C^{\ast}$-algebra is generated by the resolvent functions $f_{\pm}(x) = (i \pm x)^{-1}$?
2026-03-29 14:18:17.1774793897
Generator of complex-valued functions vanishing at infinity
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Like you noted in a comment, this is a consequence of the complex version of the Stone–Weierstrass theorem. The hypotheses are easily checked.