= Banach subalgebra generated by one element
For $x$ in a complex unital <Banach algebra> $A$, the closed unital <subalgebra> it generates is $B=\overline{\mathbb C[1,x]}$. Its <character space> is identified with $\sigma_B(x)$ by $\varphi\mapsto\varphi(x)$: <polynomial> density forces uniqueness, and <spectrum equals character values in a commutative Banach algebra> gives existence.
The ambient algebra matters. <Spectrum of a polynomial-generated Banach subalgebra> shows that $\sigma_B(x)$ is the <full spectrum> of $x$ in $A$, including all bounded complementary components of $\sigma_A(x)$. For example, in $A=C(\mathbb T)$ the coordinate $x(z)=z$ has <spectrum> the <unit circle>, whereas $B$ is the boundary realization of the <disk algebra> and $\sigma_B(x)$ is the closed <unit disk>. Consequently it is the complement of $\sigma_B(x)$, rather than necessarily that of $\sigma_A(x)$, which is connected.
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