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For $x$ in a unital complex algebra $A$, the spectrum of an element isFor a nonunital algebra one uses its unitization.

Now let $A$ be a Banach algebra. The invertible group is open, so the resolvent set is open and the spectrum is closed. If $|\lambda |>\Vert x\Vert$, the Neumann seriesconverges, so ${\sigma }_A(x)$ is contained in the closed disc of radius $\Vert x\Vert$ and is therefore compact.

If the spectrum were empty, $R(\lambda )=(\lambda 1-x{)}^{-1}$ would be an entire $A$-valued function. For each $f\in A^{\ast }$, the scalar function $f(R(\lambda ))$ is bounded: it tends to zero at infinity by the Neumann series and is bounded on every compact disc. The Liouville theorem makes it identically zero. Since the Hahn-Banach theorem separates points, this would give $R(\lambda )=0$, contradicting its invertibility. Hence the spectrum is nonempty.

Solved by gpt-5.6-sol high.

For a unital Banach algebra, $\phi (a)$ belongs to ${\sigma }_A(a)$: otherwise $a-\phi (a)1$ would be invertible, while applying $\phi$ to its inverse identity would give $0=1$. Thereforeso every character of an algebra is continuous and has norm one. The nonunital case follows by extending the character to the unitization of an algebra.

The Gelfand topology on ${\Phi }_A$ is the weak-star topology inherited from $A^{\ast }$: a net ${\phi }_i$ converges to $\phi$ exactly when ${\phi }_i(a)\to \phi (a)$ for every $a\in A$. If $A$ is unital, ${\Phi }_A$ lies in the weak-star compact dual unit ball by the Banach-Alaoglu theorem. The equationsdefine a weak-star closed subset, so ${\Phi }_A$ is compact.

Solved by gpt-5.6-sol high.

The unitization of $C_0(\mathbb{R})$ is $C({\mathbb{R}}^{+})$ on the one-point compactification of $\mathbb{R}$. The supplied homeomorphism identifies ${\mathbb{R}}^{+}$ with the circle $\mathbb{T}$. Every character $\phi$ of $C_0(\mathbb{R})$ extends to the unital characterBy part d, $\tilde{\phi }$ is evaluation at a point of ${\mathbb{R}}^{+}$. Evaluation at the point at infinity vanishes on $C_0(\mathbb{R})$ and cannot restrict to the nonzero character $\phi$. The point is therefore some $x\in \mathbb{R}$, and $\phi ={\delta }_x$. Conversely each ${\delta }_x$ is plainly a character, so

Solved by gpt-5.6-sol high.

The spectrum of $a$ in a unital complex algebra $A$ isFor a nonunital algebra it is defined in the unitization of an algebra.