2.2.4. Spectral seminorm [00MP]

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2.2.4. Spectral seminorm

Definition 2.28.

The spectral algebra seminorm $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ of an algebra seminorm $\vvvert\mathord{\cdot}\vvvert$ on a $k$ -algebra $A$ is the one defined by

\forall f\in\mathcal{A},\quad\vvvert f\vvvert_{\mathrm{sp}}:=\lim_{\begin{subarray}{c}n\to\infty\end{subarray}}\vvvert f^{n}\vvvert^{\frac{1}{n}}.

Note that the triangle inequality for $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ follows from sub-multiplicativity of $\vvvert\mathord{\cdot}\vvvert$ . In general, $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ is only a seminorm even if $\vvvert\mathord{\cdot}\vvvert$ is a norm.

Remark 2.29.

The existence of limit is guaranteed by the (multiplicative) Fekete lemma for the sub-multiplicative sequence $\{\vvvert f^{n}\vvvert\}_{n\in\mathbb{N}}$ . The spectral seminorm is sub-multiplicative, and is bounded by the original seminorm $\vvvert\mathord{\cdot}\vvvert$ . Moreover, it is power-multiplicative by construction.

Proposition 2.30.

Let $\mathcal{A}$ be a $k$ -Banach algebra. For any $f\in A$ , one has ([Ber, Theorem 1.3.1])

\vvvert f\vvvert_{\mathrm{sp}}=\max_{z\in\mathfrak{M}(A)}|f|_{z}

Definition 2.31.

Let $\mathcal{A}$ be a Banach $k$ -algebra. The radical of $\mathcal{A}$ is the null-space of its spectral seminorm $\mathfrak{n}(\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}})$ . A Banach $k$ -algebra with radical equal to $\{0\}$ is called semi-simple. Elements in the radical are said to be quasi-nilpotent (or topological nilpotent).

Remark 2.32.

The radical of $\mathcal{A}$ contains the nil-radical of $A$ ; in other words, nilpotent elemtents are quasi-nilpotent. If $\mathcal{A}$ is semi-simple, then $A$ is reduced. The converse may not be true.

Let $\mathcal{A}=(A,\vvvert\mathord{\cdot}\vvvert)$ be a $k$ -Banach algebra. The spectral seminorm $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ defines a quotient norm on the quotient $k$ -algebra $\mathcal{A}/\text{rad}(\mathcal{A})$ , still denoted by $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ . The quotient norm $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ is bounded by the quotient norm of $\vvvert\mathord{\cdot}\vvvert$ . The uniformization $\mathcal{A}^{u}$ of $\mathcal{A}$ is defined to be the Banach $k$ -algebra of separated completion of $(\mathcal{A}/\text{rad}(\mathcal{A}),\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}})$ . Conversely, if $\vvvert\mathord{\cdot}\vvvert$ is a power-multiplicative Banach algebra norm on $A$ with radical $\{0\}$ , then it is said to be uniform.

Obviously, $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ is bounded by $\vvvert\mathord{\cdot}\vvvert$ . It is important to note that the converse may not be true in general. In other words, $\vvvert\cdot\vvvert_{\mathrm{sp}}$ may not be complete on $\mathcal{A}/\text{rad}(\mathcal{A})$ . Yet one still has the following statement

Proposition 2.33.

$\mathfrak{M}(\mathcal{A})$ is canonically homeomorphic to $\mathfrak{M}(\mathcal{A}^{u})$ . ([Ber, Corollary 1.3.3, 1.3.4])