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Characters on a unital Banach algebra are continuous

Statement

Let $A$ be a nonzero unital complex Banach algebra (Unital Banach algebra) and let $χ$ be a character on $A$ (Character and maximal ideal space). Then:

$χ$ is unital: $χ (1) = 1$ ;
$χ (a) \in σ_{A} (a)$ for every $a \in A$ (Spectrum and resolvent set in a Banach algebra);
$∣ χ (a) ∣ \leq ∥ a ∥$ for every $a \in A$ ; consequently $χ$ is bounded linear with $∥ χ ∥ = 1$ , and in particular $χ$ is continuous.

No Hahn–Banach theorem, no spectral-radius formula and no existence of characters is used: the argument runs on the Neumann series alone and is a theorem of ZF.

Facts & Assumptions

Given: A nonzero unital complex Banach algebra $A$ , a character $χ$ on $A$ , and an element $a \in A$ .

[L1]

$A$ is a complex vector space with associative bilinear multiplication, a complete submultiplicative norm, a unit $1$ with $1 x = x 1 = x$ and $∥ 1 ∥ = 1$ , and $0 \neq 1$ because $A$ is nonzero (Unital Banach algebra).

[L2]

A character is nonzero, complex-linear and multiplicative; in particular $χ (1 \cdot 1) = χ (1)^{2}$ and $χ (a \cdot 1) = χ (a) χ (1)$ (Character and maximal ideal space).

[L3]

$b \in A$ is invertible exactly when $b c = c b = 1$ for some $c \in A$ , and $z \in σ_{A} (a)$ exactly when $z 1 - a$ is not invertible (Spectrum and resolvent set in a Banach algebra).

[L4]

If $∥ y ∥ < 1$ then $1 - y$ is invertible, with inverse $\sum_{n \geq 0} y^{n}$ (Neumann series).

Proof

technique · direct

1.1

$χ (1) = χ (1 \cdot 1) = χ (1)^{2}$ by multiplicativity [L2], so $χ (1) \in {0, 1}$ ; if $χ (1) = 0$ then $χ (a) = χ (a \cdot 1) = χ (a) χ (1) = 0$ for every $a$ , contradicting that $χ$ is nonzero [L2], so $χ (1) = 1$ .

L1L2algebra

2.1

Put $λ : = χ (a)$ and suppose $λ \notin σ_{A} (a)$ , so that $a - λ 1$ is invertible with two-sided inverse $b$ [L3]. Then $1 = χ (1) = χ (b (a - λ 1)) = χ (b) (χ (a) - λ χ (1)) = χ (b) \cdot 0 = 0$ by [step 1.1], [L2] and linearity, a contradiction; hence $χ (a) \in σ_{A} (a)$ .

step 1.1L2L3

3.1

Suppose $∣ λ ∣ > ∥ a ∥$ where $λ = χ (a)$ . Then $∥ a / λ ∥ = ∥ a ∥ / ∣ λ ∣ < 1$ , so $1 - a / λ$ is invertible by [L4], and therefore $λ 1 - a = λ (1 - a / λ)$ is invertible with inverse $λ^{- 1} (1 - a / λ)^{- 1}$ ; by [L3] this says $λ \notin σ_{A} (a)$ , contradicting [step 2.1]. Hence $∣ χ (a) ∣ \leq ∥ a ∥$ for every $a \in A$ .

step 2.1L1L3L4algebra

4.1

By [step 1.1] $χ (1) = 1$ and by [step 3.1] $∣ χ (a) ∣ \leq ∥ a ∥$ for all $a$ , so $χ$ is bounded linear with $∥ χ ∥ \leq 1$ ; since $∥ 1 ∥ = 1$ and $χ (1) = 1$ , $∥ χ ∥ = 1$ . Every bounded linear map between normed spaces is continuous, so $χ$ is continuous.

step 1.1step 3.1L1∎

Remarks

The unit hypothesis streamlines the proof, but continuity survives without it. For a nonzero unital Banach algebra the computation $χ (1)^{2} = χ (1)$ forces unitality. A character on a nonunital Banach algebra $B$ extends to the algebraic sum-norm unitization $C \oplus B$ , with product $(λ, a) (μ, b) = (λ μ, λ b + μ a + a b)$ , by $χ^{+} (λ, a) = λ + χ (a)$ . This is a unital character on a unital Banach algebra, so the theorem applied to $χ^{+}$ shows that the original character is continuous as well.
Choice-free. Steps 1.1–2.2 use only the algebra axioms, the definition of the spectrum and the Neumann series; no selection from nonempty sets and no separation theorem occurs.
Where the bound is used. Part 3 is what puts $Δ (A)$ inside the dual unit ball and identifies the pointwise-evaluation topology with the weak-star subspace topology; this is the standard automatic-continuity statement for characters.

Depends on

Used by

Gelfand transform of a Banach algebra need not be isometric Counterexample
C zero of a locally compact space Example
Character space of the disc algebra Example
Gelfand transform of ell one of Z Example
Characters of continuous functions are evaluations Lemma
Characters on a unital commutative C star algebra preserve star Lemma
Character space of the unitization is one-point compactification Theorem
Gelfand transform is a contractive unital homomorphism Theorem
Locally compact Gelfand duality Theorem
Maximal ideal space is compact Hausdorff Theorem
Maximal ideals and characters of a commutative Banach algebra Theorem
Spectrum as character values Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Vahid Shirbisheh, Lectures on C-star Algebras, v2 — Theorem 3.1.11 proof and Proposition 3.1.12(i), printed pp. 54–61 (standard reference, not scraped)
Theo Bühler and Dietmar A. Salamon, Functional Analysis — Chapter 5 §5.5.1, printed pp. 258–262 (standard reference, not scraped)