Uniform Algebra

Definition.

A commutative, unital Banach algebra $(𝒜,\parallel \cdot \parallel )$ is called uniform Banach algebra (for short: uB algebra) if for all $f\in 𝒜$ we have

$\parallel f^2\parallel$$={\parallel f\parallel }^2$

In what follows we will show that the Gelfand transform ${\mathrm{\Gamma }}_{𝒜}:𝒜\mapsto \widehat{𝒜}$ of a commutative, unital Banach algebra $𝒜$ is an isometry if and only if $𝒜$ is a uniform Banach algebra.

Denote by $M(𝒜)$ the space of (continuous) characters on $𝒜$.Recall that for all $f\in 𝒜$ the spectrum $\sigma (f)$ of $f$ is identical with the range $\widehat{f}(M(𝒜))$ and the spectral radius $r(f)={\parallel \widehat{f}\parallel }_{M(𝒜)}={lim}_{n\to \mathrm{\infty }}{\parallel f^n\parallel }^{\frac{1}{n}}$

Proposition 1. A Banach algebra $𝒜$ is uniform if and only if it’s Gelfand transform ${\mathrm{\Gamma }}_{𝒜}:𝒜\to \widehat{𝒜},f\mapsto \widehat{f}$ is isometric.

Proof.

If $f\mapsto \widehat{f}$ is an isometry we have $\parallel f^2\parallel ={\parallel {\widehat{f}}^2\parallel }_{M(𝒜)}={\parallel \widehat{f}\parallel }_{M(𝒜)}^2={\parallel f\parallel }^2$.

Conversely assume $\parallel f^2\parallel ={\parallel f\parallel }^2$ for all $f\in 𝒜$. Then by induction we have $\parallel f^{2^n}\parallel ={\parallel f\parallel }^{2^n}$ for all $n\in \mathbb{N}$. Hence $\parallel f\parallel ={\parallel f^{2^n}\parallel }^{\frac{1}{2^n}}\to {\parallel \widehat{f}\parallel }_{M(𝒜)}$.∎

The following characterization is also often given as the definition of a uB algebra.

Proposition 2. A Banach algebra $𝒜$ is uniform iff it is topologically and algebraically isomorphic to a closed, pointseparating subalgebra of $C(X)$ for $X$ a compact Hausdorff space.

Proof.

Since $\widehat{𝒜}$ separates the points of the compact, nonempty space $M(𝒜)$ we see that a uB algebra $𝒜$ must have this property.

Conversely, let $𝒜$ be a closed pointseparating subalgebra of $C(X)$. Then clearly ${\parallel f\parallel }_X^2={\parallel f^2\parallel }_X$ for all $f\in 𝒜$.∎