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# Uniform Algebra

###### Definition.

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

$\displaystyle\|f^{2}\|$ | $\displaystyle=\|f\|^{2}$ |

In what follows we will show that the Gelfand transform $\Gamma_{{\mathcal{A}}}\colon\mathcal{A}\mapsto\hat{\mathcal{A}}$ of a commutative, unital Banach algebra $\mathcal{A}$ is an isometry if and only if $\mathcal{A}$ is a uniform Banach algebra.

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

Proposition 1. A Banach algebra $\mathcal{A}$ is uniform if and only if itβs Gelfand transform $\Gamma_{{\mathcal{A}}}\colon\mathcal{A}\to\hat{\mathcal{A}},f\mapsto\hat{f}$ is isometric.

###### Proof.

If $f\mapsto\hat{f}$ is an isometry we have $\|f^{2}\|=\|\hat{f}^{2}\|_{{M(\mathcal{A})}}=\|\hat{f}\|_{{M(\mathcal{A})}}^{2% }=\|f\|^{2}$.

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

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

Proposition 2. A Banach algebra $\mathcal{A}$ 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 $\hat{\mathcal{A}}$ separates the points of the compact, nonempty space $M(\mathcal{A})$ we see that a uB algebra $\mathcal{A}$ must have this property.

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

# References

- (Gamelin 2005)
Theodore W. Gamelin
*Uniform Algebras*, Oxford University Press, New Edition, 2005

## Mathematics Subject Classification

46J40*no label found*46J10

*no label found*

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