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 $r(f)=\|\hat{f}\|_{M(\mathcal{A})}=\lim_{n\to\infty}\|f^{n}\|^{\frac{1}{n}}$

A Banach algebra $\mathcal{A}$ is uniform if and only if itβs 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
Title Uniform Algebra UniformAlgebra 2013-03-22 19:04:20 2013-03-22 19:04:20 karstenb (16623) karstenb (16623) 5 karstenb (16623) Definition msc 46J40 msc 46J10