# Banach Lecture 8

This lecture (banach8) was given on 9/14/09.

### Excerpts

This lecture presented many examples of Gelfond Theory.

1. Consider again $B = l^1(\mathbb{Z})$.  It was shown $\mathscr{M}(B)$ can be identified with $\mathbb{T}$.  Writing $x \in B$ as $x = \sum_{n \in \mathbb{Z}} x(n)e_n$ it follows that

$\hat x (e^{it}) = \sum_{n \in \mathbb{Z}} x(n)e^{int}$

conclusion: $\hat x$ is a function on the unit circle with absolutely convergent Fourier series.  An so define the Weiner Algebra

$W = \widehat{l^1(\mathbb{Z})}$

it become a banach algebra.

Thm I.8.1 (D.Levy) Let $f \in W$ and $h$ holom. on open set containing $f(\mathbb{T})$, then $h\circ f \in W$.

proof: apply thm I.7.2.

Cor. (N Weiner) If $f \in W$, then $1/f \in W$.

Rmk: this is hard result to prove without Gelfond theory.

Some more examples included determining $\mathscr{M}(B)$ for $B$ equal

1. $l^1(\mathbb{Z})$
2. $l^1(\mathbb{Z}_{\ge 0})$
3. $L^1(\mathbb{T})$
4. the Disk algebra = $A = \{ f \in C(\overline{\mathbb{D}})| f|_{\mathbb{D}} \mbox{ is holom.} \}$, norm is sup norm.

Thm. I.8.2 Let $f_1, ..., f_m \in A$ have no common zero in the closed unit disk.  Then there exists $g_1, ..., g_m \in A$ s.t $\sum_i f_ig_i = 1$.