Banach Lecture 6

the lecture Banach6

Excerpts

First came a review of wk-* topologies on the dual of a Banach space.  For example

Alaoglu’s Thm: Let $X$ be a Banach space and give $X^*$ the wk-* topology.  The closed unit ball in $X^*$ is compact.

It follows that all closed balls in $X^*$ are wk-* compact and that all wk-* closed and bounded sets are compact.

For a unital Banach algebra, the Gelfond topology on $\mathscr{M}(B) \subset$ unit ball in $X^*$ is the induced topology from the wk-* topology.  Using the thm above it can be shown that $\mathscr{M}(B)$ is a compt. Hausdorff space.  $\mathscr{M}(B)$ with its Gelfond topology is called the Gelfond space of $B$ or the maximal ideal space or the structure space or the spectrum.

In the nonunital case, construct Gelfond space of $B_1 = B \oplus \mathbb{C}$ and idenitify $\mathscr{M}(B)$ with $\mathscr{M}(B_1) - \{\phi_0\}$ where $\phi_0$ is the functional with $B$ as its kernel.

The Gelfond transformation is a map $B \to C(\mathscr{M}(B))$ which sends $lates a \to \hat a$ where $\hat a \colon \mathscr{M}(B) \to \mathbb{C}$ send $\phi \mapsto \langle \phi, a \rangle$.