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.

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