Re: About characters on H^{oo}

On Wed, 22 Aug 2007 07:13:36 EDT, jane
<jane1806@xxxxxxxxxx> wrote:

Let H^{oo} be the Banach algebra of all bounded on
the unit disk holomorphic functions,defined on the
unit disk with the natural product, addition.

It is clear that for any fixed |a| < 1, f -> f(a) is
a homomorphicm of H^{oo} into C, i.e. character.

I'm interested in the following question: are there
any other characters on H^{oo}, different from those
ones above ?

Yes. Simple proof: First, since H^oo is a ring with
identity, any
(proper) ideal is contained in a maximal (proper)
ideal.
Second, the result that might be called the
fundamental theorem
of commuttative Banach algebras shows that the
maximal ideals
are in natural 1-1 correspondence with the
characters:
If A is a Banach algebra with identity and I is a
maximal
ideal then A/I is what might be called a "Banach
field",
except it's not called that because the theorem says
the only "Banach field" is C. So the natural
homomorphism
from A onto A/I is actually a complex homomorphism
(people don't usually use the word "character" for
this).

So: Let I be the closed ideal generated by the
function
z-1. This is contained in some maximal ideal, and
that
maximal ideal is the kernel of a complex homomorphism
which is not evaluation at any point of the disk.

In general, if A is a Banach algebra with identity
then the space of all complex homomorphism is a
compact Hausdorff space, if you give it the
appopriate weak topology. In the case you ask
about the unit disk is dense in the maximal
ideal space, but this is far from trivial;
it's the "corona theorem" of Carleson, a very
big result.

Thanks.

Thanks a lot.


************************

David C. Ullrich
.

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