C*-algebras 5: Characters

In this video we start going over the concept of characters, which is an essential component in the construction of the Gelfand transform for Banach algebras. This is going to culminate in the Gelfand representation (or what I've been calling the commutative Gelfand-Naimark theorem for a while now) in a later video, hopefully finished before Christmas.
Book recommendation 00:00-01:06
Lecture outline 01:06-02:24
Characters, definition 02:24-03:40
The norm of characters 03:40-06:05
Characters and maximal ideals 06:05-13:12
Characters and weak* topology 13:12-15:27
Character space is locally compact and Hausdorff 15:27-21:24
Recap 21:24-23:07
Book recommendation:
books.google.se/books?id=omvi...
Weak topology:
en.wikipedia.org/wiki/Weak_to...
F. Riesz's theorem:
en.wikipedia.org/wiki/F._Ries...
The Banach-Alaoglu theorem:
en.wikipedia.org/wiki/Banach%...
NOTE: I recently discovered that the audio for this video has only been recorded for the right channel. I will resolve it as soon as I find the time.

[@gustavomezzovilla7248]

THANKS for these amazing series! Im now recently studing Fredholm operators over Hilbert C*-modules for my graduetion thesis and there are a lot of details i dont remember from C*-world. That series is really helping me a lot to identify "where things are". Your division of topics are really helpfull in that sense.

[@harrisonbennett7122]

Thank you so much for this series! It is really amazing, you deserve more reccommendation. An amazing complement to my module in Harmonic Analysis!

[@harrisonbennett7122]

P.s I hope you make more of these :)

[@Hjtrne]

I have another attempt at a comment. At 16:55, I don't think it's correct that fi(A) is weak*-closed in the non-unital case, since one of the limit points is the 0-function (or, the restriction to A of fi_0, using the upcoming term), and this doesn't belong to fi(A), which contains only non-zero elements.
This happens under the heading of 'unital case', but contrasting with the following paragraph, I think this first paragraph is meant to apply generally.

[@TheArmchairIntellectual]

I think this is an entirely valid point, and I can definitely see how the way I wrote it can make it seem like the first paragraph applies to the general case rather than just the unital case. But yeah, that slide is really just applicable to the unital case, and one should not think that the first paragraph necessarily applies to the nonunital case. In hindsight I should've probably written "Since" instead of "In the case where" in that second paragraph in order to avoid any potential confusion, but that's easy to say after the fact of course.

[@Hjtrne]

At 05:45 this isn't a contradiction unless fi is already known to be bounded, I think?

[@TheArmchairIntellectual]

It's a contradiction as long as b is a well-defined element in A, since this implies that phi(b) is a complex number. And b is indeed a well-defined element in A because A is a Banach algebra and hence complete with respect to its norm (and ||a||

[@Hjtrne]

@@TheArmchairIntellectual I suppose I can get that. Though the concept of an unbounded function taking only finite values is mental hurdle. Thanks.

[@TheArmchairIntellectual]

@@Hjtrne Unbounded functionals are possible to understand somewhat intuitively, although it can take some getting used to. Essentially, the concept of "unbounded" functional is only relevant in the context of infinite-dimensional normed spaces (since all linear maps between finite-dimensional spaces are bounded). For instance, suppose we have an infinite basis e_1,e_2,..., of unit vectors in some infinite-dimensional normed vector space X. Then we can define the functional f that takes e_n to n for n=1,2,.... Now, although f(x) is just a regular finite number for every vector x in X, we have that the image of the unit ball in X under f is an unbounded set (in R or C).