#in which i am forcing myself to learn Harmonic Analysis

[endofunktor]

Defining the Gelfand Transformation

So. I’m trying something new here. They say the best way to learn stuff is to teach stuff. And I am desperately trying to learn some Harmonic Analysis, which is way out of my comfort zone since I consider myself more of an algebra person. So here we are. The Gelfand Transformation.

The Gelfand Transformation is the application of a very algebraic concept to topics in Analysis -- namely, the concept of Evaluation Homomorphisms.

As an example, have a look at your average Ring of Polynomials K[X] over a commutative ring K. Now, fix some z in K. Then there exists a map

This is actually a ring homomorphism! Let’s check: 1. The product of polynomials is actually defined exactly so that the product of the evaluation coincides with the evaluation of the product. Here, of course, we need the commutativity of K.

2. The same holds for the sum of polynomials.

3. (optional because people don’t agree on the definition of a ring homomorphism) If K has a One, then so has K[X] and 1 in K[X] evaluated in z is just 1 in K.

Now. Because we are doing Analysis, we are not just looking at commutative rings, but at Banach algebras. What is that, I hear you ask? Basically A K-Algebra is a K-vector space B with a ring-like product structure that is compatible with scalar multiplication. For example, take the ring of n x n-matrices or K[X]. If your are doing Analysis, you want K = C. Then you can put a norm on it. If it is compatible with your algebra product, i.e.

then it is called a Normed Algebra. If it is complete, it is a Banach Algebra.

In a commutative banach algebra A, we can first define the Spectrum of A as the set of maximal ideals, but we quickly discover that maximal ideals are just kernels of non-trivial Algebra homomorphisms (basicallly continuous linear maps s.t. f(xy) = f(x)f(y) ) from A to C. This is a 1:1-correspondence, so

And now, for every a in A, we can define

our Evaluation Homomorphism.

That is not yet our Gelfand Transformation! We have to put some work into the spectrum sigma(A). By our identification, it can be thought of a subset of the dual space A’ of A (consisting of all continuous linear maps) and thus can be equipped with the subspace topology. We will skip the details here, but sigma(A) turns out to be a compact Hausdorff space and  Гa turns out to be continuous. Now we define the set of continuous functions (as usual) as C(sigma(A)). And now we can define the Gelfand Transformation as follows.

where Гa is defined as above.

I think that will be it for today. We will look at basic properties and first applications maybe tomorrow? I don’t know, but I fear I am on a tight schedule here since I have to do an exam on this at some point.

Anyway, if you have questions about anything, feel free to ask! Especially if you want more details or feel like I’m making a mistake.

[thousandmaths]

Twenty

So I don’t really remember why I was thinking about this but it’s flooding back and I’m going to forget it in an hour so I have to write it now.

------

My major advisor (not my thesis advisor) in undergrad was a fellow named Michael Orrison. He spends his time thinking about representation theory and algebraic statistics. It’s pretty cool stuff. 

During my four years in college, I’ve took three courses from him. One was Discrete, a sort of intro-to-proofs course, and I am relatively confident in saying that it was my favorite course I will ever take in my life*. One was Algebra I, which was… challenging.

[ * although, I have to admit, 9th grade history does come pretty close. ]

The last was a special topics course about harmonic analysis on finite groups, which is a stone’s throw from his research. In that class I learned three things. I learned that there is a difference between being good at presentations and being good at lectures. I learned why combinatorialists care about representation theory. And I also learned this:

At some point in the last third of the course, he takes a break from lecturing to get all starry-eyed. His #passion has been out in full force during this lecture. He’s reached a conclusion, and he turns to the class and tells us about his journey into this field. I couldn’t recount it; I don’t remember any of it. I do remember that he conceptualized the work in his discipline as a story, a story that he and his colleagues were privileged to be allowed to read.

In passing, in transition to a more important thing, he said “I’ve been thinking about this story now for 20 years, and…”. And somehow, at that moment, the incredible enormity of that statement resonated with me.

I think of myself as a pretty committed individual. I have projects that my parents started in grade school under my name and I still now actually run them. I played piano from kindergarden to graduation. I spent long enough drawing cartoons in a forum that they let me moderate it. I’m writing a blog where I’m trying to write 1000 posts about math 1000 days (it’s pretty cool you should check it out). And yet—

20 years.

I was not quite 21 years old at the time. 20 years isn’t a length of time that I have been “doing” anything. 20 years is a length of time that I have been “existing”. The idea of doing something for 20 years, is, still, mind-numbing. Literally. My brain shuts down as it tries to imagine.

How old are you? 

What fraction of your life is 20 years?

(Someone will read this who wasn’t alive 20 years ago. Someone will read this whose parents didn’t know each other 20 years ago.)

Orrison has been thinking about that story since I was pooping in diapers. Every single moment of embarrassment or irritation, every time I ran away from home, every stupid game I got obsessed with, every argument with a friend, my whole damn creative life (and longer) was spent thinking about this story.

And it occurs to me: the fact that I find this inspiring (instead of, you know: deeply, existentially terrifying) is one of the best indications I’ve ever had that I would be in the math game for the long haul.

[maybrandon]

Youtube Reiki Positive Energy Music Awesome Unique Ideas

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Reiki Ideals

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Reiki Master Business Cards

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[potterzachary]

Reiki Crystal Symbols Eye-Opening Ideas

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Reiki Symbol Akashic Records

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What Is A Reiki Crystal Grid

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[endofunktor]

Quick Update on Harmonic Analysis

I do have a draft on the use of the Gelfand Transform but I also have an exam about that on Tuesday, so wish me luck.

Anyway.

I have noticed that in usual (id est: finite) algebra, a lot of proofs about character theory include a term like this

for some function f: G → K from G to some field K or whatever. Then some argument happens and suddenly we consider a term like this

for some g in G. And because x ↦ gx is a bijection,

follows. On the other hand: Consider a topological group G, id est a group with Hausdorff topology where multiplication (with respect to the product topology) and inversion are continuous. Now we’re suddenly doing analysis! But in analysis, nothing is finite and suddenly, we are considering terms like this

and we would like to have

but that is not true in general! Whether or not that is true depends heavily on the (implied) measure dx. The natural question is: Can we construct a well-behaved left-invariant measure on G? Enter Alfréd Haar, Hungarian mathematician, who introduced this measure (which is subsequently called Haar measure) in 1933, the year of his death (I know what you think, but he died of stomach cancer).

There will be a post on Haar measures in the future. And the one about the Gelfand transform. I promise. Consider this another teaser.

[endofunktor]

Using the Gelfand Transformation -- Part 2

Okay, I promised applications, but like every good algebraist, I have to postpone them another day.

We will have a look at a variation of Wiener’s lemma soon, but there are some details that are still a bit unclear to me.

Instead, have this neat little fun fact.

Start with a compact Hausdorff space X. Consider the space C(X) of all continuous functions X → C. This is a Banach algebra (with point-wise multiplication) over C. Thus, we can consider the spectrum σ(C(X)), the set of maximal ideals in C(X). I claim that every maximal ideal is of the form

This is the kernel of the delta function in the point x in X. It evaluates continiuous functions f in said point x. (And yes, we are calling it a “function” because going from C(X) to C it is. It is, of course, not a function X → C.)

The delta function is an algebra morphism (check that if you don’t believe me!), but is it continuous? Yes, because

To prove that every proper ideal I is contained in a maximal ideal Ix of the form above, we use a neat topological argument to construct an invertible element in I (i.e. a function that is nowhere zero) which gives a contradiction to I being proper.

Assume that I is not contained in some Ix. Then for all x in X there exists a function fx in I such that

Because I is an ideal, we can multiply fx with its conjugate (which is in C(X)) and obtain another element in I. So,

Now, if we could just sum all such positive functions |fx|^2 up (over x in X), we would obtain a function g that is nowhere zero. However, X is not finite, so we’ll have to use the next best property: compactness.

The sets

form an open cover of X and thus -- by compactness! -- there exists a finite subcover, i.e. a set S of finitely many fx such that for all z in X there exists fx in S such that fx(z) is not zero. Now, those we can sum up! Then the function

is in I and has no zero. Thus, it is invertible and by that, I can not be a proper ideal. A contradiction.

Now, what was all that for? We now know that

as sets. Neat! Remember that the Gelfand Transformation goes from A to C(σ(A)) which in the case of A = C(X) reduces to a map C(X) to C(X). Let’s look at it side by side. Left: the general case and right: A = C(X).

So it turns out, Γf = f, and the Gelfand Transformation is just the identity.

[endofunktor]

Using the Gelfand Transformation -- Part 1

So. Yesterday we defined the Gelfand Transformation. I want to talk through three basic properties and then show some applications.

Remember that

Basic Properties of the Gelfand Transformation

1. The Gelfand Transformation is a continuous algebra homomorphism, i.e.

as well as

and

for a suitable M. (I’m omitting quantors for readability)

We see quickly that the Gelfand Transformation inherits its algebra homomorphism properties from the fact that σ(A) consists already of algebra homomorphisms. I will show linearity as an example:

Very straight forward computation.

For continuity, let’s do two lemmata:

(a) Let λ in C. If | λ | > || x ||, then λ - x is invertible ( λ in A means λ times the identity).

To prove this, have a look at this von-Neumann-like calculation:

Thus,

is the inverse of λ - x.

(b) For all φ in Hom(A, C) and a in A,

Assume the contrary. Then by (a), φ(a) - a is invertible. Thus,

as φ( λ ) = λ for all λ in C. However, invertible elements can not be mapped to zero. (except when φ = 0, but even then the assertion holds.) A contradiction.

Using this, we can find an upper bound for || Γa ||. It is

thus M = 1.

2. An element a in A is invertible iff Γa in C(σ(A)) is invertible. Here, the vector space C(X) (where X is compact) becomes a Banach algebra by point-wise multiplication.

This proof is actually quite algebraic. :)

An element a in A is non-invertible iff it lies in a maximal ideal. Then there exists a φ in Hom(A, C)\{0} such that

and thus Γa is non-invertible in C(σ(A)).

3. I will now introduce the notion of the spectrum of an element of A.

You may be familiar with the spectrum of a matrix being the set of eigenvalues: Let M be an n x n -matrix over C with eigenvalue λ . Then there exists v in C^n such that

which is equivalent to M being non-invertible,

This property is elevated to the general definition of the Spectrum of an Element a in A.

If A is the algebra of n x n -matrices, these definitions coincide. We immediatly recognize that both, the spectrum of an element and that of an algebra relate closely to non-invertibility.

First, we claim

which follows immediately from (2) as

Also, the spectrum of a in A computes as follows.

This is again because if  a - λ is non-invertible, it is contained in a maximal ideal and thus there exists a φ in Hom(A, C)\{0} such that

and thus φ(a) = λ and vice versa.

Okay. I will postpone further applications of the Gelfand Transformation. I have a strict non-working policy for weekends, so the next post on this topic will probably be on Monday.

I think this was a wonderfully-algebraic chapter except for maybe one estimate and one Neumann-series.

As always, feel free to correct me or ask questions!