Let's read: $C^*$-algebra, quantum mechanics, Pascual Jordan

A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details. ---- Hermann Weyl

After 3 courses in analysis I have finally reached Banach algebras, I feel exhausted by the climb and yet there's more theory ahead until the true pinnacle: the theory of $C^*$-algebra, for quantum mechanics!

Let's read, what is being promised?

$C^*$-algebras were first considered in quantum mechanics to model algebras of physical observables... began with Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators. These papers considered a special class of $C^*$-algebras which are now known as von Neumann algebras.

So experiments in quantum mechanics → weird behavior of observables in quantum mechanics → Heisenberg matrix mechanics → Jordan purified it into Jordan algebra → von Neumann generalized to rings of operators → von Neumann algebra generalized to $C^*$-algebras.

Abstract linear algebra

We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context. ---- Hermann Weyl

In the beginning there was the $\mathbb{R}^3$, and it was nice. Then it got generalized to vector spaces, finite dimensional, then infinite dimensional. Set theory came along and by Axiom of Choice, all vector spaces are classified up to isomorphism by their base field and basis cardinality. Boring.

In the beginning there was a naked vector space without geometry, but this cannot be, because geometry is everywhere in our life. Close things easily touch and far things don't. How to deal with this? Use topology on the vector space. Then we get topological vector spaces.

Don't forget the norms, and inner products! Physicists use them all the time, can't do without them. So add them in. Adding the norm gives a normed space, and adding the inner product gives inner product space.

Things are boring in finite dimensions, but too terrifying in uncountable dimensions, what to do? Countable dimensions are the way to go. Countable-infinite sums are defined on many of those topological vector spaces already, because the topology is induced by a metric.

With topology and infinite sums completeness really becomes important. Infinite sums should not fall off the edge of the world, or fall into a hole in the fabric of spacetime, so completeness of the space has to fill in all the holes and fence off all the edges. Completing normed spaces gives Banach spaces, and completing inner product spaces gives Hilbert spaces.

Linear operators become tricky now, because to play with the topology well, they must be continuous, so we usually only consider bounded linear operators. Then we get the space of operators. Unlike the vector space underneath, where the vectors can only add and multiply with scalars, the operators themselves can also multiply with each other through functional composition. This gives them an algebra structure!
(This is what von Neumann meant when he said "rings of operators", or in modern language, "operator algebras".)

Great, now we get Banach algebras from algebraizing the behavior of operators on Banach spaces (more on "algebrization" later), and forgetting all about from what Banach space the Banach algebra originally come from. But wait, I take it back! I want to keep the Banach space. Sorry, it's gone. Whoops. Not a problem. By Gelfand representation, we can recover the underlying Banach space from the Banach algebras.

Great, so it doesn't matter too much whether we do Banach space or Banach algebra, since they are mutually derivable. And as a true algebraist, when in doubt, go for more abstraction, so we go to Banach algebras.

Given a Banach algebra, a Banach space can be recovered by Gelfand representation. If it turns out that this space in fact allows an inner product, then it's a Hilbert space, and so the original Banach algebra allows an "adjoint" operation.

Basically, consider the Banach algebra $B(X)$ of bounded operators over a Banach space $X$, then the adjoint of any $f\in B(X)$ is $f^* : X^* \to X^*$, defined by $f^*(\phi) = \phi \circ f$. Then if the space is actually Hilbert, then $X^* \cong X$, and so $f^*\in B(X^*)\cong B(X)$, and so the star operator becomes a function too $B(X) \to B(X)$, and it satisfies some algebraic properties. One algebrization later, we get the Banach-* algebra!

Still there? Just one more step to go. To reach the $C^*$-algebra, what's needed is just one more axiom:
$$\forall f\in B(X), \|f^{*}f\|=\|f\|\|f^{*}\|$$
and that's it, any Banach-* algebra that satisfies this additional theorem is a $C^*$-algebra.

But why?? It turns out that such $C^*$-algebras are the "right" way to algebrize how operators behave in quantum mechanics. That's why we spent so much time climbing this mountain of abstraction.

But really, there should not be such a terribly harsh climb. I think that what we are doing now, this harsh climb to a rigorous form of quantum mechanics, only is a symptom of our own human bad understanding. In the right understanding, it'd be a lot easier. Think about how difficult it was to do physics with only Euclidean geometry in Newton's time! I tried reading his book and it was terrifying.

Algebrize everything

In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics. ---- Hermann Weyl

This is a parable of algebraists:

Some algebraists on the planet of Mars launched an expedition to Earth, and over the ranges they see a group of physicists dancing in a circle while wearing funny hats, exchanging their positions in a weird pattern. They video recorded everything for several months and went home to analyze. The algebraists noticed that there are only 4 kinds of hats, and the exchange pattern can be described by the addition properties of mod-p numbers. Then they published their finding The Algebra of the Dance of Physicists, and anyone following the recipe can reproduce the dance accurately.

They don't trouble themselves as figuring out why the physicists dance. The physicists did dance with a meaning underneath it, it is how they find out how to cut their pie fairly. It doesn't matter to the algebraists, and it won't matter anyway since there are no pies on Mars.

But the physicists didn't gain nothing either. They read the paper and saw that the algebraists found a shortcut to calculate the results of a dance, without doing the dance, and they can now eat their pie without it getting cold.

Later, the algebraists were passing physicist territory again and the physicists invited them to eat some pie but the algebraists didn't care, they just wanted to see more dancing. And not the pie-dance, since it's already studied and published. Not the muon-dance either, that's not new! It was already found as a generalization from the projective limit of the pie-dance-algebra.

Interpretation:

Algebras usually come from the patterns of how mathematical symbols move and change during a calculation of some other, more "meaningful" problem. Instead of solving the problem the old way every time, algebraization divides the problem-solving into three parts:

1. Turn the problem into a symbolic form.
2. Do symbolic manipulation on the symbols, using as many algebraic shortcuts as you can get.
3. Turn the symbolic form back to the solution.

Algebraists specialize in part two, taking an algebra and try to find what it does, what its ideals are, whether it is amenable, find out as many properties and tricks as possible.

They do part one when they want more algebras to play with. They usually do this by launching an expedition into analysis, geometry, or physics. Part three is usually easy as it's just part one reversed.

Algebraists enjoy part two as its own reward, and others who gets their field algebraized might benefit from the shortcuts and structures discovered by algebra, but algebraists might not care that much.

Sometimes algebraists generalize algebraic structures they discover in the wild, and it turned out the generalized structures are also in the wild as related to the original structure's wild form. It's a triumph of the unreasonable effectiveness of mathematics.

Applications of $C^*$-algebra

All the really hard work, starting from quantum mechanics and abstracting further and further, it'd better be worth it... And indeed it was, kinda. I actually don't know much about how $C^*$-algebras are like. I never lived with them, only barely know their definitions. I'm still uninitiated in them. But I can always do a literature search instead!

$C^*$-algebras in numerical analysis, Albrecht Böttcher:

... $C^*$-algebra techniques can... solve some concrete problems of Numerical Analysis. I focus my attention on several questions concerning the asymptotic behavior of large Toeplitz matrices. This limitation ignores the potential and the triumphs of $C^*$-algebra methods in connection with large classes of other operators and plenty of different approximation methods, but it allows me to demonstrate the essence of the $C^*$- algebra approach and to illustrate it with nevertheless nontrivial examples.

The idea of applying $C^*$-algebras to problems of numerical analysis emerged in the early 1980’s... Meanwhile the application of $C^*$-algebras to numerical analysis has grown to a big business. 

what are Toeplitz matrices though??

Typical problems modelled by Toeplitz matrices include the numerical solution of certain differential and integral equations, the computation of splines, time series analysis, signal and image processing, Markov chains, and queuing theory.

Oh okay. I see how embarrassingly practical they are.

But really, they came from quantum mechanics, they had better come back to it! And indeed they do. This is so out of my expertise I'll just put some links here and maybe come back at another time.

• mathoverflow: Quantum mechanics formalism and $C^*$-algebras
• Local quantum field theory: 

... an application to local quantum physics of $C^*$-algebra theory. Because of this it is also known as algebraic quantum field theory (AQFT)...

• Algebraic Quantum Field Theory (2006), Hans Halvorson, Michael Mueger:

... only a small proportion of those who work on QFT work on algebraic QFT (AQFT). However, there are particular reasons why philosophers, and others interested in fundational issues, will want to study the “algebraic” approach...

... studying the foundations of a theory requires that the theory has a mathematical description. (... satisfied in the case of statistical mechanics, special and general relativity, and nonrelativistic quantum mechanics.) ... having such a description greatly facilitates our ability to draw inferences securely and efficiently. 

So, philosophers of physics have taken their object of study to be theories, where theories correspond to mathematical objects... It is for this reason that AQFT is of particular interest for the foundations of quantum field theory. In short, AQFT is our best story about where QFT lives in the mathematical universe, and so is a natural starting point for foundational inquiries. 

• Introduction to Algebraic Quantum Field Theory (1990), Khoruzhii, Sergei Sergeyevich:

The algebraic approach to quantum field theory is not just a special mathematical formalism; it is a particular school of thought in physics... aims at describing relativistic quantum systems by means of the theory of algebras and in terms of observables and states treated as fundamental physical objects... Initially, it was stated as an axiomatic formalism. Later on it ceased to be purely axiomatic... However, since the results of the preceding stage were mainly general and rigorous theorems, they are incorporated as a firm basis in new investigations.

One can reconstruct the totality of experimental data on a quantum system, if the total set of the system states and the set of all its observables are given, as well as probability distributions of values allowed for each observable in each state. In this respect, the formalism based upon observables and states provides a complete description of the physical system.

Pascual Jordan

[Material for this section taken from Pascual Jordan, Glory and Demise and his legacy in contemporary local quantum physics (2003), Bert Schroer.]

Finally I'll mention Pascual Jordan, a great mathematician and discoverer of Quantum Field Theory. Jordan, as mentioned, in 1932 gave a more mathematical foundation to quantum mechanics by algebraizing the behavior of operators in Heisenberg's matrix mechanics (among others). 

But that's just the first step. Operators abstracted from matrices is just the first step on the way to a full formal theory of reality. Quantum mechanics of particles is just the first-quantization. Particles are still particles in a Euclidean spacetime, they can't be born or be killed. They must be, because real particles are born and killed all the time.

This calls for the second-quantization, where particles are turned into excitations in a quantum field. But this is still not enough, because a quantum field still lives in the familiar, classical picture of space-time. The algebraist's work is not complete if the algebra has any shred of class or geometry left.

In a 1929 conference in Kharkov, Jordan gave a speech:

Man wird wohl in Zukunft den Aufbau in zwei getrennten Schritten ganz vermeiden muessen, und in einem Zuge, ohne klassisch-korrespondenzmaessige Kruecken, eine reine Quantentheorie der Elektrizitaet zu formulieren versuchen. Aber das ist Zukunftsmusik.

In the future one perhaps will have to avoid the construction in two separated steps and rather have to approach the problem of formulating, without the crutches of classical correspondences, a pure quantum theory of electricity [a pure quantum field theory] in one fell swoop. But this is music of the future.

And so Jordan went on and developed Quantum Field Theory.

Unfortunately, he did not get a Nobel prize despite being nominated a few times. One possible reason was his support of Nazi regime.