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## Operations and Lawvere theories

Groups are in particular sets equipped with two operations: a binary operation (the group operation) $(x_1, x_2) \mapsto x_1 x_2$ and a unary operation (inverse) $x_1 \mapsto x_1^{-1}$. Using these two operations, we can build up many other operations, such as the ternary operation $(x_1, x_2, x_3) \mapsto x_1^2 x_2^{-1} x_3 x_1$, and the axioms governing groups become rules for deciding when two expressions describe the same operation (see, for example, this previous post).

When we think of groups as objects of the category $\text{Grp}$, where do these operations go? They’re certainly not morphisms in the corresponding categories: instead, the morphisms are supposed to preserve these operations. But can we recover the operations themselves?

It turns out that the answer is yes. The rest of this post will describe a general categorical definition of $n$-ary operation and meander through some interesting examples. After discussing the general notion of a Lawvere theory, we will then prove a reconstruction theorem and then make a few additional comments.

## Banach algebras, the Gelfand representation, and the commutative Gelfand-Naimark theorem

Banach algebras abstract the properties of closed algebras of operators on Banach spaces. Many basic properties of such operators have elegant proofs in the framework of Banach algebras, and Banach algebras also naturally appear in areas of mathematics like harmonic analysis, where one writes down Banach algebras generalizing the group algebra to study topological groups.

Today we will develop some of the basic theory of Banach algebras, our goal being to discuss the Gelfand representation of a commutative Banach algebra and the fact that, for commutative C*-algebras, this representation is an isometric isomorphism. This implies in particular a spectral theorem for self-adjoint operators on a Hilbert space.

This material can be found in many sources; I am working from Dales, Aiena, Eschmeier, Laursen and Willis’ Introduction to Banach Algebras, Operators, and Harmonic Analysis.

Below all vector spaces are over $\mathbb{C}$, all algebras are unital, and all algebra homomorphisms preserve units unless otherwise stated. In the context of Banach algebras, the last two assumptions are not standard, but in practice non-unital Banach algebras are studied by adjoining units first, so we do not lose much generality.

## Hilbert spaces (and dagger categories)

Hilbert spaces are a particularly nice class of Banach spaces. They axiomatize ideas from Euclidean geometry such as orthogonality, projection, and the Pythagorean theorem, but the ideas apply to many infinite-dimensional spaces of functions of interest to various branches of mathematics. Hilbert spaces are also fundamental to quantum mechanics, as vectors in Hilbert spaces (up to phase) describe (pure) states of quantum systems.

Today we’ll develop and discuss some of the basic theory of Hilbert spaces. As with the theory of Banach spaces, there are (at least) two types of morphisms we might want to talk about (unitary operators and bounded operators), and we will discuss an elegant formalism that allows us to talk about both. Things written by John Baez will be cited excessively.

## Banach spaces (and Lawvere metrics, and closed categories)

One annoying feature of the abstract theory of vector spaces, and one that often trips up beginners, is that it is not possible to make sense of an infinite sum of vectors in general. If we want to make sense of infinite sums, we should probably define them as limits of finite sums, so rather than work with bare vector spaces we need to work with topological vector spaces over a topological field, usually $\mathbb{R}$ or $\mathbb{C}$ (but sometimes fields like $\mathbb{Q}_p$ are also considered, e.g. in number theory). Common and important examples include spaces of continuous or differentiable functions.

Today we’ll discuss a class of topological vector spaces which is convenient to work with but which still covers many examples of interest, namely Banach spaces. The material in the first half of this post is completely standard and can be found in any text on functional analysis.

In the second half of the post we discuss a category of Banach spaces such that two Banach spaces are isomorphic in this category if and only if they are isometrically isomorphic but which still allows us to talk about bounded linear operators between Banach spaces, and to do this we briefly discuss Lawvere metrics; this material can be found on the nLab.

## ab, ba, and the spectrum

Let $a, b$ be two $n \times n$ matrices. If $a, b$ don’t commute, then $ab \neq ba$; however, the two share several properties. If either $a$ or $b$ is invertible, then $ab$ is conjugate to $ba$, so in particular they have the same characteristic polynomial.

What if neither $a$ nor $b$ are invertible? As it turns out, $ab$ and $ba$ still have the same characteristic polynomial, although they are not conjugate in general (e.g. we might have $ab = 0$ but $ba$ nonzero). There are several ways of proving this result, which implies in particular that $ab$ and $ba$ have the same eigenvalues.

What if $a, b$ are linear transformations on an infinite-dimensional vector space? Do $ab$ and $ba$ still have the same eigenvalues in an appropriate sense? As it turns out, the answer is yes, and the key lemma in the proof is an interesting piece of “noncommutative high school algebra.”

## Estimating roots

In lieu of a real blog post, which will have to wait for at least another two weeks, let me offer an estimation exercise: bound, as best you can, the unique positive real root of the polynomial

$\displaystyle x^{10000} + x^{100} - 1$.

The intermediate value theorem shows that $x \in (0, 1)$, which was the subject of a recent math.SE question that provided the inspiration for this question. I provide a stronger lower bound on $x$ using elementary inequalities and entirely by hand in an answer to the linked question, although I don’t try to improve the upper bound.

## Fractional linear transformations and elliptic curves

The following two lemmas might be encountered in a basic course in complex analysis (the first in a basic course in group theory, even).

Lemma 1: Fix a field $F$. The group of fractional linear transformations $PGL_2(F)$ acts triple transitively on $\mathbb{P}^1(F)$ and the stabilizer of any triplet of distinct points is trivial.

Lemma 2: The group of fractional linear transformations on $\mathbb{P}^1(\mathbb{C})$ preserving the upper half plane $\mathbb{H} = \{ z \in \mathbb{C} | \text{Im}(z) > 0 \}$ is $PSL_2(\mathbb{R})$.

I used to only know extremely boring computational proofs of both of these statements. However, I now know better! Today I’d like to give shorter and conceptual proofs of both of these, and then briefly discuss how they come about in the study of elliptic curves (a subject I’d like to talk about in more detail once this semester is over).

## Optimizing parameters

I came across a fun problem recently that gave me a good opportunity to exercise my approximation muscles.

Problem: Compute $\displaystyle \lim_{n \to \infty} \frac{n + \sqrt{n} + \sqrt[3]{n} + ... + \sqrt[n]{n}}{n}$, if it exists.

The basic approach to such sums is that the first few terms contribute to the sum because they are large and the rest of the terms contribute to the sum because there are a lot of them, so it makes sense to approximate the two parts of the sum separately. This is an important idea, for example, in certain estimates in functional analysis.

Since $\sqrt[k]{n} \ge 1, k \ge 2$ it follows that the limit, if it exists, is at least $\lim_{n \to \infty} \frac{2n-1}{n} = 2$. In fact, this is the precise value of the limit. We’ll show this by giving progressively sharper estimates of the quantity

$\displaystyle E_n = \frac{1}{n} \sum_{k=2}^{n} \left( \sqrt[k]{n} - 1 \right)$.

In the discussion that follows I’m going to ignore a lot of error terms to simplify the computations.

## Functoriality

I wanted to talk about the geometric interpretation of localization, but before I do so I should talk more generally about the relationship between ring homomorphisms on the one hand and continuous functions between spectra on the other. This relationship is of utmost importance, for example if we want to have any notion of when two varieties are isomorphic, and so it’s worth describing carefully.

The geometric picture is perhaps clearest in the case where $X$ is a compact Hausdorff space and $C(X) = \text{Hom}_{\text{Top}}(X, \mathbb{R})$ is its ring of functions. From this definition it follows that $C$ is a contravariant functor from the category $\text{CHaus}$ of compact Hausdorff spaces to the category $\mathbb{R}\text{-Alg}$ of $\mathbb{R}$-algebras (which we are assuming have identities). Explicitly, a continuous function

$f : X \to Y$

between compact Hausdorff spaces is sent to an $\mathbb{R}$-algebra homomorphism

$C(f) : C(Y) \to C(X)$

in the obvious way: a continuous function $Y \to \mathbb{R}$ is sent to a continuous function $X \xrightarrow{f} Y \to \mathbb{R}$. The contravariance may look weird if you’re not used to it, but it’s perfectly natural in the case that $f$ is an embedding because then one may identify $C(X)$ with the restriction of $C(Y)$ to the image of $f$. This restriction takes the form of a homomorphism $C(Y) \to C(X)$ whose kernel is the set of functions which are zero on $f(X)$, so it exhibits $C(X)$ as a quotient of $C(Y)$.

Question: Does every $\mathbb{R}$-algebra homomorphism $C(Y) \to C(X)$ come from a continuous function $X \to Y$?

## Extracting the diagonal

Needless to say, I have been very, very busy. But enough about me.

Suppose you are given a bivariate generating function

$\displaystyle F(x, y) = \sum_{m, n \ge 0} f(m, n) x^m y^n$

in “closed form,” where I’ll be vague about what that means. Such a generating function may arise, for example, from counting lattice paths in $\mathbb{Z}_{\ge 0}^2$; then $f(m, n)$ might count the number of paths from $(0, 0)$ to $(m, n)$. If the path is only constrained by the fact that its steps must come from some set $S \subset \mathbb{Z}_{\ge 0}^2$ of steps containing only up or left steps, then we have the simple identity

$\displaystyle F_S(x, y) = \frac{1}{1 - \sum_{(a, b) \in S} x^a y^b}$.

Question: When can we determine the generating function $\displaystyle D_F(x) = \sum_{n \ge 0} f(n, n) x^n$ in closed form?

I’d like to discuss an analytic approach to this question that gives concrete answers in at least a few important special cases, including the generating function for the central binomial coefficients, which is our motivating example.