The n-Category Café

First here’s some notation.

• For any finite set $I$, we have the real vector space $\mathbf{R}^I$.
• For any injection $f: I \to J$ between finite sets, there is the induced linear map $f_*: \mathbf{R}^I \to \mathbf{R}^J$, got by reindexing as dictated by $f$ and padding out with $0$s.
• For any $x \in \mathbf{R}^I$ and $y \in \mathbf{R}^J$, there is an element $x \otimes y \in \mathbf{R}^{I \times J}$, whose $(i, j)$-coordinate is $x_i y_j$. (I call it $x \otimes y$ because if you identify $\mathbf{R}^{I \times J}$ with $\mathbf{R}^I \otimes \mathbf{R}^J$, that’s what it is.)

A norm system is a sensible way of assigning a norm to each vector space $\mathbf{R}^I$, where $I$ ranges over finite sets. In other words, it consists of a specified norm $\Vert\cdot\Vert$ on $\mathbf{R}^I$ for each finite set $I$, such that whenever $f: I \to J$ is an injection of finite sets and $x \in \mathbf{R}^I$, then $\Vert f_*(x) \Vert = \Vert x \Vert$. All that says is that if you pad out $x$ with some zeros, and switch the order of the coordinates round, it doesn’t change the norm.

Example:  For each $p \in [1, \infty]$ there’s a norm system $\Vert\cdot\Vert_p$ given by the usual formulas: if $p \lt \infty$ then $$\Vert x \Vert_p = (\sum_{i \in I} |x_i|^p )^{1/p}$$ (for finite sets $I$ and $x \in \mathbf{R}^I$), and $\Vert x \Vert_\infty = \max_{i \in I} |x_i|$.

A norm system is multiplicative if $\Vert x \otimes y \Vert = \Vert x \Vert \Vert y \Vert$ for all finite sets $I$ and $J$, $x \in \mathbf{R}^I$, and $y \in \mathbf{R}^J$.

Example:  For each $p \in [1, \infty]$, the norm system $\Vert\cdot\Vert_p$ is multiplicative.

Theorem (Aubrun and Nechita) The only multiplicative norm systems are $\Vert\cdot\Vert_p$ ($p \in [1, \infty]$).

I find this amazing, both in itself and because it wasn’t known half a century ago. In fact it’s only the first of two theorems in their paper; the second concerns the $L_p$ norms (as opposed to what we’ve just been discussing, the $\ell_p$ norms). Anyway, I’ll stick to the first one.

I’ll say a bit more later about why I find this amazing, but first I should point out that my phrasing of Aubrun’s and Nechita’s theorem is a bit different from theirs.

How Aubrun and Nechita put it  Someone reading my phrasing might think: That’s a bit extravagant. A norm on $\mathbf{R}^I$ for every finite set $I$? Well, the axioms imply that if $I \cong J$ then the norm on $\mathbf{R}^I$ determines the norm on $\mathbf{R}^J$, so we might as well just consider one finite set of each cardinality. That is, instead of taking this huge system of norms, we take just one permutation-invariant norm on each of the spaces $\mathbf{R}^n$ ($n \in \mathbf{N}$), such that $\Vert (x_1, \ldots, x_n, 0) \Vert = \Vert (x_1, \ldots, x_n) \Vert$.

That’s fine, but there’s a small price to pay: in order to state the multiplicativity axiom, you have to choose a bijection between $\{1, \ldots, n\} \times \{1, \ldots, m\}$ and $\{1, \ldots, n m\}$ for each $n$ and $m$.

But you still might find that extravagant. The axioms imply that the norm on $\mathbf{R}^{n + 1}$ determines the norm on $\mathbf{R}^n$, so you might as well just work with $c_{00}$, the space of real sequences $(x_n)_{n = 1}^\infty$ that are $0$ in all but finitely many places. So instead of having a whole family of norms, you have just one norm; it’s a permutation-invariant norm on $c_{00}$.

That’s still fine, but there’s again a price to pay: in order to state the multiplicativity axiom, you have to choose a bijection between $\mathbf{Z}^+ \times \mathbf{Z}^+$ and $\mathbf{Z}^+$ (where $\mathbf{Z}^+$ is the set of positive integers). It doesn’t matter which you choose, in the sense that if multiplicativity holds for one choice then it holds for all of them. But you do have to choose one.

That, then, is what Aubrun and Nechita do: they characterize $\Vert\cdot\Vert_p$ ($p \in [1, \infty]$) as the only norms on $c_{00}$ that are permutation-invariant and multiplicative. They don’t mention what I’ve called ‘norm systems’.

Something impressive  Here’s a consequence of Aubrun and Nechita’s theorem that is, I think, quite non-obvious. At least, I completely failed to prove it without the aid of their theorem; maybe you can do better.

The $p$-norms have a special property — which, in what follows, I will call the special property. Given $$x^1 \in \mathbf{R}^{n_1}, \ldots, x^k \in \mathbf{R}^{n_k},$$ write $$x^1; \ldots; x^k \in \mathbf{R}^{n_1 + \cdots + n_k}$$ for their concatenation. The special property is that $$\Vert x^1; \ldots; x^k \Vert_p$$ is determined by $\Vert x^1 \Vert_p, \ldots, \Vert x^k \Vert_p$ alone.

How? By the formula $$\Vert x^1; \ldots; x^k \Vert_p = \Vert   (\Vert x^1 \Vert_p, \ldots, \Vert x^k \Vert_p)   \Vert_p.$$ One perspective on the formula is this: it says that there’s a monoid structure on $[0, \infty)$ whose $k$-fold multiplication is $$(y_1, \ldots, y_k) \mapsto \Vert (y_1, \ldots, y_k) \Vert_p$$ ($y_i \in [0, \infty)$). This is kind of obvious when $p \lt \infty$, since the operation $y \mapsto \Vert y\Vert_p$ is what you get when you transport addition — itself a monoid structure on $[0, \infty)$ — across the bijection $( )^p: [0, \infty) \to [0, \infty)$.

The Aubrun–Nechita theorem implies that any multiplicative norm system has this special property (because it must be one of the $p$-norms). But can you prove this directly, without the aid of their theorem? I couldn’t. I made a bit of progress, but the progress I made more or less reproduced the beginning of their proof of the theorem itself. (It didn’t reproduce the end, which is an application of Cramér’s large deviation theorem.)

Actually, I think that if you can prove that any multiplicative norm system has the special property, then you can probably build an alternative proof of the Aubrun–Nechita theorem, as follows.

There’s a fairly ancient body of work on ‘generalized means’, for which the classic text is Hardy, Littlewood and Pólya’s 1934 book Inequalities. A generalized mean of numbers $x_1, \ldots, x_n$ is something like the arithmetic mean, $\sum (1/n) x_i$, or the harmonic mean, $(\sum (1/n) x_i^{-1})^{-1}$, or more generally $(\sum (1/n) x_i^p)^{1/p}$ for any real $p$. The limiting case $p \to 0$ is the geometric mean, $\prod x_i^{1/n}$. You can also change the uniform weighting $1/n, \ldots, 1/n$ to some non-uniform weighting.

Part of that ancient body of work is a collection of theorems characterizing generalized means. Obviously generalized means and $p$-norms are closely related, and when I first saw Aubrun and Nechita’s paper I thought I could deduce their result from these classical theorems. But the step in the deduction that I couldn’t fill in was what I just mentioned: showing directly that a multiplicative norm system has the special property.

Incidentally, I first saw news of this paper at John’s blog. We were discussing characterizations of the Rényi entropies, which are closely related to generalized means and $p$-norms. Mark Meckes mentioned this paper. It would be nice to use Aubrun and Nechita’s characterization of the $p$-norms to produce new characterizations of generalized means and Rényi entropies.