There is a conjecture that I’ve been thinking about for a while, which I do not have the right knowledge at the moment to fully formalize but seems to be on to something.
The conjecture is that an appropriate foundation for C*-algebras would permit infinitesimals, and infinitesimals in C*-algebras would lead to a form of synthetic stochastic-differential geometry.
I give some motivation for why this conjecture might hold in https://www.youtube.com/watch?v=5bBPz4GpBIw.
Here I want to give an update on this, and see if anyone can give me pointers on how to go further.
Specifically, consider the \mathbb{R}-algebra \mathbb{R}[\varepsilon^p]_{0 \leq p \leq 1}, finitely presented by generators ε^p for 0 < p \leq 1 with the equations
I want this to play a similar central role as the ring \mathbb{R}[x]/(x^2) plays in synthetic differential geometry. This ring has a \ast-algebra structure given by the identity. Let us consider the positive elements for this \ast-algebra structure. Recall that the positive elements are those of the form a^\ast a, so in particular for \mathbb{R}[ε^p]_{0 < p \leq 1} with the identity \ast-algebra structure, these are just the elements of the form a^2.
First of all, ε^p is positive for all 0 \leq p \leq 1. As positive elements are closed under addition, this characterizes many positive elements, but are these all the positive elements?
Surprisingly (or possibly not surprisingly), this is not the case! For instance, 1 - 2ε = (1 - ε)(1 - ε) is also positive! Following this, we state the following.
Proposition 1. An element x_1ε^{p_1} + \cdots + x_nε^{p_n} with p_1 < \cdots < p_n and x_i \neq 0 \in \mathbb{R} (where we identify ε^0 with 1) is positive if and only if x_1 is positive.
To prove this, we must first prove the following lemma.
Lemma 2. Any element of \mathbb{R}[ε^p]_{0<p\leq 1} with non-zero constant term is a unit.
Proof. Without loss of generality, it suffices to prove this for 1 + d where d is purely infinitesimal. Then
which converges because d is nilpotent. We are done.
Proof sketch of proposition 1. We prove this by induction on n. For n=1, this is clearly true. Now, assume that this is true for i=1,\ldots,n and consider x_1ε^{p_1} + \cdots + x_{n+1}ε^{p_{n+1}}. First assume that x_1 is positive. Then there exists some a such that a^2 = x_1 ε^{p_1} + \cdots + x_n ε^{p_n} . We consider two cases. First, assume that p_{n+1} > 1/2. Notice that the “largest” term of a must be \sqrt{x_1}ε^{p_1/2}, and thus a/ε^{p_1/2} exists and is a unit by Lemma 2, so ε^{p_1/2}/a exists as well. Consequently, x_{n+1} ε^{p_{n+1}}/a exists, as p_{n+1} > p_1/2, and we have
If p_{n+1} \leq 1/2, then
and we may perform the same construction again with a := a + \frac{x_{n+1}ε^{p_{n+1}}}{2a}? (haven’t finished this yet)
On the other hand, if x_1 is negative then clearly x_1ε^{p_1} + \cdots + x_nε^{p_n} cannot be positive, because it is easy to show that the leading term of a^2 will always be positive. We are done.
The natural consequence of Proposition 1 is that the ordering on \mathbb{R}[ε^p]_{0 < p \leq 1} induced by positivity is the lexicographic order, which is intuitive from an “order of magnitude” standpoint; if we interpret these as actual polynomials, then this is the order that we get “eventually” as ε → 0.
Now, this is all quite tantalizing to me because the definition of a liquid vector space involves L^p spaces for 0 < p < 1. Is there a connection here? I have no idea. Is there a “good” liquid vector space structure on \mathbb{R}[ε^p]_{0<p\leq 1}? Should I instead be thinking about \mathbb{R}[ε^p]_{0<p<1}? Does this generalize to other sorts of Weil-algebras? If anyone can forward these questions along to someone who knows condensed mathematics I would be greatly appreciative.