A noncommutative moduli space

Supernatural numbers also appear in noncommutative geometry via James Glimm’s characterisation of a class of simple $C^*$-algebras, the UHF-algebras.

A uniformly hyperfine (or, UHF) algebra $A$ is a $C^*$-algebra that can be written as the closure, in the norm topology, of an increasing union of finite-dimensional full matrix algebras

$M_{c_1}(\mathbb{C}) \subset M_{c_2}(\mathbb{C}) \subset … \quad \subset A$

Such embedding are only possible if the matrix-sizes divide each other, that is $c_1 | c_2 | c_3 | … $, and we can assign to $A$ the supernatural number $s=\prod_i c_i$ and denote $A=A(s)$.

In his paper On a certain class of operator algebras, Glimm proved that two UHF-algebras $A(s)$ and $B(t)$ are isomorphic as $C^*$-algebras if and only if $s=t$. That is, the supernatural numbers $\mathbb{S}$ are precisely the isomorphism classes of UHF-algebras.

An important invariant, the Grothendieck group $K_0$ of $A(s)$, can be described as the additive subgroup $\mathbb{Q}(s)$ of $\mathbb{Q}$ generated by all fractions of the form $\frac{1}{n}$ where $n$ is a positive integer dividing $s$.

A “noncommutative space” is a Morita class of $C^*$-algebras, so we want to know when two $UHF$-algebras $A(s)$ and $B(t)$ are Morita-equivalent. This turns out to be the case when there are positive integers $n$ and $m$ such that $n.s = m.t$, or equivalently when the $K_0$’s $\mathbb{Q}(s)$ and $\mathbb{Q}(t)$ are isomorphic as additive subgroups of $\mathbb{Q}$.

Thus Morita-equivalence defines an equivalence relation on $\mathbb{S}$ as follows: if $s=\prod p^{s_p}$ and $t= \prod p^{t_p}$ then $s \sim t$ if and only if the following two properties are satisfied:

(1): $s_p = \infty$ iff $t_p= \infty$, and

(2): $s_p=t_p$ for all but finitely many primes $p$.

That is, we can view the equivalence classes $\mathbb{S}/\sim$ as the moduli space of noncommutative spaces associated to UHF-algebras!

Now, the equivalence relation is described in terms of isomorphism classes of additive subgroups of the rationals, which was precisely the characterisation of isomorphism classes of points in the arithmetic site, that is, the finite adèle classes

$\mathbb{S}/\sim~\simeq~\mathbb{Q}^* \backslash \mathbb{A}^f_{\mathbb{Q}} / \widehat{\mathbb{Z}}^*$

and as the induced topology of $\mathbb{A}^f_{\mathbb{Q}}$ on it is trivial, this “space” is usually thought of as a noncommutative space.

That is, $\mathbb{S}/\sim$ is a noncommutative moduli space of noncommutative spaces defined by UHF-algebras.

The finite integers form one equivalence class, corresponding to the fact that the finite dimensional UHF-algebras $M_n(\mathbb{C})$ are all Morita-equivalent to $\mathbb{C}$, or a bit more pompous, that the Brauer group $Br(\mathbb{C})$ is trivial.

Multiplication of supernaturals induces a well defined multiplication on equivalence classes, and, with that multiplication we can view $\mathbb{S}/\sim$ as the ‘Brauer-monoid’ $Br_{\infty}(\mathbb{C})$ of simple UHF-algebras…

(Btw. the photo of James Glimm above was taken by George Bergman in 1972)

3 related new math-sites

F_un Mathematics

Hardly a ‘new’ blog, but one that is getting a new life! On its old homepage you’ll find a diagonal banner stating ‘This site has moved’ and clicking on it will guide you to its new location : cage.ugent.be/~kthas/Fun.

From now on, this site will be hosted at the University of Ghent and maintained by Koen Thas. So, please update your bookmarks and point your RSS-aggregator to the new feed.

Everyone interested in contributing to this blog dedicated to the mathematics of the field with one element should contact Koen by email.


Though I may occasionally (cross)post at F_un mathematics, my own blog-life will center round a new blog to accompany the master-course ‘seminar noncommutative geometry’ I’m running at Antwerp University this semester. Its URL is noncommutative.org and it is called :

Here, angs is short for Antwerp Noncommutative Geometry Seminar and the additions @t resp. + are there to indicate we will experiment a bit trying to find useful interactions between the IRL seminar, its blog and social media such as twitter and Google+.

The seminar (and blog) are scheduled to start in earnest september 30th, but I may post some prep-notes already. This semester the seminar will try to decode Smirnov’s old idea to prove the ABC-conjecture in number theory via geometry over the field with one element and connect it with new ideas such as Borger’s $\mathbb{F}_1$-geometry using $\lambda$-rings and noncommutative ideas proposed by Connes, Consani and Marcolli.

Again, anyone willing to contribute actively is invited to send me an email or to comment on ‘angst’, tweet about it using the hashtag #angs (all such tweets will appear on the frontpage) or share its posts on Google+.

Noncommutative Arithmetic Geometry Media Library

Via the noncommutative geometry blog a new initiative maintained by Alain Connes and Katia Consani was announced : the Noncommutative Arithmetic Geometry Media Library.

This site is dedicated to maintain articles, videos, and news about meetings and activities related to noncommutative arithmetic geometry. The website is still `under construction’ and the plan is to gradually add more videos (also from past conferences and meetings), as well as papers and slides.

eBook ‘geometry and the absolute point’ v0.1

In preparing for next year’s ‘seminar noncommutative geometry’ I’ve converted about 30 posts to LaTeX, centering loosely around the topics students have asked me to cover : noncommutative geometry, the absolute point (aka the field with one element), and their relation to the Riemann hypothesis.

The idea being to edit these posts thoroughly, add much more detail (and proofs) and also add some extra sections on Borger’s work and Witt rings (and possibly other stuff).

For those of you who prefer to (re)read these posts on paper or on a tablet rather than perusing this blog, you can now download the very first version (minimally edited) of the eBook ‘geometry and the absolute point’. All comments and suggestions are, of course, very welcome. I hope to post a more definite version by mid-september.

I’ve used the thesis-documentclass to keep the same look-and-feel of my other course-notes, but I would appreciate advice about turning LaTeX-files into ‘proper’ eBooks. I am aware of the fact that the memoir-class has an ebook option, and that one can use the geometry-package to control paper-sizes and margins.

Soon, I will be releasing a LaTeX-ed ‘eBook’ containing the Bourbaki-related posts. Later I might also try it on the games- and groups-related posts…

Art and the absolute point (3)

Previously, we have recalled comparisons between approaches to define a geometry over the absolute point and art-historical movements, first those due to Yuri I. Manin, subsequently some extra ones due to Javier Lopez Pena and Oliver Lorscheid.

In these comparisons, the art trend appears to have been chosen more to illustrate a key feature of the approach or an appreciation of its importance, rather than giving a visual illustration of the varieties over $\mathbb{F}_1$ the approach proposes.

Some time ago, we’ve had a couple of posts trying to depict noncommutative varieties, first the illustrations used by Shahn Majid and Matilde Marcolli, and next my own mental picture of it.

In this post, we’ll try to do something similar for affine varieties over the absolute point. To simplify things drastically, I’ll divide the islands in the Lopez Pena-Lorscheid map of $\mathbb{F}_1$ land in two subsets : the former approaches (all but the $\Lambda$-schemes) and the current approach (the $\Lambda$-scheme approach due to James Borger).

The former approaches : Francis Bacon “The Pope” (1953)

The general consensus here was that in going from $\mathbb{Z}$ to $\mathbb{F}_1$ one looses the additive structure and retains only the multiplicative one. Hence, ‘commutative algebras’ over $\mathbb{F}_1$ are (commutative) monoids, and mimicking Grothendieck’s functor of points approach to algebraic geometry, a scheme over $\mathbb{F}_1$ would then correspond to a functor

$h_Z~:~\mathbf{monoids} \longrightarrow \mathbf{sets}$

Such functors are described largely by combinatorial data (see for example the recent blueprint-paper by Oliver Lorscheid), and, if the story would stop here, any Rothko painting could be used as illustration.

Most of the former approaches add something though (buzzwords include ‘Arakelov’, ‘completion at $\infty$’, ‘real place’ etc.) in order to connect the virtual geometric object over $\mathbb{F}_1$ with existing real, complex or integral schemes. For example, one can make the virtual object visible via an evaluation map $h_Z \rightarrow h_X$ which is a natural transformation, where $X$ is a complex variety with its usual functor of points $h_X$ and to connect both we associate to a monoid $M$ its complex monoid-algebra $\mathbb{C} M$. An integral scheme $Y$ can then be said to be ‘defined over $\mathbb{F}_1$’, if $h_Z$ becomes a subfunctor of its usual functor of points $h_Y$ (again, assigning to a monoid its integral monoid algebra $\mathbb{Z} M$) and $Y$ is the ‘best’ integral scheme approximation of the complex evaluation map.

To illustrate this, consider the painting Study after Velázquez’s Portrait of Pope Innocent X by Francis Bacon (right-hand painting above) which is a distorded version of the left-hand painting Portrait of Innocent X by Diego Velázquez.

Here, Velázquez’ painting plays the role of the complex variety which makes the combinatorial gadget $h_Z$ visible, and, Bacon’s painting depicts the integral scheme, build up from this combinatorial data, which approximates the evaluation map best.

All of the former approaches more or less give the same very small list of integral schemes defined over $\mathbb{F}_1$, none of them motivically interesting.

The current approach : Jackson Pollock “No. 8” (1949)

An entirely different approach was proposed by James Borger in $\Lambda$-rings and the field with one element. He proposes another definition for commutative $\mathbb{F}_1$-algebras, namely $\lambda$-rings (in the sense of Grothendieck’s Riemann-Roch) and he argues that the $\lambda$-ring structure (which amounts in the sensible cases to a family of endomorphisms of the integral ring lifting the Frobenius morphisms) can be viewed as descent data from $\mathbb{Z}$ to $\mathbb{F}_1$.

The list of integral schemes of finite type with a $\lambda$-structure coincides roughly with the list of integral schemes defined over $\mathbb{F}_1$ in the other approaches, but Borger’s theory really shines in that it proposes long sought for mystery-objects such as $\mathbf{spec}(\mathbb{Z}) \times_{\mathbf{spec}(\mathbb{F}_1)} \mathbf{spec}(\mathbb{Z})$. If one accepts Borger’s premise, then this object should be the geometric object corresponding to the Witt-ring $W(\mathbb{Z})$. Recall that the role of Witt-rings in $\mathbb{F}_1$-geometry was anticipated by Manin in Cyclotomy and analytic geometry over $\mathbb{F}_1$.

But, Witt-rings and their associated Witt-spaces are huge objects, so one needs to extend arithmetic geometry drastically to include such ‘integral schemes of infinite type’. Borger has made a couple of steps in this direction in The basic geometry of Witt vectors, II: Spaces.

To depict these new infinite dimensional geometric objects I’ve chosen for Jackson Pollock‘s painting No. 8. It is no coincidence that Pollock-paintings also appeared in the depiction of noncommutative spaces. In fact, Matilde Marcolli has made the connection between $\lambda$-rings and noncommutative geometry in Cyclotomy and endomotives by showing that the Bost-Connes endomotives are universal for $\lambda$-rings.