Connes Embeddings and von Neumann Regular Closures of Group Algebras

This is an interesting paper of Gabor Elek's which touches on some things I've posted about recently - especially (i) the Atiyah conjecture and (ii) the idea (which shows up in the work of Ara et al) that one can use some kind of "asymptotic rank" instead of "asymptotic trace" in some contexts where you want to build "continuous dimension" type invariants.

Exchange rings and translation algebras

Recall that a ring $R$ is von Neumann regular if given any $x\in R$ there is $y\in R$ such that $xyx=x$. (Examples: fields; matrix algebras; various rings of unbounded operators, where $y$ is "the inverse of $x$ away from the kernel".) A ring $R$ is called an exchange ring if, for every $x\in R$, there is an idempotent $e\in R$ such that $e\in xR$ and $(1-e)\in (1-x)R$. Von Neumann regular rings are examples of exchange rings. There is a developed algebraic theory of these rings, with links to real rank 0 C*-algebras among other things.

I just became aware of the paper Ara, P., K. C O'Meara, and F. Perera. Gromov translation algebras over discrete trees are exchange rings. Transactions of the American Mathematical Society 356, no. 5 (2004): 2067–2079 (electronic). In this paper it is shown that the algebraic translation algebra associated to a tree, with coefficients in a von Neumann regular ring, must be an exchange ring. The tree hypothesis is used in various places, but the authors don't know, apparently, whether there are examples of metric spaces $X$ for which the translation algebra is not an exchange ring. (The plane might be a good example to start with.)

C*-algebras, foliations and K-theory

In 1980, Alain Connes gave a course entitled "C*-algebras, foliations and K-theory". Jean Renault was a student in the course at that time and took notes, and photocopies of his meticulously handwritten manuscript have been passed around generations of students. I must have acquired mine some time around 1988.

The notes describe projective modules, Morita equivalence, K-theory, non-unital algebras and multipliers, quasi-isomorphisms, smooth subalgebras and "holomorphic closure", Bott periodicity, crossed products, the Thom isomorphism for crossed products, and the beginnings of noncommutative geometry. Its fascinating to see how early some of these ideas were germinating, and what they looked like at taht early stage.

In our seminar last year we assigned graduate students to read and lecture on various parts of the manuscript. This led to a rough English translation, which I'm now in the process of tidying up. I hope to post a more polished version to the arXiv before too long. I'm grateful to Alain and Jean for encouraging this project.

You can find scans (not too legible) of the lecture notes here.

Bounded groups

Today's post arises from a paper by my Penn State colleague Dima Burago and his collaborators, Conjugation-invariant norms on groups of geometric origin. (October 7, 2007). http://arxiv.org/abs/0710.1412.

The basic definition is a very simple one: say that a group $G$ is bounded if it has finite diameter with respect to any bi-invariant metric.
(Of course it is enough to consider the distance from a group element $g$ to the identity element, and this function $|g|$ then becomes a \emph{conjugation invariant norm} on the group, hence the title of the paper. A group is then bounded if every conjugation-invariant norm is bounded from above. One can also ask whether every conjugation-invariant norm is bounded from below (away from the identity element). When a norm is bounded from above and below and can be said to be trivial: it is equivalent to the norm which equals 1 on all non-identity elements. If every conjugation-invariant norm is trivial the group is called meager.

The basic phenomenon studied in the paper is the surprising fact that conjugation-invariant norms on large groups of diffeomorphisms tend to be trivial. In fact the main result is that the identity component of the diffeomorphism group of $M$ is meager whenever $M$ is a sphere or a closed connected 3-manifold. However the paper begins by proving some of the basic facts about bounded groups, and this part was already very interesting to me.

The groups $SL(n,R)$, $n\ge 2$, and $SL(n,Z)$, $n\ge 3$, are bounded. In the real case this follows from "a suitable version of the Gauss elimination process" to quote the paper... you have to show that any element of $SL(n,R)$ can be written as the product of a bounded number of conjugates of a bounded number of generators (namely, the elementary matrices with a 1 off the diagonal). Takes a moment's thought. Over the integers one needs also the bounded generation of $SL(n,Z)$ for $n\ge 3$ (Carter, David, and Gordon Keller. “Bounded Elementary Generation of SLn(O).” American Journal of Mathematics 105, no. 3 (June 1983): 673-687.)

An abelian group is bounded if and only if it is finite. This seems obvious, but there is no finite generation restriction here, so some care is needed in applying structure theory.

If $G$ surjects onto an unbounded group $H$, then $G$ is unbounded The basic idea is simple enough - take a conjugation-invariant norm on $H$ and pull it back via the surjection - but unfortunately the result need not be a norm (it vanishes on the whole kernel not just on the identity), so one needs to develop a more flexible theory of unbounded "quasi-norms" and prove that the existence of unbounded norms and quasi-norms are equivalent.