
\titleofpresentation{The Atiyah conjecture and embedding group rings
into division rings}

\presenter{Peter A. Linnell}  %  Name just one speaker, please; collaborators
			%  can be credited in the body of the abstract.

%  [Body of abstract goes here.]

Let $G$ be a discrete group and let $\ell^2(G)$ denote the Hilbert
space $\{\sum_{g\in G} a_gg \mid a_g \in
\mathbb{C},\ \sum_{g \in G}|a_g|^2 < \infty\}$.
There is a well defined multiplication $\ell^2(G) \times \ell^2(G)
\to \ell^{\infty}(G) = \{\sum_{g\in G}
\mid a_g \in \mathbb{C},\ \sup_{g\in G} |a_g| < \infty\}$ given by
\[
\sum_{g \in G} a_gg \sum_{h\in G} b_hh =\sum_{g,h \in G} a_gb_h gh =
\sum_{g\in G}\bigl(\sum_{x \in G} a_{gx^{-1}}b_x\bigr)g.
\]
Then the group von Neumann algebra $\mathcal{N}(G)$ of $G$ is
$\{\alpha \in \ell^2(G) \mid \alpha \ell^2(G) \subseteq \ell^2(G)\}$.
Let $\mathcal{B}(\ell^2(G))$ denote the bounded linear operators on
$\ell^2(G)$, acting on the left.  Then $\mathcal{N}(G)$
can be considered as a $\mathbb{C}$-subalgebra of
$\mathcal{B}(\ell^2(G))$, and it can also
be described as $\{\alpha \in \mathcal{B}(\ell^2(G)) \mid (\alpha v)g
= \alpha (vg)$ for all $g \in G$ and $v \in \ell^2(G)\}$,
or alternatively as the weak closure
of $\mathbb{C}G$ in $\mathcal{B}(\ell^2(G))$.  Let $\mathcal{U}(G)$
denote the closed densely defined unbounded linear operators
affiliated to $\mathcal{N}(G)$.  Then $\mathcal{U}(G)$ is a quotient
ring containing $\mathcal{N}(G)$; more precisely $\mathcal{N}(G)$ is
a subring of $\mathcal{U}(G)$ and every element of $\mathcal{U}(G)$
can be written in the form $\alpha\beta^{-1}$, where $\alpha,\beta
\in \mathcal{N}(G)$ and $\beta$ is a non-zerodivisor in
$\mathcal{N}(G)$.  Algebraically we have
\[
\mathbb{C}G \subseteq \mathcal{N}(G) \subseteq \ell^2(G) \subseteq
\mathcal{U}(G),
\]
where the ring operations are compatible with these inclusions; the
only caution is that $\ell^2(G)$ is closed under multiplication if
and only if $G$ is finite.  Furthermore
$\mathcal{U}(G)$ is a finite von Neumann regular ring.
One version of the Atiyah conjecture states that if $G$ is torsion
free, then there is a division ring $D$ such that $\mathbb{C}G
\subseteq D \subseteq \mathcal{U}(G)$.  I shall consider this
conjecture, especially in the case $G$ is a congruence subgroup.  
I shall also consider cases when $G$ has torsion.  As a byproduct, I
hope to present a result concerning embedding a group ring into a
finite von Neumann regular ring, that uses a theorem of Brian
Hartley.


\presenteraddress{Virginia Tech, Blacksburg}  % The name of a university is fine.

\email{linnell@math.vt.edu}

\web{www.math.vt.edu/people/linnell/}  % Those without homepages should delete this.

