**
Christopher
Schafhauser,
Department
of
Pure
Mathematics,
University
of
Waterloo**

"An Embedding Theorem for C*-algebras"

A
C*-algebra
consists
of
an
algebra of
bounded
linear
operators
acting
on
a Hilbert
space
which
is
closed the
adjoint
operation
(roughly,
the
transpose)
and
is complete
in
a
certain
metric.
Typical
examples
include
the
ring
of
*
n* × *n* complex
matrices and
the
ring
*
C*(*X*) of
representation
of
continuous
functions
from
a
compact
space
*
X* to
the
complex
numbers.
Many
more
interesting
examples
arise
from
various
dynamical
objects
(e.g. group
and
group
actions)
and
from
various
geometric/topological
constructions.

The structure of finite dimensional C*-algebras is well understood: they are finite direct sums of complex matrix algebras. The class of approximately finite-dimensional (AF) C*-algebras, ones which may be written as (the closure of) an increasing union of finite-dimensional subalgebras, are also well understood: they are determined up to isomorphism by their module structure. However, the class of sub-algebras of AF-algebras is still rather mysterious; it includes, for instance, all commutative C*-algebras and all C*-algebras generated by amenable groups. It is a long-standing problem to find an abstract characterization of subalgebras of AF-algebras.

I will discuss this class of algebras and a recent partial answer to this AF-Embedding problem. My goal is to keep the talk accessible and mostly expository. In particular, I won't assume any prior knowledge of operator algebras or functional analysis.

MC 5403