Abstracts for Ken Davidson's talks

Colloquium
Operator theory meets algebraic geometry

I will discuss how to study commuting sets of operators
on Hilbert space which satisfy polynomial relations.
Under a natural norm constraint, there is a universal
operator algebra that models this.  In an effort to
classify these algebras up to isomorphism, one must
deal with the variety associated to the polynomial relations.
Classification up to  isometric isomorphism is very nice.
But the algebraic isomorphism problem raises many difficulties.


Operator Algebra Talks
1. The Choquet boundary of an operator system

I will review Arveson's approach to non-commutative dilation theory.
In particular, I will explain about the role of boundary representations
and the C*-envelope. While the existence of the C*-envelope was
established early on in the theory, and is now a central theme, the
existence of boundary representations has been a mystery. Recently
Matt Kennedy and I solved this problem, showing that there are
always enough boundary representations to determine the C*-envelope
as envisaged by Arveson 45 years ago.


2. Semicrossed products

I will explain the semicrossed product, which is a universal operator
algebra that encodes the information in a pair $(A,\alpha)$, where
$A$ is a operator algebra and $\alpha$ is a completely isometric
endomorphism. I will discuss issues related to dilation theory and
determination of the C*-envelope. In particular, I will mention some
recent work with Kakariadis and other work with Katsoulis.


3. Dilation theory and commutant lifting

I will discuss a recent paper with Katsoulis proposing a somewhat
different take on what an appropriate version of the commutant lifting
theorem should be for arbitrary operator algebras. A more restrictive class
of dilations makes it more likely that a commutant lifting theorem can hold.
We show that this leads to corresponding lifting theorems for representations
of semicrossed products.