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.