#
The Kadison-Singer Problem

September 25 to September 29, 2006
at the

American Institute of Mathematics,
Palo Alto, California

organized by

Pete Casazza,
Richard Kadison,
and David Larson

## Original Announcement

This workshop will be devoted
to the Kadison-Singer Problem and its
relationship to various areas of research in
mathematics and engineering.
The 1959 Kadison-Singer Problem in C*-algebras has
defied the best efforts of some of the most talented
mathematicians of our time. Recently it was shown that
this problem is equivalent to fundamental unsolved problems
in a dozen areas of research in Pure Mathematics,
Applied Mathematics and
Engineering, including: operator theory, Banach space theory,
harmonic analysis, and signal processing.

The specific purpose of this workshop is to bring together
some of the top minds in the various areas of research
impacted by the Kadison-Singer Problem to see if we can resolve
the problem. For more realistic goals, we expect:

- The participants to release partial results they have on the problem;
- To formulate related problems that might be more tractable than the
full-scope problem, and could potentially lead to the eventual solution of
the problem;
- To map out possible "paths" that could lead to the solution -
especially interactive paths between two or more research areas; and
- To establish long term relationships between people from the diverse
areas of research impacted by Kadison-Singer which will lead to
interactive research in the future.

We anticipate having only one talk each day presenting an introduction
to one area impacted by the Kadison-Singer Problem. These talks will
form an introduction to the workshop's activities for that day.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.

### Introduction

The following papers contain an introduction to various aspects of the
Kadison-Singer problem.
Kadison-Singer from mathematical physics: an introduction by Palle Jorgensen

State extensions and the Kadison-Singer problem by Akemann and Paulsen.

Paving and the Kadison-Singer problem, by Casazza, Paulsen, and Weiss.

Frames and the Kadison-Singer problem, by Casazza, Kutyniok, Larsen, and Speegle.

### Open problems

Below one can find lists of problems around Kadison-Singer that might be
more approachable than the entire conjecture.
Problems on paving and the Kadison-Singer problem

Problems on frames and the Kadison-Singer problem

### Recent observations

These papers contain recent results that will be of interest to those
working on problems related to Kadison-Singer.
The paving property for uniformly bounded matrices, by Tropp

The paving conjecture is equivalent to the paving conjecture for triangular matrices by Casazza and Tremain

A decomposition theorem for frames and the Feichtinger conjecture, by Casazza, Kutyniok, Speegle, and Tremain

Projections and the Kadison-Singer problem, by Casazza, Edidin, Kalra, and Paulsen