Speaker: Ken Dykema (Texas A&M University)

Title: Some partial answers to natural questions about II1 factors

Abstract: The following are natural questions in a II1-factor M, because their direct analogues in Mn(C) have positive answers.

(a) Is every element in M whose trace value is zero equal to a single commutator of elements of M?

(b) If A is a masa in M and if and element x of A is positive and of norm ≤1, is there a projection p in M such that EA(p)=x, where EA$ is the trace-preserving conditional expectation onto A?

Both questions are still open, and in this talk we'll discuss some partial answers and a related question.

We'll discuss partial answers to (a), from joint work with Anna Skripka.

Question (b) is known as the carpenter problem, thus named by R.V. Kadison. It is a particular case of the Schur-Horn problem in II1-factors, which was formulated by Arveson and Kadison. We'll discuss answers to the carpenter and Schu-Horn problems in some cases, from joint work with Junsheng Fang, Don Hadwin and Roger Smith.