On dilations of CP-semigroups (my talk at the Midrasha at Weizmann Institute)

by Orr Shalit

I had the privilege of being invited this year again to give a talk at the Group Theory Seminar at the Weizmann Institute (Here is a link to last year’s talk). I am pasting here a link to the recording of the talk; the link is to the third minute, the recording started three minutes before the lecture began.

Title: Dilations of CP-semigroups via subproduct systems and superproduct systems of C*-correspondences.

Abstract:

The title is a bit of a mouthful, so let us unpack it together:

1. A C*-correspondence is a certain kind of bimodule over a C*-algebra B that has a B-valued inner product.
2. Sub and super-product systems are families of C*-correspondences that enjoy certain semigroup-like properties under the tensor product. 
3. A CP-semigroup is a family of completely positive maps that form a semigroup under composition; the most important examples are when the semigroup is given by a family (T_t) parameterized by positive real t>0.
4. By dilation of CP-semigroup we mean a specific way of exhibiting one CP-semigroup as a part of another semigroup that belongs to a better understood category, typically a semigroup of *-homomorphisms.

The problem of constructing dilations for CP-semigroups or determining their existence/uniqueness has drawn some of the best minds in operator algebras. And still there are open problems. I will describe the elaborate framework of sub and super-product systems employed by Michael Skeide and myself to attack this problem, a framework which is built on top of decades of work by ourselves and others. Using this framework we resolve some problems in the multi-parameter case; more surprisingly, we also obtain new results in the one-parameter case.