Michael Hartz

Seminar on completely positive maps

(Block seminar in September 2022)


Time and Place

Meeting dates are September 21 and 22 in HS 4.


Completely positive maps are a special class of linear maps between C*-algebras. They play a key role in certain modern approaches to the theory of operators on Hilbert space, and are also important in the theory of tensor products of C*-algebras. Remarkably, completely positive maps also appear in quantum information theory under the name of quantum channels.

The goal of this seminar is to understand the basic theory of completely positive maps. In particular, we will cover von Neumann's inequality, Stinespring's dilation theorem and Arveson's extension theorem.


Prerequisite is knowledge of basic functional analysis, including the elementary theory of C* algebras. In particular, the course is suitable for students who took Functional Analysis I last semester.


Main textbook:
Paulsen, Vern, Completely Bounded maps and Operator algebras , 2002.

Additional literature:
Blecher, David and Le Merdy, Christian, Operator algebras and their modules -- an operator space approach , 2004.
Effros, Edward and Ruan, Zhong-Jin, Operator Spaces , 2000.
Pisier, Gilles, Introduction to operator space theory , 2003.
Pisier, Gilles, Similarity problems and completely bounded maps , 2001.