Math 617 - Topics in Functional Analysis: Introduction to Operator Spaces

Instructor

Volker Runde

Office hours

TBA

Syllabus

Originally, an operator space was a closed subspace of B(H), the algebra of all bounded operators on a Hilbert space H. In his 1988 thesis, Zhong-Jin Ruan showed that operator spaces can be characterized (up to a suitable notion of isomorphism, of course) through two, very simple axioms. In recent years, it has be come fashionable to speak of operator spaces as of "quantized Banach spaces", which are the main object of study in "quantized functional analysis". Many basic principles from ordinary functional analysis - duality, the Hahn-Banach theorem - have analogs in the quantized setting whereas others - the open mapping theorem and local reflexivity, for instance - don't.

In this course, I plan to cover:

• concrete and abstract operator spaces;
• duality for operator spaces and Wittstock's Hahn-Banach theorem;
• operator space tensor products (injective, projective, and Haagerup).

Depending on the background of the students enrolled in this course, I may treat at least some of the following:
• interpolation of operator spaces;
• applications to abstract harmonic analysis;
• non-selfadjoint operator algebras.

Textbooks

None required, but the following are recommended:
1. E. G. Effros and Z.-J. Ruan, Operator Spaces. Clarendon Press, Oxford, 2000.
2. G. Pisier, Introduction to Operator Space Theory, London Mathematical Society Lecture Note Series, 294. Cambridge University Press, Cambridge, 2003.
3. V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics, 78. Cambridge University Press, Cambridge, 2002.
4. G. Wittstock, et al., What are Operator Spaces? - An Online Dictionary. Online Resource, 2001.

Grading

The grade will be based on occasional homework assignments and a project at the end of the semester.