Alexander Helemskii (Moscow State University)
alexander@helemskii.mccme.ru
Quantum functional analysis (more prosaically, `Operator space theory') is a new and rapidly developing branch of functional analysis; it is a fruitful synthesis of the theories of operator algebras and Banach space geometry.
The objects of the theory (spaces, algebras, modules) have a structure - the quantum norm - which is richer than the usual one.
Previously, the quantum norm was introduced with the help of a family of norms on matrix spaces. However, there is another way of doing this; we consider one norm on a linear space with operator coefficients. It is known that both approaches are equivalent (Pisier), but there is no systematic exposition based on the second approach.
In the present talk we shall give some concepts and results related to the `non-matricial' exposition of the theory. We shall pay particular attention to quantum tensor products and quantum duality theory.