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An Isaac Newton Institute Workshop

Trends in Noncommutative Geometry

Unbounded elements in C-star algebras

22nd December 2006

Author: S.L. Woronowicz (Warsaw University and The Isaac Newton Institute)

Abstract

In many situations we deal with unbounded operators that are in a sense related to operator algebras. This is the case in quantum mechanics, where (unbounded) Hamiltonian is related to the algebra of observables and in locally compact (non-compact) quantum groups, where matrix elements of finite-dimensional representations are related to the C-star algebra of "functions on the group". Similarly infinitesimal generators of a Lie group are related to the algebra C-star of the group. In all these cases the unboundnes is the only feature that prevents us to to include the operators to the algebra. Instead we say that the operators are affiliated with the algebra. The aim of the talk is to define the affiliation relation in the C-star algebra context and show a number of applications and properties.