Grant: Discovery Grants Program - Individual

This proposed research investigates a branch of mathematics known as operator algebras. These mathematical objects arose during the 1930's when mathematicians and physicists were trying to get a rigorous description of matter on the atomic scale. The physical theory developed for that description was Quantum Theory. Mathematically, the possible configurations or states of matter (an electron moving around an atom, for example) are described by ""Hilbert spaces""---a point in the Hilbert space for an electron in the hydrogen atom represents a possible configuration for the electron ""wave"" around the hydrogen nucleus. If we perform an experiment on the atom, and alter the state of the electron, we have applied what is known mathematically as an operator to the space of possible states (the Hilbert space). The collection of such operators is described mathematically as an algebra of operators---operator algebras. C*-algebras provide an abstract way to describe algebras of operators without referring to the underlying Hilbert space, and the study of the structure of C*-algebras becomes crucial for our understanding of operator algebras. Launched by Professor Elliott at the University of Toronto, the programme of classifying C*-algebras using K-theory---an invariant which was used earlier in algebraic geometry and algebraic topology---soon developed into an important mathematical pursuit with applications to areas such as dynamical systems and mathematical physics. Lots of naturally arising C*-algebras, for instance, the CAR algebra, rotation algebras, etc., are covered by the classification programme. However, there are still many of them which have inductive limit decomposition in terms of relatively simpler building blocks, but are still out of the reach of the classification at the present time. The main objective of the proposed research is to cover these algebras by the classification programme.