Operator Algebras and Topological Invariants

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This project is guided by the premise that the ideas and the methods of algebraic topology, understood in a broad sense, will continue to play a key role in the study of operator algebras. The underlying theme of the research is to devise new techniques and tools in the deformation theory of operator algebras. The idea of deformations of algebras is intimately related to K-theory, as is revealed by the development of E-theory. In particular one can read the K-homology of a locally compact connected space X from matricial deformations of the algebra of continuous functions on X. In what may be regarded as a counterpart of this property in condensed-matter physics, Kitaev has proposed very recently a classification of topological insulators that is based on real K-homology and that relies ultimately on a classification up to approximate unitary equivalence of matricial deformations of spaces. A first objective of the project is to study invariants of C*-algebras that integrate the primitive spectrum along with homological invariants. This is naturally tied to the idea of deformation and the theory of continuous fields and will involve the study of KK-theory and E-theory parametrized by spaces that are not necessarily Hausdorff. A second objective is to investigate the existence of deformations of group C*-algebras into matrix algebras with the property that they detect the K-theory of the group C*-algebras. In addition, the principal investigator aims to provide formulas for computing invariants associated with these deformations. A third objective is to study invariants of simple C*-algebras that capture properties not detected by K-theory or tracial states. In particular, the project will investigate rigidity properties of simple C*-algebras.The physics of elementary particles has led to new mathematical theories in which numerical functions are replaced with infinite arrays of numbers or matrices. Matrices can be multiplied, but unlike numerical multiplication, the order of the factors is essential, so that A times B is not always equal to B times A. This property of matrix multiplication, called noncommutativity, is crucial for the interpretation of very complex phenomena in quantum mechanics that might initially appear completely counter-intuitive (e.g., Heisenberg's uncertainty principle). The passage from numbers to matrices often involves a process of deformation in which spatial structures are discretized in such a way that the possible twists of surfaces (or even more complicated objects) are captured in the form of numerical invariants. This project is concerned with the theory of such deformations and connects classical geometry with noncommutative geometry, a theory that proposes a reconstruction of the general concept of space that is better aligned with the principles of quantum physics than is traditional geometry. Moreover, the deformation theory that the principal investigator will study provides mathematical tools relevant to physical phenomena that are observed in certain special materials. These materials behave as insulators in their interiors, while permitting the movement of charges on their surfaces, due to exotic metallic states localized there. Such remarkable new materials, known as "topological insulators," have already had a considerable impact on condensed matter physics and are expected to find technological applications in magneto-electronics, an emerging field in which spin currents are used instead of charge currents, and in quantum computing.