C*-algebras, Groups, and Topological Invariants

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The study of physical laws has led to the development of a refined mathematical framework where the algebra of numerical functions is subsumed by a theory based on infinite matrices. Infinite matriceal structures such as operator algebras are able to depict and model the interactions of elementary particles and the symmetries underlying quantum physics. The present project is part of a concerted effort to extend fundamental ideas and techniques of analysis and geometry to the noncommutative context of operator (matrix) algebras. The project will investigate the existence of finite-dimensional approximations of operator algebras that are sufficiently rich to capture key features of the initial data. Passing to finite-dimensional models is important since it gives access to concrete numerical invariants. Inevitably, the structure of the finite-dimensional models will be somewhat less symmetric than their infinite-dimensional counterparts. The loss of exact symmetries is an essential feature of the approximant finite models. It reflects subtle topological properties that the principal investigator aims to quantify in numerical form. The resulting invariants are related to those arising in the mathematics underpinning the physics of novel materials, such as topological insulators with crystalline symmetry.The research concerns two projects in operators algebras that have analytical and topological aspects. The first project is devoted to groups, C*-algebras, and their approximations by finite-dimensional matrix models. It will examine the existence of deformations of discrete groups and group C*-algebras into matrix algebras, the invariants that arise from these deformations, and potential topological obstructions encountered in the process. One underlying goal of the investigation is to develop a better understanding of the topological nature of quasidiagonality, a finite-dimensional approximation property that plays a central role in the structure theory of C*-algebras. The second project concerns the theory of continuous fields of C*-algebras and their generalizations to C*-algebras over general spaces. While the primitive spectrum of a C*-algebra is a fundamental invariant, it has one important limitation. It gives only a low-level description of how the ideals are glued together. A first goal of the project is to develop computable invariants that capture K-theoretical interactions between ideals and local quotients. A second goal is to further develop the generalized Dixmier-Douady theory of continuous fields of strongly self-absorbing C*-algebras due to Ulrich Pennig and the principal investigator.