NSF-BSF: C*-algebras and dynamics beyond the Elliott classification program

Operator algebras are mathematical objects which arise in a variety of situations, including geometry, dynamical systems, and quantum physics. As geometry is as- sociated to the understanding of classical physics, the study of operator algebras is associated to the theoretical underpinnings of quantum physics.

Nearly 40 years ago, G. Elliott proposed a broad program to understand and elas- sify all important class of operator algebras, now known as simple C*-algebras with finite nuclear dimension. This program has had great success, and at this point the program is viewed as nearly complete. This has led to calls for operator algebraists to understand C*—algebras which fall outside the scope of Elliott’s program.

Much of the work on the Elliott program involved studying operator algebras which can be built up from simpler building blocks, known as homogeneous C*—algebras. Part of our proposal focuses on a problem posed by Blackadar over 30 years ago, which is still open today, about how certain types of those basic building blocks call fit into one another. This is connected to another long—standing open problem. Each C*—algebra is associated with a mirror object, known as the opposite C*—algebra. In many cases, a C*—algebra cannot be distinguished from its opposite. However, there are some cases in which this symmetry breaks. So far, however, nobody has found an example of such symmetry breaking within the class of operator algebras just outside Elliott"s program. We hope to use insights we might gain from Blackadar’s problem to attempt to construct such a symn1etry—breaking example by finding complicated building blocks which cannot fit into one another.