Twisted continuous trace C*-algebras and C*-algebras not isomorphic to their opposites

PIs:

• Ilan Hirshberg, Department of Mathematics, Ben Gurion University, Israel.

• Marius Dadarlat, Department of Mathematics, Purdue University, West Lafayette, IN, USA.

• N. Christopher Phillips, Department of Mathematics, University of Oregon, Eugene, OR, USA.

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

About 30 years ago, G. Elliott proposed a broad program to understand and classify an important class of operator algebras, known as simple nuclear C*—algebras. This program has had lnuch success. However, in the last 10 years, M. Rordam and A. Toms constructed complicated examples of such objects which can be shown to violate Elliott’s predictions.

The project produced several results. Those include two papers which were published or accepted, two which are in final stages of preparation and will be submitted soon, and additional results which were not submitted for publication.

One of the results concerns the construction of an object which violates Elliott’s prediction in a way which is internal to the object itself. That it, Elliott’s classification program predicts that such an object should possess a certain symmetry, yet this symmetry does not exist. This is the first known example of such objects, and it reveals a deeper structure within such objects which wasn't known before.

Two additional results, one of which was accepted for publication, concerns opposite C*—algebras. Each C*—algebra is associated with a “mirror” object, known as the opposite C*—algebra. In many cases, a C*—algcbra 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 which Elliott's program addresses. Our original project outlined a method for constructing such a symmetry breaking object, however we discovered that the conjectured basis for our intended method cannot work. At the same time, we discovered an example of a C*-algebra which this symmetry breaking occurs in the so-called equivariant setting, that is, the C*-algebra and its opposite come equipped with a given collection other symmetries, and taking into account those symmetries, one can in fact distinguish the object from its opposite.

Additional results emanating from the project concern the notion of Rokhlin dimension for a group of symmetries of a C*-algebra, which measures a notion of regularity of those symmetries. We studied properties of such symmetries, and found obstructions for this regularity which weren't known before. Related results concerning structural aspects of such regularity properties were studied by Hirshberg's Ph.D. student, J. Orovitz, in conjunction with the other PI's and Q. Wang, a postdoctoral fellow of Phillips, and are part of Orovitz's Ph.D. dissertation.