Tracial Invariants, Classification and Ii1 Factor Representations of Popa Algebras

Using various finite dimensional approximation properties, four convex subsets of the tracial space of a unital C∗-algebra are defined. One subset is characterized by Connes’ hypertrace condition. Another is characterized by hyperfiniteness of GNS representations. The other two sets are more mysterious but are shown to be intimately related to Elliott’s classification program. Applications of these tracial invariants include: 1. An analogue of Szegö’s Limit Theorem for arbitrary self adjoint operators. 2. A McDuff factor embeds into R if and only if it contains a weakly dense operator system which is injective. 3. There exists a simple, quasidiagonal, real rank zero C∗-algebra with non-hyperfinite II1 factor representations and which is not tracially AF. This answers negatively questions of Sorin Popa and, respectively, Huaxin Lin. 4. If A is any one of the standard examples of a stably finite, non-quasidiagonal C∗-algebra and B is a C∗-algebra with Lance’s WEP and at least one tracial state then there is no unital ∗-homomorphism A → B. In particular, many stably finite, exact C∗-algebras can’t be embed into a stably finite, nuclear C∗-algebra. Dedicated to my big, beautiful family.