Let A be a unital separable amenable quasidiagonal simple C∗-algebra with real rank zero, stable rank one, weakly unperforated K0(A) and with a unique tracial state. We show that A must have tracial rank zero. Suppose also that A satisfies the Universal Coefficient Theorem. Then A can be classified by its (ordered) K-theory up to isomorphism. In particular, A must be a simple AH-algebra with no...

We present tracial analogs of the classical results of Curto and Fialkow on moment matrices. A sequence of real numbers indexed by words in non-commuting variables with values invariant under cyclic permutations of the indexes, is called a tracial sequence. We prove that such a sequence can be represented with tracial moments of matrices if its corresponding moment matrix is positive semidefini...

We study the problem of extending a state on an abelian C∗-subalgebra to a tracial state on the ambient C∗-algebra. We propose an approach that is well-suited to the case of regular inclusions, in which there is a large supply of normalizers of the subalgebra. Conditional expectations onto the subalgebra give natural extensions of a state to the ambient C∗-algebra; we prove that these extension...

The tracial analog of Hilbert’s classical result on positive binary quartics is presented: a trace-positive bivariate noncommutative polynomial of degree at most four is a sum of hermitian squares and commutators. This is applied via duality to investigate the truncated tracial moment problem: a sequence of real numbers indexed by words of degree four in two noncommuting variables with values i...

We prove that every separable tracial von Neumann algebra embeds into a II1 factor with property (T) which can be taken to have trivial outer automorphism and fundamental groups. also establish an analogous result for the extension over non-atomic probability space of countable p.m.p. equivalence relation. In addition, we obtain two new results concerning structure infinitely generic factors. T...

We continue to study topological properties of the group Homeo(X) of all homeomorphisms of a Cantor set X with respect to the uniform topology τ , which was started in [B-K 1], [B-D-K 1; 2], [B-D-M], and [M]. We prove that the set of periodic homeomorphisms is τ -dense in Homeo(X) and deduce from this result that the topological group (Homeo(X), τ) has the Rokhlin property, i.e., there exists a...

We study some general properties of tracial C∗-algebras. In the first part, we consider Dixmier type approximation theorem and characterize symmetric amenability for C∗-algebras. In the second part, we consider continuous bundles of tracial von Neumann algebras and classify some of them.