Related papers: QDQ vs. UCT

We classify all essential extensions of the form $$0 \rightarrow \W \rightarrow \D \rightarrow A \rightarrow 0$$ where $\W$ is the unique separable simple C*-algebra with a unique tracial state, with finite nuclear dimension and with…

We present a stable uniqueness theorem for non-unital C*-algebras. Generalized tracial rank one is defined for stably projectionless simple C*-algebras. Let $A$ and $B$ be two stably projectionless separable simple amenable C*-algebras with…

In this article we survey some of the recent goings-on in the classification programme of C$^*$-algebras, following the interesting link found between the Cuntz semigroup and the classical Elliott invariant and the fact that the Elliott…

This note provides some technical support to the proof of a result of W. Winter which shows that two unital separable simple amenable ${\cal Z}$-absorbing C*-algebras with locally finite decomposition property satisfying the UCT whose…

It is shown that if A is a separable, exact C*-algebra which satisfies the Universal Coefficient Theorem (UCT) and has a faithful, amenable trace, then A admits a trace-preserving embedding into a simple, unital AF-algebra with unique…

In this paper we identify QD(A,B), the quasidiagonal classes in KK_1(A,B), in terms of K_*(A) and K_*(B), and we use these results in various applications. Here is our central result. Theorem: Suppose that A is in the category of separable…

In this paper, we exhibit two unital, separable, nuclear ${\rm C}^*$-algebras of stable rank one and real rank zero with the same ordered scaled total K-theory, but they are not isomorphic with each other, which forms a counterexample to…

We define united K-theory for real C*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors -- real K-theory, complex K-theory, and self-conjugate K-theory -- and the natural…

We provide some background on the category of classifiable $\mathrm{C}^*$-algebras, whose objects are infinite-dimensional, simple, separable, unital $\mathrm{C}^*$-algebras that have finite nuclear dimension and satisfy the universal…

Let A be a separable unital nuclear purely infinite simple C*-algebra satisfying the Universal Coefficient Theorem, and such that the K_0-class of the identity is zero. We prove that every automorphism of order two of the K-theory of A is…

In this paper we associate to every reduced C*-algebraic quantum group A a universal C*-algebraic quantum group. We fine tune a proof of Kirchberg to show that every *-representation of a modified L1-space is generated by a unitary…

It is shown that if a C*-algebra has nuclear dimension $n$ then its Cuntz semigroup has the property of $n$-comparison. It then follows from results by Ortega, Perera, and Rordam that $\sigma$-unital C*-algebras of finite nuclear dimension…

We give a detailed survey of the theory of quasidiagonal C*-algebras. The main structural results are presented and various functorial questions around quasidiagonality are discussed. In particular we look at what is currently known (and…

In this article I study a number of topological and algebraic dimension type properties of simple C*-algebras and their interplay. In particular, a simple C*-algebra is defined to be (tracially) (m,\bar{m})-pure, if it has (strong tracial)…

We prove that separable C*-algebras which are completely close in a natural uniform sense have isomorphic Cuntz semigroups, continuing a line of research developed by Kadison - Kastler, Christensen, and Khoshkam. This result has several…

We show that the UCT problem for separable, nuclear $\mathrm C^*$-algebras relies only on whether the UCT holds for crossed products of certain finite cyclic group actions on the Razak-Jacelon algebra. This observation is analogous to and…

We examine the ranks of operators in semi-finite C*-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple C*-algebra whose extreme tracial boundary is nonempty and finite contains…

In this note we start the study of whether the reduced C*-algebra of an inverse semigroup is quasi-diagonal, making explicit use of the inner structure of this class of semigroups in order to produce quasi-diagonal approximations. Given a…

A universal coefficient theorem is proved for C*-algebras over an arbitrary finite T_0-space X which have vanishing boundary maps. Under bootstrap assumptions, this leads to a complete classification of unital/stable real-rank-zero…

We introduce a notion of approximate ideal structure for a $C^*$-algebra, and use it as a tool to study $K$-theory groups. The notion is motivated by the classical Mayer-Vietoris sequence, by the theory of nuclear dimension as introduced by…