Mikael Rordam

On the structure of simple C∗-algebras tensored with a UHF-algebra, II

Let D be a simple unital C∗-algebra, let B be a UHF-algebra, and put A = B ⊗ D. It is proved that if p, q ϵ A are projections and τ(p) < τ(q) for all quasitraces τ on A, then p ≲ q (in the sense of Murray and von Neumann). A more general result involving positive operators in A is also proved. If D has finitely many extremal quasi-traces, and the projections in D ⊗ K separate these, then it is proved that A has real rank zero. Finally it is proved that provided D is stably finite, then each positive state on K0(D) lifts to a quasi-trace.

Classification of Nuclear C*-Algebras. Entropy in Operator Algebras

I. Classification of Nuclear, Simple C*-algebras.- II. A Survey of Noncommutative Dynamical Entropy.

A simple C^*-algebra with a finite and an infinite projection

An example is given of a simple, unital C*-algebra which contains an infinite and a non-zero finite projection. This C*-algebra is also an example of an infinite simple C*-algebra which is not purely infinite. A corner of this C*-algebra is a finite, simple, unital C*-algebra which is not stably finite. Our example shows that the type decomposition for von Neumann factors does not carry over to simple C*-algebras. Added March 2002: We also give an example of a simple, separable, nuclear C*-algebra in the UCT class which contains an infinite and a non-zero finite projection. This nuclear C*-algebra arises as a crossed product of an inductive limit of type I C*-algebras by an action of the integers.

Approximately Central Matrix Units and the Structure of Noncommutative Tori

The property of approximate divisibility for C*-algebras is introduced and studied. Simple approximately divisible C*-algebras are shown to have nice nonstable K-theory properties. Non- rational noncommutative tori are shown to be approximately divisible. It follows that every simple noncommutative torus (in particular, every irrational rotation algebra) has stable rank one and real rank zero.

The Jiang–Su algebra revisited

Abstract We give a number of new characterizations of the Jiang–Su algebra 𝒵, both intrinsic and extrinsic, in terms of C*-algebraic, dynamical, topological and K-theoretic conditions. Along the way we study divisibility properties of C*-algebras, we give a precise characterization of those unital C*-algebras of stable rank one that admit a unital embedding of the dimension-drop C*-algebra Z n, n+1, and we prove a cancellation theorem for the Cuntz semigroup of C*-algebras of stable rank one.

THE STABLE AND THE REAL RANK OF

Suppose that A is a C*-algebra for which

{\mathcal Z}

A \cong A \otimes {\mathcal Z}

-ABSORBING C*-ALGEBRAS

, where

Classification of Nuclear, Simple C*-algebras

{\mathcal Z}

Extending states on preordered semigroups and the existence of quasitraces on C∗-algebras

is the Jiang–Su algebra: a unital, simple, stably finite, separable, nuclear, infinite-dimensional C*-algebra with the same Elliott invariant as the complex numbers. We show that: (i) The Cuntz semigroup W(A) of equivalence classes of positive elements in matrix algebras over A is almost unperforated. (ii) If A is exact, then A is purely infinite if and only if A is traceless. (iii) If A is separable and nuclear, then