Narrow Results

Suppose that A is a C*-algebra for which , where 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 if and only if A is traceless.

(iv) If A is simple and unital, then the stable rank of A is one if and only if A is finite.

We also characterize when A is of real rank zero.

Let G be an inductive limit of finite cyclic groups, and A be a unital simple projectionless C*-algebra with K₁(A) ≅ G and a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in Aut(A)/WInn(A) are conjugate, where WInn(A) means the subgroup of Aut(A) consisting of automorphisms which are inner in the tracial representation.

In the second part of this paper, we consider a class of unital simple C*-algebras with a unique tracial state which contains the class of unital simple A𝕋-algebras of real rank zero with a unique tracial state. This class is closed under inductive limits and crossed products by actions of ℤ with the Rohlin property. Let A be a TAF-algebra in this class. We show that for any automorphism α of A there exists an automorphism ᾶ of A with the Rohlin property such that ᾶ and α are asymptotically unitarily equivalent. For the proof we use an aperiodic automorphism of the Jiang-Su algebra.

Let be the Jiang–Su algebra and let τ be its unique tracial state. We prove that for all , the following statements are equivalent:

(1) a is a finite sum of commutators.

(2) a is a sum of five commutators.

(3) τ(a) = 0.

We investigate C*-algebras whose central sequence algebra has no characters, and we raise the question if such C*-algebras necessarily must absorb the Jiang–Su algebra (provided that they also are separable). We relate this question to a question of Dadarlat and Toms if the Jiang–Su algebra always embeds into the infinite tensor power of any unital C*-algebra without characters. We show that absence of characters of the central sequence algebra implies that the C*-algebra has the so-called strong Corona Factorization Property, and we use this result to exhibit simple nuclear separable unital C*-algebras whose central sequence algebra does admit a character. We show how stronger divisibility properties on the central sequence algebra imply stronger regularity properties of the underlying C*-algebra.