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.

The smallest primitive ideal spaces for which there exist counterexamples to the classification of non-simple, purely infinite, nuclear, separable C*-algebras using filtrated K-theory, are four-point spaces. In this article, we therefore restrict to real rank zero C*-algebras with four-point primitive ideal spaces. Up to homeomorphism, there are ten different connected T₀-spaces with exactly four points. We show that filtrated K-theory classifies real rank zero, tight, stable, purely infinite, nuclear, separable C*-algebras that satisfy that all simple subquotients are in the bootstrap class for eight out of ten of these spaces.

We investigate when discrete, amenable groups have $C^{\ast }$-algebras of real rank zero. While it is known that this happens when the group is locally finite, the converse is an open problem. We show that if $C^{\ast }(G)$ has real rank zero, then all normal subgroups of $G$ that are elementary amenable and have finite Hirsch length must be locally finite.

The conjecture that the real rank of multiplier algebra of every AF-algebra is zero was formally made by L. G. Brown and G. K. Pedersen in [4]. The main purpose of this note is to prove the conjecture. However, we also show that this Weyl-von Neumann type theorem holds for C*-algebras with stable (FU) and stable rank one. In particular, this Weyl-von Neumann type theorem holds for C*-algebras of inductive limits of finite direct sums of basic building blocks with real rank zero and trivial K₁-groups considered by George A. Elliott recently.

We show that all trivial (unital and essential) extensions of C (X) by a σ-unital purely infinite simple C*-algebra A with K₁(A) = 0 are unitarily equivalent, provided that X is homeomorphic to a compact subset of the real line or the unit circle. Therefore all (unital and essential) extensions of such can be completely determined by Ext(B, A). An invariant is introduced to classify all such trivial (unital and essential) extensions of C (X) by a σ-unital C*-algebra A with the properties that RR (M (A)) = 0 and C (A) is simple.

In this paper, we consider the real rank zero $C^{\ast }$-algebras which can be written as an inductive limit of the Elliott–Thomsen building blocks and prove a decomposition result for the connecting homomorphisms; this technique will be used in the classification theorem.

In this paper, using ordered total K-theory, we give a K-theoretic classification for the real rank zero inductive limits of direct sums of generalized dimension drop interval algebras.

Let ${𝒜}$ and ${ℬ}$ be two unital $C^{\ast }$-algebras with unit $I$. It is shown that the mapping $\varphi :{{𝒜}}_s\to {{ℬ}}_s$ which preserves arithmetic mean and Jordan triple product is a difference of two Jordan homomorphisms provided that $0\in \mathrm{\text{Ran}}\varphi$. The structure of $\varphi$ is more refined when $\varphi (I)\ge 0$ or $\varphi (I)\le 0$. Furthermore, if ${𝒜}$ is a $C^{\ast }$-algebra of real rank zero and $\varphi :{𝒜}\to {ℬ}$ is additive and preserves absolute value of product, then $\varphi ={\varphi }_1\oplus {\varphi }_2$ such that ${\varphi }_1$ (respectively, ${\varphi }_2$) is a complex linear (respectively, antilinear) ∗-homomorphism.