Distance between unitary orbits of unitary elements in $$C^*$$-algebras of real rank zero

AF-embeddings into [formula omitted]-algebras of real rank zero

We prove a number of fundamental facts about the canonical order on projections in C*-algebras of real rank zero. Specifically, we show that this order is separative and that arbitrary countable collections have equivalent (in terms of their lower bounds) decreasing sequences. Under the further assumption that the order is countably downwards closed, we show how to characterize greatest lower bounds of finite collections of projections, and their existence, using the norm and spectrum of simple product expressions of the projections in question. We also characterize the points at which the canonical homomorphism to the Calkin algebra preserves least upper bounds of countable collections of projections, namely that this occurs precisely when the span of the corresponding subspaces is closed.

We obtain an order sharp estimate for the distance from a given bounded operator A A on a Hilbert space to the set of normal operators in terms of ‖ [ A , A ∗ ] ‖ \|[A,A^*]\| and the distance to the set of invertible operators. A slightly modified estimate holds in a general C ∗ C^* -algebra of real rank zero.

Let Ext ( C ( X ), A ) be the set of unitarily equivalence classes of essential C *-algebra extensions of the following form: 0 → A → E → C ( X ) → 0, where A is a nonunital separable simple C *-algebra of real rank zero, stable rank one with unique normalized trace and X is a finite CW complex. We show that there is a bijection [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]: Ext ( C ( X ), A ) → KK ( C ( X ), M ( A )/ A ), where M ( A ) is the multiplier algebra of A . In particular, we determine when an extension is actually splitting. We also, in a more general setting, give a condition when an essential extension is quasidiagonal.

Almost Commuting Unitaries and Classification of Purely Infinite SimpleC*-Algebras

On the Range of Certain ASH Algebras of Real Rank Zero

Classification of Homomorphisms from C (X) to Simple C *-Algebras of Real Rank Zero

A unital C*-algebra A is said to have cancellation of projections if the semigroup D(A) of Murray–von Neumann equivalence classes of projections in matrices over A is cancellative. It has long been known that stable rank one implies cancellation for any A, and some partial converses have been established. In this paper it is proved that cancellation does not imply stable rank one for simple, stably finite C*-algebras. 2000 Mathematics Subject Classification 46L80 (primary), 46L85 (secondary).

On the Classification of C ∗ -Algebras of Real Rank Zero, II

Let H and K be two complex Hilbert spaces and 𝒜 ⊂ B(H) and ℬ ⊂ B(K) be two unital C*-algebras. It is shown that if φ : 𝒜 → ℬ is an additive surjective mapping satisfying φ(|A|) = |φ(A)| for every A ∈ 𝒜 and φ(I) is a projection, then the restriction of mapping φ to both 𝒜s and 𝒜sk is a Jordan *-homomorphism onto corresponding set in ℬ, where 𝒜s and 𝒜sk denote the set of all self-adjoint and skew-self-adjoint elements, respectively. Furthermore, if ℬ is a C*-algebra of real-rank zero then φ is a ℂ-linear or ℂ-antilinear *-homomorphism.

Let A be a C∗-algebra with stable rank one and real rank zero. In this paper, it is shown that the usual distance dU defined on the approximate unitary equivalence classes (or unitary orbits) of the positive elements in A is equal to the distance dW defined on morphisms from Cuntz semigroup of C0(0,1] to the Cuntz semigrout of A.

In this paper we give a representation theorem for the Cuntz monoid S(A) of a σ-unital C*-algebra A with real rank zero and stable rank one, which allows to prove several Riesz decomposition properties on the monoid. As a consequence, it is proved that the comparability conditions (FCQ), stable (FCQ) and (FCQ+) are equivalent for simple C*-algebras with real rank zero. It is also shown that the Grothendieck group [Formula: see text] of S(A) is a Riesz group, and lattice-ordered under some additional assumptions on A.

We prove that a crossed product algebra arising from a minimal dynamical system on the product of the Cantor set and the circle has real rank zero if and only if that system is rigid. In the case that cocycles take values in the rotation group, it is also shown that rigidity implies tracial rank zero, and in particular, the crossed product algebra is isomorphic to a unital simple AT-algebra of real rank zero. Under the same assumption, we show that two systems are approximately $K$-conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps $C(X\times \T)$ onto $C(X\times \T)$.

We say that a unital C ∗ \mathrm {C}^{*} -algebra A A has the approximate positive factorization property (APFP) if every element of A A is a norm limit of products of positive elements of A A . (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of C ∗ \mathrm {C}^{*} -algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial K 1 K_{1} , and stable rank 1. In the unital case, the K 0 K_{0} group separates the tracial states. The APFP passes to matrix algebras, and if I I is an ideal in A A such that I I and A / I A/I have the APFP, then so does A A . We also give some new examples of C ∗ \mathrm {C}^{*} -algebras with the APFP, including type I I 1 \mathrm {II}_{1} factors and infinite-dimensional simple unital direct limits of homogeneous C ∗ \mathrm {C}^{*} -algebras with slow dimension growth, real rank zero, and trivial K 1 K_{1} group. Simple direct limits of homogeneous C ∗ \mathrm {C}^{*} -algebras with slow dimension growth which have the APFP must have real rank zero, but we also give examples of (nonsimple) unital algebras with the APFP which do not have real rank zero. Our analysis leads to the introduction of a new concept of rank for a C ∗ \mathrm {C}^{*} -algebra that may be of interest in the future.

This dissertation focuses on finite group actions with the tracial Rokhlin property and the structure of the corresponding crossed products. It consists of two major parts. For the first part, we study several different aspects of finite group actions with certain versions of the Rokhlin property. We are able to give an explicit characterization of product-type actions with the tracial Rokhlin property or strict Rokhlin property. We also show that, in good circumstances, the actions with the tracial Rokhlin property are generic. In the second portion of this dissertation, we introduce the weak tracial Rokhlin property for actions on non-simple C*-algebras. The main results are as follows. Let A be a unital non-simple C*-algebra and α be an action of G on A with the weak tracial Rokhlin property. Assume that the crossed product C*: G, A, α) is simple. Suppose A has either of the following property: tracial rankk, stable rank one, real rank zero. Then C*: G, A, α) has the same property.