Distance between unitary orbits in C∗-algebras with stable rank one and real rank zero

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 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.

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).

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 $C(X\times \T)$.

Alternative characterisations of nuclear dimension and decomposition rank in terms of finite dimensional subalgebras and approximate partitions of unity are given for C*-algebras with real rank zero. These characterisations aid the understanding of the two concepts, as well as the difference between them, and lead to elementary proofs of known, important results.

Let be a C*-algebra with real rank zero that has the stable weak cancellation property. Let be an ideal of such that is stable and satisfies the corona factorization property. We prove thatis a full extension if and only if the extension is stenotic and K-lexicographic. As an immediate application, we extend the classification result for graph C*-algebras obtained by Tomforde and the first named author to the general non-unital case. In combination with recent results by Katsura, Tomforde, West, and the first named author, our result may also be used to give a purely K-theoretical description of when an essential extension of two simple and stable graph C*-algebras is again a graph C*- algebra.

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.

A construction method is presented for a class of simple C*-algebras whose basic properties -including their real ranks- can be computed relatively easily, using linear algebra. A numerical invariant attached to the construction determines whether a given algebra has real rank 0 or 1 . Moreover, these algebras all have stable rank 1, and each nonzero hereditary sub-C*-algebra contains a nonzero projection, yet there are examples in which the linear span of the projections is not dense. (This phenomenon was first exhibited by Blackadar and Kumjian.) The construction also produces easy examples of simple C*-algebras with real rank 0 and stable rank 1 for which Ko fails to be unperforated

Let $\Omega $ be a compact subset of $\mathbb {C}$ and let A be a unital simple, separable $C^*$ -algebra with stable rank one, real rank zero, and strict comparison. We show that, given a Cu-morphism ${\alpha :\mathrm { Cu}(C(\Omega ))\to \mathrm {Cu}(A)}$ with , there exists a homomorphism $\phi : C(\Omega )\to A$ such that $\mathrm {Cu}(\phi )=\alpha $ . Moreover, if $K_1(A)$ is trivial, then $\phi $ is unique up to approximate unitary equivalence. We also give classification results for maps from a large class of $C^*$ -algebras to A in terms of the Cuntz semigroup.

We classify certain almost multiplicative morphisms up to approximate unitary equivalence and use this result to prove the following: Let A and B be two unital separable simple C *-algebras of real rank zero, stable rank one, with weakly unperforated K 0 -groups and with unique normalized quasi-traces. Suppose that both A and B are locally AH and ( K * ( A ), K * ( A ) + ,[1 A ]) ≅ ( K * ( B ), K * ( B ) + ,[1 B ]). Then A is isomorphic to B .

Distance between unitary orbits of normal elements in simple [formula omitted]-algebras of real rank zero

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.