Cornel Pasnicu

Purely infinite C*-algebras of real rank zero

Abstract We show that a separable purely infinite C*-algebra is of real rank zero if and only if its primitive ideal space has a basis consisting of compact-open sets and the natural map K 0(I) → K 0(I/J) is surjective for all closed two-sided ideals J⊂I in the C*-algebra. It follows in particular that if A is any separable C*-algebra, then A ⊗𝒪2 is of real rank zero if and only if the primitive ideal space of A has a basis of compact-open sets, which again happens if and only if A ⊗ 𝒪2 has the ideal property, also known as property (IP).

The projection property

In this paper we prove that the ideal property and the projection property do not conincide in general even in the separable case (despite the fact that, as we proved before, they are the same for GAH algebras-and, in particular, for AH algebras-and for separable LB algebras). We also study the behaviour of the projection property with respect to several natural operations. 2000 Mathematics Subject Classification. 46L05, 46L99. We begin by introducing the following definition. Definition 1. A C*-algebra is said to have the projection property if each of its ideals has an approximate unit consisting of projections. In the present paper all the ideals are closed and two-sided. In this paper we shall study the projection property: we shall prove that it differs from the ideal property (even in the separable case) despite of a lot of ‘‘possible evidence’’ for the contrary conclusion and we shall also study the behaviour of the projection property with respect to some natural operations. It is obvious that the projection property is stronger than the ideal property, whose study was suggested by G.A. Elliott. Recall that a C*-algebra has the ideal property if each of its ideals is generated (as an ideal) by projections. The class of the C*-algebras with the ideal property (studied in [20], [11–19]) is interesting since it contains two important classes of algebras: the real rank zero C*-algebras ([4]) and the simple, unital C*algebras. Therefore, the C*-algebras with the ideal property are very important in Elliott’s classification program; (see [6]). On the other hand, in [13] we proved, in particular, that for AH algebras the ideal property and the projection property coincide. We generalized this fact in [15], where we showed, in particular, that these two properties coincide for the class of GAH algebras (note that a GAH algebra is an inductive limit C*-algebra lim ! An, where for each n N, An= Lkn i1⁄41 An and each A i n is a unital C*-algebra whose proper ideals have no nonzero projections [15]; obviously, an AH algebra is a GAH algebra). Recently we generalized these results in the separable case, proving that the ideal property and the projection property coincide for the class of separable LB algebras (see Definition 16 below and [18]). Hence, the following question is both natural and interesting: Glasgow Math. J. 44 (2002) 293–300. # 2002 Glasgow Mathematical Journal Trust. DOI: 10.1017/S0017089502020104. Printed in the United Kingdom Question 2. Do the ideal property and the projection property coincide? The following result gives two ways of rephrasing the projecton property: Proposition 3. Let A be a C*-algebra. Then, the following conditions are equivalent. (1) A has the projection property. (2) Each ideal of A is an inductive limit of unital C*-algebras (3) Each ideal of A is an inductive limit of unital hereditary C*-algebras of A. Proof. (1) (3). Let I be an ideal of A. Since A has the projection property, let ðeiÞi2 be an approximate unit of projections for I. Now, it is not difficult to see that I 1⁄4 [ i2 eiAei 1⁄4 lim ! eiAei and that each eiAei is a unital hereditary C*-subalgebra of A. (3) (2) is obvious. (2) (1). Let I be an ideal of A. By hypothesis, I 1⁄4 lim ! ðIi; i;jÞ, where each Ii (i 2 ) is a unital C*-algebra. Since the quotient of a unital C*-algebra is unital, we may suppose that the canonical homomorphisms Ii ! Ij; i j; i; j 2 and Ii ! I; i 2 are inclusions, and hence that: I 1⁄4 [

The ideal property in crossed products

We describe the lattice of the ideals generated by projections and prove a characterization of the ideal property for large classes of crossed products of commutative C*-algebras by discrete, amenable groups; some applications are also given. We prove that the crossed product of a C*-algebra with the ideal property by a group with the ideal property may fail to have the ideal property; this answers a question of Shuzhou Wang.

The Weak Ideal Property and Topological Dimension Zero

Following up on previous work, we prove a number of results for C*-algebras with the weak ideal property or topological dimension zero, and some results for C*-algebras with related properties. Some of the more important results include: The weak ideal property implies topological dimension zero. For a separable C*-algebra~A, topological dimension zero is equivalent to RR (O_2 \otimes A) = 0, to D \otimes A having the ideal property for some (or any) Kirchberg algebra~D, and to A being residually hereditarily in the class of all C*-algebras B such that O_{\infty} \otimes B contains a nonzero projection. Extending the known result for Z_2, the classes of C*-algebras with topological dimension zero, with the weak ideal property, and with residual (SP) are closed under crossed products by arbitrary actions of abelian 2-groups. If A and B are separable, one of them is exact, A has the ideal property, and B has the weak ideal property, then A \otimes_{min} B has the weak ideal property. If X is a totally disconnected locally compact Hausdorff space and A is a C_0 (X)-algebra all of whose fibers have one of the weak ideal property, topological dimension zero, residual (SP), or the combination of pure infiniteness and the ideal property, then A also has the corresponding property (for topological dimension zero, provided A is separable). Topological dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C*-algebras including all separable locally AH algebras. The weak ideal property does not imply the ideal property for separable Z-stable C*-algebras. We give other related results, as well as counterexamples to several other statements one might hope for.

Inductive limits of C(X)-modules and continuous fields of AF-algebras

Abstract Let X be a compact connected space and (Ai)i = 1∞, a sequence of finite-dimensional C∗-algebras. Each inductive limit fx103-1, with C(X)-linear connecting ∗-homomorphisms, is ∗-isomorphic as C(X)-module to the C∗-algebra defined by a certain continuous field E L of AF-algebras. We classify the C∗-algebras L for which E L has simple fibres. In the general case the classification is given in the category of the C∗-algebras which are C(X)-modules.

On approximately inner automorphisms of certain crossed product c⋆-algebras

Let G be a compact connected topological group having a dense subgroup isomorphic to Z. Let C(G) xZ be the crossed product C*-algebra of cc C(G) with Z, where Z acts on G by rotations. Automorphisms of C(G) >i Z leaving invariant the canonical copy of C(G) are shown to be approximately inner iff they act trivially on K1 (C(G) xi Z). Let G be a compact abelian topological group. An element s E G is called a generator if the group algebraically generated by s is dense in G. G is called monothetic if it has at least one generator. If in addition G is connected, this is equivalent to saying that the topology of G has a base of cardinality < c. Moreover if G is second countable then the set of generators is measurable and its Haar measure equals 1. (See [4], Theorems 24.15, 24.27.) From now on, G is a monothetic compact connected infinite topological group and s E G is a fixed generator. Let A = C(G) be the C*-algebra of all complex-valued continuous functions on G. We consider the action oc: Z -* Aut(A) given by (cxk (a))(x) = a(s kx), foraEA, xEG and the corresponding crossed product C*-algebra A x Z (see [5, 8]). Denote cc by AutA (A x Z) the closed subgroup cc {3 E Aut(A x Z): /B(A) = A} cc where Aut(A x Z) has the topology of pointwise norm convergence. Note that cc AutA(A X Z) = {,8 E Aut(A x Z): ,B(A) c A}, since A is a maximal abelian cc cc self-adjoint subalgebra in A x Z (see [8], Proposition 4.14). We prove the cc following. Received by the editors May 4, 1989; presented at the OATE 2 Conference, Craiova, Romania, August 28-September 8, 1989, organized by INCREST. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L40; Secondary 46L80. ? 1990 American Mathematical Society 0002-9939/90

Reduction of topological stable rank in inductive limits of

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