The Weak Ideal Property and Topological Dimension Zero

Abstract

Abstract. 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:• e weak ideal property implies topological dimension zero.• For a separable C*-algebra A, topological dimension zero is equivalent to RR(O ⊗ A) = , to D ⊗ 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 ∞ ⊗ B contains a nonzero projection.• Extending the known result for Z , the classes of C*-algebras with residual (SP), which are residually hereditarily (properly) in nite, or which are purely in nite and have the ideal property, are closed under crossed products by arbitrary actions of abelian -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 ⊗ min B has the weak ideal property.• If X is a totally disconnected locally compact Hausdor space and A is a C (X)-algebra all of whose bers have one of the weak ideal property, topological dimension zero, residual (SP), or the combination of pure in niteness 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.• e 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.

Relating properties of crossed products to those of fixed point algebras

“…For separable C*-algebras, topological dimension zero is known to be of the form residually hereditarily in C for an upwards directed class C of C*-algebras. See Theorem 2.10 of [19]. This is probably true in general, but we don't need to know this to prove that it satisfies the conditions of Corollary 1.5.…”

“…The proofs are in [18]. A convenient summary, with explicit references, is given at the end of Section 1 of [19].…”

“…For topological dimension zero, assuming A is separable, the chain of implications is the same, but now uses, in the same order as above, Theorem 3.6 of [19], Theorem 3.17 of [17], Lemma 3.3 of [17], and [24]. When G is a finite abelian 2-group, all the properties under consideration have the form "residually hereditarily in an upwards directed class of C*-algebras", and we use the same reasoning, now applying, in order, Theorem 3.2(ii) of [19], Corollary 3.3(ii) of [19], Proposition 5.10(2) of [18], and [24]. We expect the restriction to 2-groups to be unnecessary.…”

“…For the rest of (4), use Proposition 2.6 of [7] to see that topological dimension zero passes to extensions and quotients, and use Lemma 2.6 of [19] and Lemma 3.6 of [17] to see that it passes to closures of increasing unions of ideals.…”

On invariants of C*-algebras with the ideal property

“…It is obvious that both simple, unital C * -algebras and real rank zero C * -algebras have the ideal property. There are many other examples of C * -algebras arising from dynamical systems which have the ideal property (see [20], [31], [32], [33], [34], [35], etc.). In 1995, K. Stevens classified all unital approximately divisible AI algebras with the ideal property ( [39]).…”