Reduction of the dimension of nuclear C*-algebras

Abstract

We show that for a large class of C*-algebras A, containing arbitrary direct limits of separable type I C*-algebras, the following statement holds: If A ∈ A and B is a simple projectionless C*-algebra with trivial K-groups that can be written as a direct limit of a system of (nonunital) recursive subhomogeneous algebras with no dimension growth then the stable rank of A ⊗ B is one. As a consequence we show that if A ∈ A and W is the C*-algebra constructed in [12] then the stable rank of A ⊗ W is one. We also prove the following stronger result: If A is separable C*-algebra that can be written as a direct limit of C*-algebras of the form C0(X) ⊗ Mn, where X is locally compact and Hausdorff, then A ⊗ W can be written as a direct limit of a sequence of 1-dimensional noncommutative CW-complexes.

Finite group actions on certain stably projectionless C*-algebras with the Rohlin property

“…Robert's classification theorem shows that if A is a simple approximately finite dimensional (AF) algebra with a unique tracial state and no unbounded traces, then [41]). Note that every simple inductive limit C * -algebra of 1-dimensional NCCW complexes has strict comparison and stable rank one.…”

Crossed products by actions of finite groups with the Rokhlin property

“…It follows that B ⊗ K is the direct limit of a sequence of ideals of 1-dimensional NCCW-complexes. Each of these ideals is a direct limit of a sequence of 1-dimensional NCCW-complexes by [37,Lemma 3.11]. Hence, B ⊗ K can be locally approximated by 1-dimensional NCCW-complexes.…”

Geometric structure of dimension functions of certain continuous fields

“…For given ǫ > 0 we choose δ 0 > 0 satisfying the conclusion of [35,Lemma 3.4] for ǫ 2 , i.e., for any positive contraction a and any unitary u with ua − a < δ 0 there is a path of unitaries (u t ) t∈[0,1] in A such that u 0 = u, u 1 = 1 A , and u t a − a < ǫ 2 for all t ∈ [0, 1]. (It follows from our assumptions and [33, Theorem 2.10] that U(B ∼ ) is connected for each hereditary subalgebra B of A, which is needed for the application of [35,Lemma 3.4]. ) Find 0 < δ ≤ δ 0 2 such that whenever a − b < δ, then |a| − |b| < δ 0 2 .…”