𝐾-stability of continuous 𝐶(𝑋)-algebras

Abstract

We show that the property of being rationally Kstable passes from the fibers of a continuous C(X)-algebra to the ambient algebra, under the assumption that the underlying space X is compact, metrizable, and of finite covering dimension. As an application, we show that a crossed product C*-algebra is (rationally) K-stable provided the underlying C*-algebra is (rationally) K-stable, and the action has finite Rokhlin dimension with commuting towers. Given a compact Hausdorff space X, a continuous C(X)-algebra is the section algebra of a continuous field of C*-algebras over X. Such algebras form an important class of non-simple C*-algebras, and it is often of interest to understand those properties of a C*-algebra which pass from the fibers to the ambient C(X)-algebra. Given a unital C*-algebra, we write U n (A) for the group of n × n unitary matrices over A. This is a topological group, and its homotopy groups π j (U n (A)) are termed the nonstable K-theory groups of A. These groups were first systematically studied by Rieffel [20] in the context of noncommutative tori. Thomsen [26] built on this work, and developed the notion of quasi-unitaries, thus constructing a homology theory for (possibly non-unital) C*-algebras. Unfortunately, the nonstable K-theory for a given C*-algebra is notoriously difficult to compute explicitly. Even for the algebra of complex numbers, these groups are naturally related to the homotopy groups of spheres π j (S n), which are not known for many values of j and n. It is here that rational homotopy theory has proved to be useful to topologists and, in this paper, we employ this tool in the context of C*-algebras.

Rational -Stability of Continuous -Algebras

“…The next lemma is an analogue of [23,Lemma 2.7], and is a consequence of that result and Proposition 3.9.…”

“…Let A and B be separable C*-algebras. A * -homomorphism ϕ : A → B is said to be sequentially split if, for every compact set F ⊂ A, and for every > 0, there exists a * -homomorphism ψ = ψ F, : B → A such that ψ • φ(a) − a < for all a ∈ F. [23] The next theorem, due to Gardella et al [11,Proposition 4.11] is an important structure theorem that allows one to prove permanence results concerning crossed products with finite Rokhlin dimension (with commuting towers). THEOREM 4.4.…”

“…A C*-algebra A is said to be K-stable if the homotopy groups π j (U n (A)) are naturally isomorphic to the K-theory groups K j+1 (A), and rationally K-stable if the analogous statement holds for the rational homotopy groups (see Definition 2.3). In [23], we proved that, for a continuous C(X)-algebra, the properties of being K-stable passes from the fibres to the whole algebra, provided the underlying space X is metrizable and has finite covering dimension. The goal of this paper is to prove an analogous result for rational K-stability.…”

$K$-Stability of A$\mathbb{T}$-Algebras

“…Note that, for a K-stable C*-algebra, G k (A) ∼ = K k+1 (A) and for a rationally K-stable C * -algebra, F m (A) ∼ = K m+1 (A) ⊗ Q. A variety of interesting C*-algebras are known to be K-stable (see [14,Remark 1.5]). Clearly, K-stability implies rational K-stability.…”

“…Thomsen [17] had proved that the Cuntz algebras and simple, infinite dimensional AF-algebras have this property, and he termed this phenomenon K-stability. A variety of interesting C*-algebras are now known to have this property (see [14,Remark 1.5]). Similarly, we say that a C*-algebra A is rationally K-stable if the groups π k ( U n (A)) ⊗ Q are all naturally isomorphic to one another.…”