Lifting Solutions to Perturbing Problems in 𝐶*-Algebras

Crossed products by finite group actions with the Rokhlin property

“…We therefore present a systematic framework for proving that a C*-algebra with a local approximation property by homomorphic images of a suitable class of semiprojective C*-algebras can in fact be written as a direct limit of algebras in the class. Our result is an analog of Lemma 15.2.2 of [21], where the local approximation property uses injective homomorphic images. Results of this kind for specific cases are already implicit in the literature; our contribution is primarily to systematize the method.…”

“…(2) For every A ∈ C, every n ∈ N, and every nonzero projection p ∈ M n (A), the corner pM n (A)p is finitely generated, and is semiprojective in the sense of Definition 14.1.3 of [21].…”

“…It is easy to prove that these operations all preserve Condition (2) of Definition 1.2, using the following three facts: a finite direct sum of semiprojective C*-algebras is semiprojective (Theorem 14.2.1 of [21]), a finite direct sum of finitely generated C*-algebras is finitely generated (easy), and a corner in a finite direct sum is a finite direct sum of corners in the summands.…”

Extensions of Hilbert C*-modules II

“…It is a Hilbert C * -module version of a result which states that a C * -algebra is projective if and only if it is corona projective ( [7], Theorem 10.1.9). Again, the proof below follows the proof for C * -algebras without changes.…”

Tracial Equivalence for $C^*$-Algebras and Orbit Equivalence for Minimal Dynamical Systems

“…We note that we may identify G n with j n (K 0 (B )), where j n : B → A is the embedding. Note that B is semi-projective in the sense in [14]. Therefore, for any η > 0 and for all large n, there is an injective homomorphism h n : B → A such that…”