C*-Algebras Generated by Stable Relations

Presentations of certain C*-algebras, such as (C0(0, 1) ⊗ Mn)∼, by generators and relations are proven to be stable. By the weakest definition of stability, a set p1, ..., pn of star-polynomials is stable if operators a1, ..., an which almost satisfy these relations, meaning ||pi(a1, ..., an)|| ≤ϵ, can be perturbed within C*(a1,..., an) to satisfy these relations exactly. Homotopy and perturbation results for these C*-algebras are corollaries. In particular, we prove some new results about the "dimension-drop" C*-algebras An = {ƒ∈C([0, 1], Mn) | ƒ(0), ƒ(1) ∈ CI}. We show that An is semiprojective, answering a question of Effros and Kaminker. Also, we show that whenever φ : An → B is a ∗-homomorphism approximately contained in a sub-C*-algebra B0 ⊆ B, there is a nearby ∗-homomorphism of An into B0. This answers a question raised by Elliott. We prove technical results regarding Mn(C0(0, 1]) which should be useful elsewhere. First, we show this is projective, meaning a star-homomorphism Mn(C0(0, 1]) → B/J always lifts to a star-homomorphism Mn(C0(0. 1]) → B. Secondly, we show that for A finitely generated, Mn(A) ≅ Mn(C0(0, 1])∗ C0(0, 1]A if the amalgamation is done properly.