Semiprojectivity and properly infinite projections in graph C⁎-algebras

Abstract

We give a complete description of which unital graph C * -algebras are semiprojective, and use it to disprove two conjectures by Blackadar. To do so, we perform a detailed analysis of which projections are properly infinite in such C * -algebras.

Corners of Cuntz-Krieger algebras

“…He showed in [6,Proposition 2.7] that a full unital corner of a semiprojective C * -algebra is semiprojective. Recently, S. Eilers and T. Katsura showed in [11] that a corner of a unital graph C * -algebra that is semiprojective is also semiprojective. Corollary 4.11 is a special case of their results since every Cuntz-Krieger algebra is isomorphic to a unital semiprojective graph C * -algebra.…”

“…Semiprojectivity is easy in our case since the graphs are finite. Thus we do not need any results from [11].…”

Automorphisms of Cuntz–Krieger algebras

“…(2) A 1 = B 1 ⊗ K, where B 1 is a unital purely infinite graph algebra with exactly one non-trivial ideal. The fact that A 1 is semiprojective follows from the results in [4].…”

“…Set k = p s1 1 · · · p s ℓ ℓ . Let z ∈ H e1 n, 4 . Set x = h 1,n,out n,4,e1 (z) ∈ F e1 n, 5 and let x = β e1 n,5 (x) ∈ F e1 1,2 .…”

“…Let t be such that n i,t < r < n i,t+1 (and t = 0 if r < n i,1 and t = k i if n i,ki < r). Let z ∈ H e1 n, 4 and set x = h 1,n,out n,4,e1 (z) ∈ F e1 n,5 . Note that there exists y ∈ F e1 1,5 and z 1 , .…”