The noncommutative topology of one-dimensional spaces (Q1099373)

If \(X\) and \(Y\) are compact topological spaces, the unital star-homomorphisms from \(C(X)\) to \(C(Y)\) satisfy certain homotopy properties when X is an absolute neighborhood retract. We show that two of these properties still hold when \(C(Y)\) is replaced by a ``noncommutative space'', i.e. an arbitrary unital \(C^ *\)-algebra, but only under the additional assumption that \(X\) is one-dimensional. Specifally, we show that \(C(X)\) is semiprojective and that two unital star-homomorphisms from \(C(X)\) to a \(C^ *\)-algebra A are homotopic whenever they are close.