A note on quasi-diagonal \(C^*\)-algebras and homotopy (Q1814239)

A complete set of obstructions to the quasi-diagonality of a \(C^*\)- algebra is still unknown. Various examples suggest that the obstructions are from noncommutative topology. In this note we prove that a \(C^*\)- algebra which is homotopically dominated by a quasi-diagonal \(C^*\)- algebra is quasi-diagonal. This, clearly, confirms the topological nature of quasi-diagonality. Let us also mention that our result improves a result of Eberhard Kirchberg, who had shown that the suspension of a \(C^*\)-algebra has an extension by an ideal of block-diagonal compact operators, which is quasi-diagonal.