On the structure of centrally biprojective \(C^*\)-algebras (Q1803037)

Let \(A\) be a \(C^*\)-algebra with a centre \(Z(A)\). The algebra \(A\) is called centrally biprojective if \(A\) is a projective element in the category of \((A,Z(A))\) bimoduls. The structure of such algebras is described. The main result says that any centrally biprojective algebra \(A\) such that \(A\cdot Z(A)\) is dense in \(A\) is isomorphic to a \(c_ 0\)- direct sum of countably many homogeneous \(C^*\)-algebras of finite rank. (\(A\) is called \(n\)-homogeneous \((n<\infty)\) if every irreducible representation of \(A\) has its value in the algebra of all \(n\times n\)- matrices.) On the other hand it is shown that every \(C^*\)-algebra with spectrum of zero topological dimension which is a \(c_ 0\)-direct sum of \(n\)-homogeneous algebras is centrally biprojective.