For a \(C^*\) -algebra \({\mathcal {A}},\) its scattered radical \({\mathcal {R}}_s({\mathcal {A}})\) is the largest scattered ideal of \({\mathcal {A}};\) an ideal is scattered if its elements all have countable spectrum. We say that \({\mathcal {A}}\) is scattered if \({\mathcal {R}}_s({\mathcal {A}})={\mathcal {A}}.\) In this paper, we first show that any scattered von Neumann algebra is finite dimensional and then obtain a complete characterization of scattered radical of von Neumann algebras. Furthermore, we give a topological characterization of \({\mathcal {R}}_s(C(M)),\) that is, \({\mathcal {R}}_s(C(M))=\{f\in C(M): f(P(M))=0\},\) where M is a Hausdorff compact space and P(M) is the largest perfect subset of M. Finally, we show that \({\mathcal {R}}_s({\mathcal {A}}\otimes _{\min } {\mathcal {B}})={\mathcal {R}}_s({\mathcal {A}})\otimes _{\min } {\mathcal {R}}_s({\mathcal {B}})\) if \({\mathcal {A}},{\mathcal {B}},\) satisfying one of the following conditions: (i) \({\mathcal {A}},{\mathcal {B}}\) are \(C^*\) -algebras and \({\mathcal {A}},{\mathcal {B}}\) are exact; (ii) \({\mathcal {A}},{\mathcal {B}}\) are \(C^*\) -algebras and \({\mathcal {A}}\) or \({\mathcal {B}}\) is nuclear.
