The scattered radical of some C∗-algebras

For a C∗-algebra A, its scattered radicalRs(A) is the largest scattered ideal of A; an ideal is scattered if its elements all have countable spectrum. We say that A is scattered if Rs(A)=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 Rs(C(M)), that is, Rs(C(M))={f∈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 Rs(A⊗minB)=Rs(A)⊗minRs(B) if A,B, satisfying one of the following conditions: (i) A,B are C∗-algebras and A,B are exact; (ii) A,B are C∗-algebras and A or B is nuclear.