A Note on Scattered C*-algebras and the Radon-nikodym Property

Recently, Jensen [3], Lazar [4] and Rothwell [6] have investigated in detail the structure of scattered C*-algebras which can be regarded as the most elementary generalization of the finite dimensional involutive algebras. A C*-algebra si is called scattered if every positive functional of si is the sum of a sequence of pure functionals. The object of this note is to give a short proof of a natural conjecture, stemming from a splinter group discussion at the 1980 Sheffield British Mathematical Colloquium, that a C*-algebra is scattered if, and only if, its dual possesses the Radon-Nikodym property. This result generalizes the well-known fact that a compact Hausdorff space Q. has no non-empty perfect subset if, and only if, C(Q)* has the Radon-Nikodym property [5]. We also prove that a von Neumann algebra is a direct sum of type I factors if, and only if, its predual possesses the Radon-Nikodym property. I would like to thank K. F. Ng and M. L. Rothwell for useful discussions. I would also like to thank the referee for suggesting a simpler version as well as Theorem 4.