Minimal conditional expectations for inclusions with atomic centres

We study the set of minimal conditional expectations for inclusions of von Neumann algebras with atomic centres. Contrary to the case of finite dimensional centres, the set of minimal conditional expectations with scalar index consists, in general, of more than one element. Some calculations relative to the connection between minimal expectations and entropy are also done. The last section is devoted to the existence of conditional expectations preserving a Markov trace. Simple examples show that all possibilities can occur.