Remarks on free products with respect to non-tracial states

Abstract

Since the appearance of D. Voiculescu's free probability theory detailed study on free products of von Neumann algebras (based on his theory) has been carried by several authors, and free group factors are their typical examples. Indeed, his theory gives us very powerful tools to analyze free group factors. On the other hand, at the beginning of the 80's (i.e., before the appearance of D. Voiculescu's theory) S. Popa studied in detail free group factors and related type II1 factors and got several deep results ([P1], [P2]). He used there some techniques for type II1 factors related to his notion ``orthogonal pairs''. Then, at the mid 90's L. Ge generalized them to more general (but type II1) setting and solved some problems on maximal injective subalgebras ([G]). In this note, we will try to generalize the techniques to more general and not necessary type II1 setting. These generalizations are rather natural, but enable us to obtain some results on free products with respect to non-tracial states. In fact, as applications a factoriality and type classification result for free products and a result on (normal) conditional expectations from free products onto their subalgebras will be obtained. Based on the latter result we will discuss the recent question posed by M. Izumi, R. Longo and S. Popa ([ILP]): Let M L N be von Neumann algebras with a normal conditional expectation from M onto N. If N 0 \M ˆ C1, does there exist a normal conditional expectation from M onto L ?