Introduction

This chapter is devoted to the construction of irreducible hyperfinite subfactors R in a separable II1 factor N with suitable additional properties available for R in its embedding in N. All these results depend on inductive matrix methods that were developed by Popa [136]. The method has already been used extensively in Chapter 12 for the construction of singular and semiregular masas.

In Section 13.2, a basic method is presented to show that an irreducible hyperfinite subfactor exists in each separable II1 factor. Section 13.3 shows that if A is a Cartan masa in a separable II1 factor N, then there is an irreducible hyperfinite subfactor R in N with A ⊆ R and A Cartan in R (see [141]). Section 13.4 discusses the basic theory of property Γ factors, a topic which we will revisit in greater depth in Appendix A. This is applied in Section 13.5 to prove a useful result (Theorem 13.5.4) on the existence of a masa in a Γ factor that is Cartan in an irreducible hyperfinite subfactor and that contains unitaries that can be used in the Γ condition. This theorem combines methods from [140] and from [30, Theorem 5.3] that give the Γ condition.