A Unique Decomposition Result for Ht Factors with Torsion Free Core

We prove that II1 factors M have a unique (up to unitary conjugacy) crossproduct type decomposition around “core subfactors” N ⊂ M satisfying the property HT of ([P1]) and a certain “torsion freeness” condition. In particular, this shows that isomorphism of factors of the form Lα(Z)⋊Γ, for torsion free, non-amenable subgroups Γ ⊂ SL(2,Z) and α = e, t 6∈ Q, implies isomorphism of the corresponding groups Γ. Let M be a type II1 factor and B ⊂ M a von Neumann subalgebra. Recall from (Section 2 in [P1]) that M has the property H relative to B if the identity map on M can be approximated pointwise by regular (i.e., subunital, subtracial) completely positive maps on M that “vanish at infinity” relative to B. Also, B ⊂ M is a rigid inclusion (or (M,B) has the relative property (T)) if regular completely positive maps on M that are close (pointwise) to the identity on M follow uniformly close to the identity on B (cf. 4.2 in [P1]). If B ⊂ M satisfies both these conditions, then it is called a HT inclusion and B is called a HT subalgebra of M . (N.B. In fact, the notation used in [P1] to designate this property is “HTs”, while “HT” is used for a slightly weaker condition. We opted for the notation “HT” in this paper for the benefit of simplicity.) We refer to ([P1]) for the detailed definitions, as well as for notations and terminology used hereafter. In ([P1]), one primarily studies HT Cartan subalgebras of M , i.e., HT inclusions B ⊂ M with B maximal abelian in M and with the normalizer NM (B) = {u ∈ U(M) | uBu = B} generating M . Thus, the main technical result in ([P1]) shows the uniqueness, up to unitary conjugacy, of the HT Cartan subalgebras in separable II1 factors. Such unique decomposition results are crucial for the calculation of invariants of II1 factors (e.g., the fundamental and automorphism groups) and ultimately for their classification. Thus, one application of the uniqueness result for HT Cartan Supported in part by NSF Grant 0100883. Typeset by AMS-TEX 1