Coarse decomposition of II1 factors

Abstract

We prove that any separable II1 factor M admits a coarse decomposition over the hyperfinite II1 factor R—that is, there exists an embedding R↪M such that L2M⊖L2R is a multiple of the coarse Hilbert R-bimodule L2R⊗‾L2Rop. Equivalently, the von Neumann algebra generated by left and right multiplication by R on L2M⊖L2R is isomorphic to R⊗‾Rop. Moreover, if Q⊂M is an infinite-index irreducible subfactor, then R↪M can be constructed to be coarse with respect to Q as well. This implies the existence of maximal abelian ∗-subalgebras that are mixing, strongly malnormal, and with infinite multiplicity, in any given separable II1 factor.