Coarse decomposition of II1 factors

Abstract

We prove that any separable ${\mathrm{II}}_1$ factor M admits a coarse decomposition over the hyperfinite ${\mathrm{II}}_1$ factor R—that is, there exists an embedding $R\hookrightarrow M$ such that $L^{2}M\ominus L^{2}R$ is a multiple of the coarse Hilbert R-bimodule $L^{2}R\bar{\otimes }{L}^2{R}^{\mathrm{op}}$. Equivalently, the von Neumann algebra generated by left and right multiplication by R on $L^{2}M\ominus L^{2}R$ is isomorphic to $R\bar{\otimes }{R}^{\mathrm{op}}$. Moreover, if $Q\subset M$ is an infinite-index irreducible subfactor, then $R\hookrightarrow M$ can be constructed to be coarse with respect to Q as well. This implies the existence of maximal abelian ${}^{\ast }$-subalgebras that are mixing, strongly malnormal, and with infinite multiplicity, in any given separable ${\mathrm{II}}_1$ factor.