1-Bounded entropy and regularity problems in von Neumann algebras

Abstract

We define and investigate the singular subspace |$\mathcal{H}_{s}(N\subseteq M)$| of an inclusion of tracial von Neumann algebras. The singular subspace is a canonical |$N$|-|$N$| subbimodule of |$L^{2}(M)$| containing the quasinormalizer introduced in [37], the one-sided quasinormalizer introduced in [12], and the wq-normalizer introduced in [15] (following upon work in [26] and [39]). We then obtain a weak notion of regularity (called spectral regularity) by demanding that the singular subspace of |$N\subseteq M$| generates |$M.$| By abstracting Voiculescu’s original proof of absence of Cartan subalgebras in [51] we show that there can be no diffuse, hyperfinite subalgebra of |$L(\Bbb F_{n})$| which is spectrally regular. Our techniques are robust enough to repeat this process by transfinite induction and rule out chains of spectrally regular inclusions of algebras starting from a diffuse, hyperfinite subalgebra and ending in |$L(\Bbb F_{n}).$| We use this to prove some conjectures made by Galatan-Popa in their study of smooth cohomology of |$\textrm{II}_{1}$|-factors (see [15]). Our results may be regarded as a consistency check for the possibility of existence of a “good” cohomology theory of |$\textrm{II}_{1}$|-factors. We can also use our techniques to show that if |$U_{t}$| is a one-parameter orthogonal group on a real Hilbert space |$\mathcal{H}$| and the spectral measure of its generator is singular with respect to the Lebesgue measure, then the continuous core of the free Araki-Woods factor |$\Gamma(\mathcal{H},U_{t})''$| is not isomorphic to |$L(\Bbb F_{t}\overline{\otimes} B(\ell^{2}(\Bbb N))$| for any |$t\in (1,\infty].$| In particular, |$\Gamma(\mathcal{H},U_{t})''\not\cong \Gamma(L^{2}(\Bbb R,m),\lambda_{t})''$| where |$m$| is Lebesgue measure and |$\lambda$| is the left regular representation. This was previously only known when the spectral measure of the generator of |$U_{t}$| had all of its convolution powers singular with respect to Lebesgue measure. We give similar applications to crossed products by free Bogoliubov actions in the spirit of [23].