# Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems

Research paper by Yuhei Suzuki

Indexed on: 05 May '18Published on: 05 May '18Published in: arXiv - Mathematics - Operator Algebras

#### Abstract

Let $\Gamma$ be a discrete group satisfying the Haagerup--Kraus approximation property (AP). For any free action $\alpha \colon \Gamma \curvearrowright X$ and its extension $\pi \colon Y \rightarrow X$, we prove that every intermediate C*-algebra of the associated crossed product inclusion arises from an intermediate extension of $\pi$. We also establish the measurable analogue of this result (with no requirement on $\Gamma$). As an application, by combining with Margulis's factor theorem, we obtain (new) maximal amenable subalgebras inside full factors. Notably their intermediate von Neumann algebras are completely parametrized by a subset lattice of simple restricted roots. We also give their non-commutative variants. As applications, we provide (i) infinite index endomorphisms without intermediate C*-algebras on all Kirchberg algebras, (ii) realizations of the closed subgroup lattices as a subfactor lattice, (iii) further examples of maximal amenable subalgebras of non-amenable factors, (iv) new examples of ambient nuclear C*-algebras without intermediate C*-algebras.