Irreducibility and weak spectral gap in free product von Neumann algebras

Abstract

Consider a free product $(M,\phi )=(M_1,{\phi }_1)\ast (M_2,{\phi }_2)$ of non-trivial von Neumann algebras. Using amalgamated free product techniques, we establish irreducibility and weak spectral gap results for free product subalgebras and the centralizer subalgebra of the free product state. In particular, it is shown that, under a mild dimension constraint, the inclusion of the diffuse summand of these subalgebras into the corresponding corner of $M^{\omega }$ is irreducible for any free ultrafilter $\omega \in \beta \mathbb{N}\setminus \mathbb{N}$. We also obtain analogous irreducibility results for graph products of von Neumann algebras.

16 pages