Exotic full factors via weakly coarse bimodules
David Gao, David Jekel, Srivatsav Kunnawalkam Elayavalli, Gregory Patchell
TL;DR
The paper develops a new, $(T)$-independent method for producing full von Neumann factors via explicit bimodule computations in iterated amalgamated free products, enabling fullness transfer along weakly coarse inclusions. Central to the approach is a detailed analysis of bimodule decompositions that arise in the $3$-handle construction, which yields full II$_1$ factors from arbitrary inputs and, in particular, a full $ actor of $L( obreak[1] _2)$ that is not elementarily equivalent to $L( obreak[1] _2)$ and contains no diffuse $(T)$ subalgebras. The results extend to type $_1$ factors produced by the same construction under suitable state-preserving conditions, showing fullness without relying on $(T)$ phenomena in the Type $$ setting. Overall, the work provides new invariants and techniques for distinguishing algebras up to elementary equivalence and expands the landscape of exotic full factors beyond the tracial setting.
Abstract
We are able to explicitly compute the bimodule structure of von Neumann algebra inclusions in handle constructions, which arise as inductive limits of iterated amalgamated free products not elementarily equivalent to $L(\mathbb{F}_2)$. Our computation is achieved via identifying delicate normal form decompositions in amalgamated free products built in an iterated fashion. Using these techniques, we are able to show that the handles constructions are always full, without any need to appeal to Property (T) phenomena which was essential in all previous works. Furthermore our bimodule machinery works in the setting of arbitrary von Neumann algebras equipped with faithful normal states, yielding examples of full $\mathrm{III}_1$ factors via handle constructions.