Exotic full factors via weakly coarse bimodules

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.

💡 Research Summary

This paper presents a significant advance in the construction of “exotic” full factors in von Neumann algebra theory, specifically focusing on factors that are not elementarily equivalent to the free group factor L(F₂). The central innovation is a new method for proving the “fullness” of factors obtained through an iterative “3-handle construction,” which completely bypasses the need for Property (T)—a crucial and limiting requirement in all prior works.

The handle construction is an inductive process that builds a larger factor by repeatedly taking amalgamated free products over certain subalgebras. Its goal is to create a factor with a controlled “sequential commutation diameter” (a graph-theoretic invariant related to its ultrapower), ensuring it is distinct from L(F₂). The major historical hurdle has been proving that the resulting limit factor remains full (i.e., has trivial central sequence algebra). Previous approaches (