Some quasinilpotent generators of the hyperfinite $\mathrm{II}_1$ factor

TL;DR

This paper constructs specific quasinilpotent operators within the hyperfinite II_1 factor that generate the entire algebra, possess non-trivial invariant subspaces, and have computable moments, contributing to the hyperinvariant subspace problem.

Contribution

It introduces a new class of quasinilpotent generators in the hyperfinite II_1 factor with detailed properties and moment formulas, advancing understanding of their structure and invariant subspaces.

Findings

Operators are quasinilpotent and generate the entire hyperfinite II_1 factor.

They have non-trivial, closed, invariant subspaces affiliated to the algebra.

Explicit combinatorial and analytical formulas for their moments are provided.

Abstract

For each sequence $\{c_n\}_n$ in $l_{1}(\N)$ we define an operator $A$ in the hyperfinite $\mathrm{II}_1$-factor $\mathcal{R}$. We prove that these operators are quasinilpotent and they generate the whole hyperfinite $\mathrm{II}_1$-factor. We show that they have non-trivial, closed, invariant subspaces affiliated to the von Neumann algebra and we provide enough evidence to suggest that these operators are interesting for the hyperinvariant subspace problem. We also present some of their properties. In particular, we show that the real and imaginary part of $A$ are equally distributed, and we find a combinatorial formula as well as an analytical way to compute their moments. We present a combinatorial way of computing the moments of $A^{*}A$.