Cyclic Hilbert spaces and Connes' embedding problem

TL;DR

This paper introduces cyclic Hilbert spaces as a new structure related to $II_1$-factors and explores their potential to embed into $L^2$ spaces, aiming to address Connes' embedding problem.

Contribution

It defines cyclic Hilbert spaces and formulates a key embedding problem that could lead to an algorithm for Connes' embedding conjecture.

Findings

Formulated the problem of embedding cyclic Hilbert spaces into $L^2(M,\tau)$.

Made initial progress towards solving the embedding problem.

Established a framework connecting cyclic spaces to Connes' conjecture.

Abstract

Let $M$ be a $II_1$-factor with trace $\tau$, the linear subspaces of $L^2(M,\tau)$ are not just common Hilbert spaces, but they have additional structure. We introduce the notion of a cyclic linear space by taking those properties as axioms. In Sec.2 we formulate the following problem: "does every cyclic Hilbert space embed into $L^2(M,\tau)$, for some $M$?". An affirmative answer would imply the existence of an algorithm to check Connes' embedding Conjecture. In Sec.3 we make a first step towards the answer of the previous question.