Cyclic Hilbert spaces and Connes' embedding problem

Let $M$ be a $II_1$-factor with trace $τ$, the linear subspaces of $L^2(M,τ)$ 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 $L^2(M,τ)$, 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.

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