  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.
