Cyclic Hilbert Spaces and Connes' Embedding Problem

Let M be a II (1)-factor with trace tau, the finite dimensional subspaces of L (2)(M, tau) are not just common Hilbert spaces, but they have an additional structure. We introduce the notion of a cyclic linear space by taking these additional properties as axioms. In Sect. 3 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 Sect. 4 we make a first step towards the answer of the previous question.