Representations of the infinite Cuntz algebra

We define an intrinsic unital completely positive (ucp) map from the Cuntz algebra to the Pythagorean with finitely many generators. This permits to lift states and cyclic representations. This process restricts to the Cuntz-Toeplitz algebras and the contraction algebras. Taking inductive limits we obtain an effective machinery for constructing states on the infinite Cuntz algebra ${\mathcal{O}}_{\mathrm{\infty }}$. Combining this approach with previous techniques of the first author we obtain large families of pairwise inequivalent irreducible representations of ${\mathcal{O}}_{\mathrm{\infty }}$: one family being in an obvious bijection with the unit sphere of Hilbert’s space.