Let X be 1-connected CW complex of finite type. We construct algebraic Steenrod operations on the Hochschild cohomology  of the cobar construction on the singular chains on X with coefficients in its dual. We prove that these operations can be identified to those defined on the cohomology of the free loop space. We prove the same result with negative cyclic cohomology and the space of geometric free loops.

Hochschild (co)homology, cyclic (co)homology, bar construction and cobar construction, shc-algebra and p-shc-algebra, shc-coalgebra and p-shc-coalgebra.