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I was reading John D.S. Jones' paper "Cyclic homology and equivariant homology" where he introduces a variant of cyclic homology that is isomorphic (as modules over the ring $K[u]$) to equivariant homology of the loop space of a simply connected space with the $S^1$ action on the loops, which is what I mean by String Homology.

I have seen how the loop homology BV operator shows up in Hochschild homology. Do we have an analog of string topology operations, such as the string bracket, in cyclic homology?

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Yes, definitively, for example you can look at (my choices are completely arbitrary, I am sorry I am sure to forget plenty of very good references):

it's all about duality and the way you move operations from Hochschild homology to cyclic homology via the Connes'operator.

The situation in the algebraic context of cyclic homology is as rich as the one first described by Moira chas and Dennis Sullivan in string topology see for example (part III) for the geometric story :

http://arxiv.org/abs/0710.4141

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In particular, there is an involutive Lie bialgebra on the reduced cyclic cohomology of a Frobenius CDGA. This resembles Chas-Sullivan's involutive Lie bialgebra on $H^{S^1}_*(LM,M)$. –  Manuel Rivera May 21 at 12:23
    
And in general, the cyclic cohomology of a Frobenius CDGA is an algebra over the PROP of Sullivan diagrams. Reference: On the cyclic Deligne Conjecture, Tradler and Zeinalian. –  Manuel Rivera May 21 at 12:30
    
Great! I had trouble finding references myself, this was really helpful. –  Felix T. May 21 at 17:16

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