String topology prospectra and Hochschild cohomology

We study string topology for classifying spaces of connected compact Lie groups, drawing connections with Hochschild cohomology and equivariant homotopy theory. First, for a compact Lie group $G$, we show that the string topology prospectrum $LB{G}^{-TBG}$ is equivalent to the homotopy fixed-point prospectrum for the conjugation action of $G$ on itself, $G^{hG}$. Dually, we identify $LB{G}^{-ad}$ with the homotopy orbit spectrum $(DG{)}_{hG}$, and study ring and co-ring structures on these spectra. Finally, we show that in homology, these products may be identified with the Gerstenhaber cup product in the Hochschild cohomology of $C^{\ast }(BG)$ and $C_{\ast }(G)$, respectively. These, in turn, are isomorphic via Koszul duality.